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Space-efficient Approximation Scheme for Maximum Matching in Sparse Graphs

This document describes a space efficient approximation scheme for maximum matching in sparse graphs. It begins with an introduction to matching problems and Baker's algorithm for approximating problems on planar graphs. It notes that computing distances is difficult in logspace for planar graphs. The document then outlines previous work on matching algorithms and complexity, and states that the goal is to obtain an approximation scheme for maximum matching that runs in logspace.

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Introduction Matching Our Contribution Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs Samir Datta Raghav Kulkarni Anish Mukherjee Chennai Mathematical Institute NMI Workshop on Complexity Theory, IIT Gandhinagar November 04, 2016 Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Overview 1 Introduction 2 Matching 3 Our Contribution Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm Theorem (Baker ’83, Informal) A class of problems (many of which are NP-Hard in general) can be approximated arbitrarily close to the optimal value in linear time for planar graphs. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm Theorem (Baker ’83, Informal) A class of problems (many of which are NP-Hard in general) can be approximated arbitrarily close to the optimal value in linear time for planar graphs. Example Includes problems like maximum independent set partition into triangles minimum vertex-cover minimum dominating set Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm Theorem (Baker ’83, Informal) A class of problems (many of which are NP-Hard in general) can be approximated arbitrarily close to the optimal value in linear time for planar graphs. Example Includes problems like maximum independent set partition into triangles minimum vertex-cover minimum dominating set ... and any MSO definable properties Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm Basic Idea Partition the vertices into breadth-first search levels Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm Basic Idea Partition the vertices into breadth-first search levels Decompose the graph into successive width-k slices by deleting levels congruent to i mod k Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm Basic Idea Partition the vertices into breadth-first search levels Decompose the graph into successive width-k slices by deleting levels congruent to i mod k Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm Basic Idea Partition the vertices into breadth-first search levels Decompose the graph into successive width-k slices by deleting levels congruent to i mod k Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm Basic Idea Partition the vertices into breadth-first search levels Decompose the graph into successive width-k slices by deleting levels congruent to i mod k Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm Basic Idea Partition the vertices into breadth-first search levels Decompose the graph into successive width-k slices by deleting levels congruent to i mod k Resulting components have treewidth 3k − 1 [Boadlander] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm Basic Idea Partition the vertices into breadth-first search levels Decompose the graph into successive width-k slices by deleting levels congruent to i mod k Resulting components have treewidth 3k − 1 [Boadlander] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm Basic Idea Partition the vertices into breadth-first search levels Decompose the graph into successive width-k slices by deleting levels congruent to i mod k Resulting components have treewidth 3k − 1 [Boadlander] Solve the problem optimally in each partition in linear time Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm Basic Idea Partition the vertices into breadth-first search levels Decompose the graph into successive width-k slices by deleting levels congruent to i mod k Resulting components have treewidth 3k − 1 [Boadlander] Solve the problem optimally in each partition in linear time Union of solutions in all components is within (1 − 1/k) OPT Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm II But here we are interested in space efficient algorithms, namely algorithms running in Logspace Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm II But here we are interested in space efficient algorithms, namely algorithms running in Logspace EJT gives an algorithm for Courcelle’s theorem in Logspace Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm II But here we are interested in space efficient algorithms, namely algorithms running in Logspace EJT gives an algorithm for Courcelle’s theorem in Logspace But for the first part we need to compute distance Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm II But here we are interested in space efficient algorithms, namely algorithms running in Logspace EJT gives an algorithm for Courcelle’s theorem in Logspace But for the first part we need to compute distance Distance is NL-Complete in general undirected graphs and in UL ∩ co-UL for planar graphs. And these classes are not believed to be in Logspace. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm II But here we are interested in space efficient algorithms, namely algorithms running in Logspace EJT gives an algorithm for Courcelle’s theorem in Logspace But for the first part we need to compute distance Distance is NL-Complete in general undirected graphs and in UL ∩ co-UL for planar graphs. And these classes are not believed to be in Logspace. Question Can we get away without using distance ? Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Baker’s Algorithm II But here we are interested in space efficient algorithms, namely algorithms running in Logspace EJT gives an algorithm for Courcelle’s theorem in Logspace But for the first part we need to compute distance Distance is NL-Complete in general undirected graphs and in UL ∩ co-UL for planar graphs. And these classes are not believed to be in Logspace. Question Can we get away without using distance ? Not yet Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Overview 1 Introduction 2 Matching 3 Our Contribution Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Matching Matching A matching M ⊆ E is a set of independent edges Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Matching Matching A matching M ⊆ E is a set of independent edges A matching M is called perfect if M covers all vertices of G M of maximum size is called maximum matching Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Matching Matching A matching M ⊆ E is a set of independent edges A matching M is called perfect if M covers all vertices of G M of maximum size is called maximum matching Augmenting Paths In an alternating path the edges alternate between M and E M Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Matching Matching A matching M ⊆ E is a set of independent edges A matching M is called perfect if M covers all vertices of G M of maximum size is called maximum matching Augmenting Paths In an alternating path the edges alternate between M and E M An alternating path P is augmenting if P begins and ends at unmatched vertices, that is, M ⊕ P = (M P) ∪ (P M) is a matching with cardinality |M| + 1. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Matching Edmond’s blossom algorithm for maximum matching was one of the first examples of a non-trivial polynomial time algorithm Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Matching Edmond’s blossom algorithm for maximum matching was one of the first examples of a non-trivial polynomial time algorithm Valiant’s #P-hardness for counting perfect matchings gave surprising insights into the counting complexity classes Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Matching Edmond’s blossom algorithm for maximum matching was one of the first examples of a non-trivial polynomial time algorithm Valiant’s #P-hardness for counting perfect matchings gave surprising insights into the counting complexity classes The RNC bound for maximum matching has yielded powerful tools, such as the isolating lemma [MVV87] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Matching Edmond’s blossom algorithm for maximum matching was one of the first examples of a non-trivial polynomial time algorithm Valiant’s #P-hardness for counting perfect matchings gave surprising insights into the counting complexity classes The RNC bound for maximum matching has yielded powerful tools, such as the isolating lemma [MVV87] Bipartite Perfect Matching is in quasi-NC [FGT16] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Matching Edmond’s blossom algorithm for maximum matching was one of the first examples of a non-trivial polynomial time algorithm Valiant’s #P-hardness for counting perfect matchings gave surprising insights into the counting complexity classes The RNC bound for maximum matching has yielded powerful tools, such as the isolating lemma [MVV87] Bipartite Perfect Matching is in quasi-NC [FGT16] The best hardness known is NL-hardness [CSV84] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Matching in Planar Graphs Counting perfect matchings in planar graphs is in NC [Vaz88] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Matching in Planar Graphs Counting perfect matchings in planar graphs is in NC [Vaz88] Only the bipartite planar case is known to be in NC for finding a perfect matching [MN95] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Matching in Planar Graphs Counting perfect matchings in planar graphs is in NC [Vaz88] Only the bipartite planar case is known to be in NC for finding a perfect matching [MN95] Open Problem Is the construction version in general planar graphs in NC ? Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Matching in Planar Graphs Counting perfect matchings in planar graphs is in NC [Vaz88] Only the bipartite planar case is known to be in NC for finding a perfect matching [MN95] Open Problem Is the construction version in general planar graphs in NC ? Computing a maximum matching for bipartite planar graphs is shown to be in NC [Hoang] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Matching in Planar Graphs Counting perfect matchings in planar graphs is in NC [Vaz88] Only the bipartite planar case is known to be in NC for finding a perfect matching [MN95] Open Problem Is the construction version in general planar graphs in NC ? Computing a maximum matching for bipartite planar graphs is shown to be in NC [Hoang] Only L-hardness is known for planar graphs [DKLM10]. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Time-Space Tradeoff Removing non-determinism even for planar reachability leads to either a quasi-polynomial time blow-up or need large space (O( √ n)) [INPVW13, AKNW14] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Time-Space Tradeoff Removing non-determinism even for planar reachability leads to either a quasi-polynomial time blow-up or need large space (O( √ n)) [INPVW13, AKNW14] For general graphs it is even worse, with O(n/2 √ log n) space and polynomial time [BBRS] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Time-Space Tradeoff Removing non-determinism even for planar reachability leads to either a quasi-polynomial time blow-up or need large space (O( √ n)) [INPVW13, AKNW14] For general graphs it is even worse, with O(n/2 √ log n) space and polynomial time [BBRS] Previous Results Approximating maximum matching has been considered both in time and parallel complexity model Linear-time [DP14] and NC [HV06] approximation scheme are the best known complexity bounds here Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Time-Space Tradeoff Removing non-determinism even for planar reachability leads to either a quasi-polynomial time blow-up or need large space (O( √ n)) [INPVW13, AKNW14] For general graphs it is even worse, with O(n/2 √ log n) space and polynomial time [BBRS] Previous Results Approximating maximum matching has been considered both in time and parallel complexity model Linear-time [DP14] and NC [HV06] approximation scheme are the best known complexity bounds here But work on space efficient approximation seems limited. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Overview 1 Introduction 2 Matching 3 Our Contribution Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Results Theorem Given a planar graph and any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Results Theorem Given a planar graph and any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. This result extends to many other sparse graph classes Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Results Theorem Given a planar graph and any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. This result extends to many other sparse graph classes Some of our ideas are similar to the classical algorithm of Hopcroft-Karp for maximum matching in bipartite graphs Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Results Theorem Given a planar graph and any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. This result extends to many other sparse graph classes Some of our ideas are similar to the classical algorithm of Hopcroft-Karp for maximum matching in bipartite graphs But we consider graphs which are not necessarily bipartite Our algorithm trades off Logspace and non-bipartiteness for approximation and sparsity Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Results Theorem Given a planar graph and any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. This result extends to many other sparse graph classes Some of our ideas are similar to the classical algorithm of Hopcroft-Karp for maximum matching in bipartite graphs But we consider graphs which are not necessarily bipartite Our algorithm trades off Logspace and non-bipartiteness for approximation and sparsity Solve by reducing it to bounded degree graphs suitably. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Results Theorem Let G be a graph with degrees bounded by a constant d then for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Results Theorem Let G be a graph with degrees bounded by a constant d then for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. The main fact we use here is that any bounded degree graphs always contains a linear size matching Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Results Theorem Let G be a graph with degrees bounded by a constant d then for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. The main fact we use here is that any bounded degree graphs always contains a linear size matching Many planar graph classes, such as 3-connected planar graphs, are known to be containing a large matching Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Results Theorem Let G be a graph with degrees bounded by a constant d then for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. The main fact we use here is that any bounded degree graphs always contains a linear size matching Many planar graph classes, such as 3-connected planar graphs, are known to be containing a large matching In fact our algorithm works for any recursively sparse graph containing a large matching. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution A Brief Idea 1 Consider short augmenting paths. In a bounded degree graph, there exist linearly many short augmenting paths Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution A Brief Idea 1 Consider short augmenting paths. In a bounded degree graph, there exist linearly many short augmenting paths 2 Pick a large subset of non-intersecting augmenting paths i.e find a large independent set of in Logspace Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution A Brief Idea 1 Consider short augmenting paths. In a bounded degree graph, there exist linearly many short augmenting paths 2 Pick a large subset of non-intersecting augmenting paths i.e find a large independent set of in Logspace 3 To convert a planar graph to a bounded degree graph we delete high degree vertices Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution A Brief Idea 1 Consider short augmenting paths. In a bounded degree graph, there exist linearly many short augmenting paths 2 Pick a large subset of non-intersecting augmenting paths i.e find a large independent set of in Logspace 3 To convert a planar graph to a bounded degree graph we delete high degree vertices 4 The number of such vertices is small though possibly still linear in the graph size Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution A Brief Idea 1 Consider short augmenting paths. In a bounded degree graph, there exist linearly many short augmenting paths 2 Pick a large subset of non-intersecting augmenting paths i.e find a large independent set of in Logspace 3 To convert a planar graph to a bounded degree graph we delete high degree vertices 4 The number of such vertices is small though possibly still linear in the graph size 5 Remove small number of vertices and edges to transform the graph down to one containing a linear sized matching. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Bounded degree graphs I We deal with augmenting paths of length at most 2k + 1 Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Bounded degree graphs I We deal with augmenting paths of length at most 2k + 1 Such paths can be found in Logspace by say exhaustively listing all (2k + 1)-tuples of vertices using L-transducers Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Bounded degree graphs I We deal with augmenting paths of length at most 2k + 1 Such paths can be found in Logspace by say exhaustively listing all (2k + 1)-tuples of vertices using L-transducers If |M| differs significantly from |Mopt| then we show that there are many short augmenting paths Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Bounded degree graphs I We deal with augmenting paths of length at most 2k + 1 Such paths can be found in Logspace by say exhaustively listing all (2k + 1)-tuples of vertices using L-transducers If |M| differs significantly from |Mopt| then we show that there are many short augmenting paths Lemma If |M| < (1 − 3 k )|Mopt| for some k then there are at least 3|Mopt|/2k augmenting paths consisting of at most 2k + 1 edges. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Bounded degree graphs I We deal with augmenting paths of length at most 2k + 1 Such paths can be found in Logspace by say exhaustively listing all (2k + 1)-tuples of vertices using L-transducers If |M| differs significantly from |Mopt| then we show that there are many short augmenting paths Lemma If |M| < (1 − 3 k )|Mopt| for some k then there are at least 3|Mopt|/2k augmenting paths consisting of at most 2k + 1 edges. Form an intersection graph of these short augmenting paths by making two paths adjacent if they have a vertex in common Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Maximum matching in bounded degree graphs II Lemma A β-factor approximation to the maximum independent set can be computed in Logspace Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Maximum matching in bounded degree graphs II Lemma A β-factor approximation to the maximum independent set can be computed in Logspace Colour the paths and the largest colour class works Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Maximum matching in bounded degree graphs II Lemma A β-factor approximation to the maximum independent set can be computed in Logspace Colour the paths and the largest colour class works As the degree is bounded by some D, find at most D disjoint forests that partition the edge set Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Maximum matching in bounded degree graphs II Lemma A β-factor approximation to the maximum independent set can be computed in Logspace Colour the paths and the largest colour class works As the degree is bounded by some D, find at most D disjoint forests that partition the edge set Can be done using Reingold’s algorithm for connectivity Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Maximum matching in bounded degree graphs II Lemma A β-factor approximation to the maximum independent set can be computed in Logspace Colour the paths and the largest colour class works As the degree is bounded by some D, find at most D disjoint forests that partition the edge set Can be done using Reingold’s algorithm for connectivity Colour each forest with 2 colours and it gives D bit colours to every node Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Maximum matching in bounded degree graphs II Lemma A β-factor approximation to the maximum independent set can be computed in Logspace Colour the paths and the largest colour class works As the degree is bounded by some D, find at most D disjoint forests that partition the edge set Can be done using Reingold’s algorithm for connectivity Colour each forest with 2 colours and it gives D bit colours to every node This yields a 2D i.e. constant colouring of the graph. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Theorem In a bounded degree graph for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in L. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Theorem In a bounded degree graph for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in L. Previous lemma yields large fraction of short paths, augmentable in parallel Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Theorem In a bounded degree graph for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in L. Previous lemma yields large fraction of short paths, augmentable in parallel A L-transducer can do the augmentation and we chain (1 − 3/k)2k/β such transducers Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Theorem In a bounded degree graph for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in L. Previous lemma yields large fraction of short paths, augmentable in parallel A L-transducer can do the augmentation and we chain (1 − 3/k)2k/β such transducers At each step we increase the matching size by an additive term of |Mopt|/(2k/β) Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Theorem In a bounded degree graph for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in L. Previous lemma yields large fraction of short paths, augmentable in parallel A L-transducer can do the augmentation and we chain (1 − 3/k)2k/β such transducers At each step we increase the matching size by an additive term of |Mopt|/(2k/β) After k rounds the ratio would be at least (1 − 3/k) ≥ 1 − . Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Algorithm 1 1 Fix integer k = 3 . 2 Construct the intersection graph of augmenting paths of length at most 2k + 1 in G. 3 Let the graph be H with maximum degree ≤ D = (2k + 1)2d2k+1 4 Find at most D disjoint forests that partition the edge set. 5 Colour each forest with 2 colours, giving D bit colours to every node 6 Augment the vertex disjoint augmenting paths in parallel 7 Add the new matching to M 8 Return M Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Planar maximum matching Definition A graph is tame if all pairs of vertices (a, b) which are endpoints of a even length isolated path, support at most two of them. This can be ensured by deleting a set of edges E from G Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Planar maximum matching Definition A graph is tame if all pairs of vertices (a, b) which are endpoints of a even length isolated path, support at most two of them. This can be ensured by deleting a set of edges E from G Lemma The size of the maximum matching in G E is the same as in G. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Planar maximum matching Definition A graph is tame if all pairs of vertices (a, b) which are endpoints of a even length isolated path, support at most two of them. This can be ensured by deleting a set of edges E from G Lemma The size of the maximum matching in G E is the same as in G. Main Lemma A tame planar graph has a linear sized maximum matching. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Planar maximum matching: tame graphs One of the following is true : Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Planar maximum matching: tame graphs One of the following is true : Total length of long isolated paths in G is large enough Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Planar maximum matching: tame graphs One of the following is true : Total length of long isolated paths in G is large enough We can transform the graph by case analysis to a minimum degree 3 planar graph Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Planar maximum matching: tame graphs One of the following is true : Total length of long isolated paths in G is large enough We can transform the graph by case analysis to a minimum degree 3 planar graph Lemma A graph in which the total length of isolated paths is N has a matching of size at least N/4. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Planar maximum matching: tame graphs One of the following is true : Total length of long isolated paths in G is large enough We can transform the graph by case analysis to a minimum degree 3 planar graph Lemma A graph in which the total length of isolated paths is N has a matching of size at least N/4. Lemma A min degree 3 planar graph has a matching of size at least n/140. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Planar maximum matching III Theorem There is a LSAS for maximum matching in planar graphs. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Planar maximum matching III Theorem There is a LSAS for maximum matching in planar graphs. proof Tame the graph G to G preserving the maximum matching size. Suppose there are least αn matching edges in G Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Planar maximum matching III Theorem There is a LSAS for maximum matching in planar graphs. proof Tame the graph G to G preserving the maximum matching size. Suppose there are least αn matching edges in G Delete vertices of degree more than d from G which removes at most 6n/d many matching edges Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Planar maximum matching III Theorem There is a LSAS for maximum matching in planar graphs. proof Tame the graph G to G preserving the maximum matching size. Suppose there are least αn matching edges in G Delete vertices of degree more than d from G which removes at most 6n/d many matching edges So we have a (α − 6/d)n sized matching remaining Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Planar maximum matching III Theorem There is a LSAS for maximum matching in planar graphs. proof Tame the graph G to G preserving the maximum matching size. Suppose there are least αn matching edges in G Delete vertices of degree more than d from G which removes at most 6n/d many matching edges So we have a (α − 6/d)n sized matching remaining Taking d = 12 2α− reduces the problem to find a (1 − /2) factor approximation algorithm for bounded degree graphs. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Conclusion We showed that maximum matching can be approximated to any arbitrary constant factor for bounded degree graphs Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Conclusion We showed that maximum matching can be approximated to any arbitrary constant factor for bounded degree graphs For planar graphs we require only the following properties: Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Conclusion We showed that maximum matching can be approximated to any arbitrary constant factor for bounded degree graphs For planar graphs we require only the following properties: Sparsity: The average degree is bounded by 6. Bipartite sparsity: Even lower, i.e 4. Min-degree: The minimum degree is at least 3 Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Conclusion We showed that maximum matching can be approximated to any arbitrary constant factor for bounded degree graphs For planar graphs we require only the following properties: Sparsity: The average degree is bounded by 6. Bipartite sparsity: Even lower, i.e 4. Min-degree: The minimum degree is at least 3 So can be extended many other classes of sparse graphs Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Conclusion We showed that maximum matching can be approximated to any arbitrary constant factor for bounded degree graphs For planar graphs we require only the following properties: Sparsity: The average degree is bounded by 6. Bipartite sparsity: Even lower, i.e 4. Min-degree: The minimum degree is at least 3 So can be extended many other classes of sparse graphs bounded genus graphs, k-page graphs, 1-planar graphs, k-Apex graphs etc recursively sparse graph containing a linear size matching. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Open Problems Baker’s Theorem in Logspace ? Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Open Problems Baker’s Theorem in Logspace ? Devise an LSAS for maximum matching in general graphs Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Open Problems Baker’s Theorem in Logspace ? Devise an LSAS for maximum matching in general graphs or at least in arbitrary sparse graphs Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Open Problems Baker’s Theorem in Logspace ? Devise an LSAS for maximum matching in general graphs or at least in arbitrary sparse graphs Lower bounds in the context of approximation ? Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Open Problems Baker’s Theorem in Logspace ? Devise an LSAS for maximum matching in general graphs or at least in arbitrary sparse graphs Lower bounds in the context of approximation ? Currently we do not know of any non-trivial, even TC0 -hardness results for approximation to any factor. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Thank You Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Packing complex patterns ? H-Matching Pack disjoint copies of a fixed graph H Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Packing complex patterns ? H-Matching Pack disjoint copies of a fixed graph H Maximum planar H-matching is NP-Complete for any H containing at least three nodes. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Packing complex patterns ? H-Matching Pack disjoint copies of a fixed graph H Maximum planar H-matching is NP-Complete for any H containing at least three nodes. Approximation and hardness is known for some restricted cases Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Packing complex patterns ? H-Matching Pack disjoint copies of a fixed graph H Maximum planar H-matching is NP-Complete for any H containing at least three nodes. Approximation and hardness is known for some restricted cases We give LSAS for graphs with a small balanced separator, for packing any fixed graph H when degrees are bounded Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Packing complex patterns ? H-Matching Pack disjoint copies of a fixed graph H Maximum planar H-matching is NP-Complete for any H containing at least three nodes. Approximation and hardness is known for some restricted cases We give LSAS for graphs with a small balanced separator, for packing any fixed graph H when degrees are bounded Otherwise, Packing some special class of patterns Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
Introduction Matching Our Contribution Packing complex patterns ? H-Matching Pack disjoint copies of a fixed graph H Maximum planar H-matching is NP-Complete for any H containing at least three nodes. Approximation and hardness is known for some restricted cases We give LSAS for graphs with a small balanced separator, for packing any fixed graph H when degrees are bounded Otherwise, Packing some special class of patterns As before, the idea is to delete high degree vertices and tame the graph by removing some forbidden patterns Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in

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Space-efficient Approximation Scheme for Maximum Matching in Sparse Graphs

  • 1.
    Introduction Matching Our Contribution Space EfficientApproximation Scheme for Maximum Matching in Sparse Graphs Samir Datta Raghav Kulkarni Anish Mukherjee Chennai Mathematical Institute NMI Workshop on Complexity Theory, IIT Gandhinagar November 04, 2016 Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 2.
    Introduction Matching Our Contribution Overview 1 Introduction 2Matching 3 Our Contribution Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 3.
    Introduction Matching Our Contribution Baker’s Algorithm Theorem(Baker ’83, Informal) A class of problems (many of which are NP-Hard in general) can be approximated arbitrarily close to the optimal value in linear time for planar graphs. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 4.
    Introduction Matching Our Contribution Baker’s Algorithm Theorem(Baker ’83, Informal) A class of problems (many of which are NP-Hard in general) can be approximated arbitrarily close to the optimal value in linear time for planar graphs. Example Includes problems like maximum independent set partition into triangles minimum vertex-cover minimum dominating set Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 5.
    Introduction Matching Our Contribution Baker’s Algorithm Theorem(Baker ’83, Informal) A class of problems (many of which are NP-Hard in general) can be approximated arbitrarily close to the optimal value in linear time for planar graphs. Example Includes problems like maximum independent set partition into triangles minimum vertex-cover minimum dominating set ... and any MSO definable properties Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 6.
    Introduction Matching Our Contribution Baker’s Algorithm BasicIdea Partition the vertices into breadth-first search levels Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 7.
    Introduction Matching Our Contribution Baker’s Algorithm BasicIdea Partition the vertices into breadth-first search levels Decompose the graph into successive width-k slices by deleting levels congruent to i mod k Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 8.
    Introduction Matching Our Contribution Baker’s Algorithm BasicIdea Partition the vertices into breadth-first search levels Decompose the graph into successive width-k slices by deleting levels congruent to i mod k Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 9.
    Introduction Matching Our Contribution Baker’s Algorithm BasicIdea Partition the vertices into breadth-first search levels Decompose the graph into successive width-k slices by deleting levels congruent to i mod k Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 10.
    Introduction Matching Our Contribution Baker’s Algorithm BasicIdea Partition the vertices into breadth-first search levels Decompose the graph into successive width-k slices by deleting levels congruent to i mod k Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 11.
    Introduction Matching Our Contribution Baker’s Algorithm BasicIdea Partition the vertices into breadth-first search levels Decompose the graph into successive width-k slices by deleting levels congruent to i mod k Resulting components have treewidth 3k − 1 [Boadlander] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 12.
    Introduction Matching Our Contribution Baker’s Algorithm BasicIdea Partition the vertices into breadth-first search levels Decompose the graph into successive width-k slices by deleting levels congruent to i mod k Resulting components have treewidth 3k − 1 [Boadlander] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 13.
    Introduction Matching Our Contribution Baker’s Algorithm BasicIdea Partition the vertices into breadth-first search levels Decompose the graph into successive width-k slices by deleting levels congruent to i mod k Resulting components have treewidth 3k − 1 [Boadlander] Solve the problem optimally in each partition in linear time Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 14.
    Introduction Matching Our Contribution Baker’s Algorithm BasicIdea Partition the vertices into breadth-first search levels Decompose the graph into successive width-k slices by deleting levels congruent to i mod k Resulting components have treewidth 3k − 1 [Boadlander] Solve the problem optimally in each partition in linear time Union of solutions in all components is within (1 − 1/k) OPT Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 15.
    Introduction Matching Our Contribution Baker’s AlgorithmII But here we are interested in space efficient algorithms, namely algorithms running in Logspace Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 16.
    Introduction Matching Our Contribution Baker’s AlgorithmII But here we are interested in space efficient algorithms, namely algorithms running in Logspace EJT gives an algorithm for Courcelle’s theorem in Logspace Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 17.
    Introduction Matching Our Contribution Baker’s AlgorithmII But here we are interested in space efficient algorithms, namely algorithms running in Logspace EJT gives an algorithm for Courcelle’s theorem in Logspace But for the first part we need to compute distance Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 18.
    Introduction Matching Our Contribution Baker’s AlgorithmII But here we are interested in space efficient algorithms, namely algorithms running in Logspace EJT gives an algorithm for Courcelle’s theorem in Logspace But for the first part we need to compute distance Distance is NL-Complete in general undirected graphs and in UL ∩ co-UL for planar graphs. And these classes are not believed to be in Logspace. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 19.
    Introduction Matching Our Contribution Baker’s AlgorithmII But here we are interested in space efficient algorithms, namely algorithms running in Logspace EJT gives an algorithm for Courcelle’s theorem in Logspace But for the first part we need to compute distance Distance is NL-Complete in general undirected graphs and in UL ∩ co-UL for planar graphs. And these classes are not believed to be in Logspace. Question Can we get away without using distance ? Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 20.
    Introduction Matching Our Contribution Baker’s AlgorithmII But here we are interested in space efficient algorithms, namely algorithms running in Logspace EJT gives an algorithm for Courcelle’s theorem in Logspace But for the first part we need to compute distance Distance is NL-Complete in general undirected graphs and in UL ∩ co-UL for planar graphs. And these classes are not believed to be in Logspace. Question Can we get away without using distance ? Not yet Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 21.
    Introduction Matching Our Contribution Overview 1 Introduction 2Matching 3 Our Contribution Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 22.
    Introduction Matching Our Contribution Matching Matching A matchingM ⊆ E is a set of independent edges Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 23.
    Introduction Matching Our Contribution Matching Matching A matchingM ⊆ E is a set of independent edges A matching M is called perfect if M covers all vertices of G M of maximum size is called maximum matching Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 24.
    Introduction Matching Our Contribution Matching Matching A matchingM ⊆ E is a set of independent edges A matching M is called perfect if M covers all vertices of G M of maximum size is called maximum matching Augmenting Paths In an alternating path the edges alternate between M and E M Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 25.
    Introduction Matching Our Contribution Matching Matching A matchingM ⊆ E is a set of independent edges A matching M is called perfect if M covers all vertices of G M of maximum size is called maximum matching Augmenting Paths In an alternating path the edges alternate between M and E M An alternating path P is augmenting if P begins and ends at unmatched vertices, that is, M ⊕ P = (M P) ∪ (P M) is a matching with cardinality |M| + 1. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 26.
    Introduction Matching Our Contribution Matching Edmond’s blossomalgorithm for maximum matching was one of the first examples of a non-trivial polynomial time algorithm Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 27.
    Introduction Matching Our Contribution Matching Edmond’s blossomalgorithm for maximum matching was one of the first examples of a non-trivial polynomial time algorithm Valiant’s #P-hardness for counting perfect matchings gave surprising insights into the counting complexity classes Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 28.
    Introduction Matching Our Contribution Matching Edmond’s blossomalgorithm for maximum matching was one of the first examples of a non-trivial polynomial time algorithm Valiant’s #P-hardness for counting perfect matchings gave surprising insights into the counting complexity classes The RNC bound for maximum matching has yielded powerful tools, such as the isolating lemma [MVV87] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 29.
    Introduction Matching Our Contribution Matching Edmond’s blossomalgorithm for maximum matching was one of the first examples of a non-trivial polynomial time algorithm Valiant’s #P-hardness for counting perfect matchings gave surprising insights into the counting complexity classes The RNC bound for maximum matching has yielded powerful tools, such as the isolating lemma [MVV87] Bipartite Perfect Matching is in quasi-NC [FGT16] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 30.
    Introduction Matching Our Contribution Matching Edmond’s blossomalgorithm for maximum matching was one of the first examples of a non-trivial polynomial time algorithm Valiant’s #P-hardness for counting perfect matchings gave surprising insights into the counting complexity classes The RNC bound for maximum matching has yielded powerful tools, such as the isolating lemma [MVV87] Bipartite Perfect Matching is in quasi-NC [FGT16] The best hardness known is NL-hardness [CSV84] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 31.
    Introduction Matching Our Contribution Matching inPlanar Graphs Counting perfect matchings in planar graphs is in NC [Vaz88] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 32.
    Introduction Matching Our Contribution Matching inPlanar Graphs Counting perfect matchings in planar graphs is in NC [Vaz88] Only the bipartite planar case is known to be in NC for finding a perfect matching [MN95] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 33.
    Introduction Matching Our Contribution Matching inPlanar Graphs Counting perfect matchings in planar graphs is in NC [Vaz88] Only the bipartite planar case is known to be in NC for finding a perfect matching [MN95] Open Problem Is the construction version in general planar graphs in NC ? Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 34.
    Introduction Matching Our Contribution Matching inPlanar Graphs Counting perfect matchings in planar graphs is in NC [Vaz88] Only the bipartite planar case is known to be in NC for finding a perfect matching [MN95] Open Problem Is the construction version in general planar graphs in NC ? Computing a maximum matching for bipartite planar graphs is shown to be in NC [Hoang] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 35.
    Introduction Matching Our Contribution Matching inPlanar Graphs Counting perfect matchings in planar graphs is in NC [Vaz88] Only the bipartite planar case is known to be in NC for finding a perfect matching [MN95] Open Problem Is the construction version in general planar graphs in NC ? Computing a maximum matching for bipartite planar graphs is shown to be in NC [Hoang] Only L-hardness is known for planar graphs [DKLM10]. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 36.
    Introduction Matching Our Contribution Time-Space Tradeoff Removingnon-determinism even for planar reachability leads to either a quasi-polynomial time blow-up or need large space (O( √ n)) [INPVW13, AKNW14] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 37.
    Introduction Matching Our Contribution Time-Space Tradeoff Removingnon-determinism even for planar reachability leads to either a quasi-polynomial time blow-up or need large space (O( √ n)) [INPVW13, AKNW14] For general graphs it is even worse, with O(n/2 √ log n) space and polynomial time [BBRS] Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 38.
    Introduction Matching Our Contribution Time-Space Tradeoff Removingnon-determinism even for planar reachability leads to either a quasi-polynomial time blow-up or need large space (O( √ n)) [INPVW13, AKNW14] For general graphs it is even worse, with O(n/2 √ log n) space and polynomial time [BBRS] Previous Results Approximating maximum matching has been considered both in time and parallel complexity model Linear-time [DP14] and NC [HV06] approximation scheme are the best known complexity bounds here Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 39.
    Introduction Matching Our Contribution Time-Space Tradeoff Removingnon-determinism even for planar reachability leads to either a quasi-polynomial time blow-up or need large space (O( √ n)) [INPVW13, AKNW14] For general graphs it is even worse, with O(n/2 √ log n) space and polynomial time [BBRS] Previous Results Approximating maximum matching has been considered both in time and parallel complexity model Linear-time [DP14] and NC [HV06] approximation scheme are the best known complexity bounds here But work on space efficient approximation seems limited. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 40.
    Introduction Matching Our Contribution Overview 1 Introduction 2Matching 3 Our Contribution Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 41.
    Introduction Matching Our Contribution Results Theorem Given aplanar graph and any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 42.
    Introduction Matching Our Contribution Results Theorem Given aplanar graph and any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. This result extends to many other sparse graph classes Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 43.
    Introduction Matching Our Contribution Results Theorem Given aplanar graph and any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. This result extends to many other sparse graph classes Some of our ideas are similar to the classical algorithm of Hopcroft-Karp for maximum matching in bipartite graphs Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 44.
    Introduction Matching Our Contribution Results Theorem Given aplanar graph and any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. This result extends to many other sparse graph classes Some of our ideas are similar to the classical algorithm of Hopcroft-Karp for maximum matching in bipartite graphs But we consider graphs which are not necessarily bipartite Our algorithm trades off Logspace and non-bipartiteness for approximation and sparsity Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 45.
    Introduction Matching Our Contribution Results Theorem Given aplanar graph and any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. This result extends to many other sparse graph classes Some of our ideas are similar to the classical algorithm of Hopcroft-Karp for maximum matching in bipartite graphs But we consider graphs which are not necessarily bipartite Our algorithm trades off Logspace and non-bipartiteness for approximation and sparsity Solve by reducing it to bounded degree graphs suitably. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 46.
    Introduction Matching Our Contribution Results Theorem Let Gbe a graph with degrees bounded by a constant d then for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 47.
    Introduction Matching Our Contribution Results Theorem Let Gbe a graph with degrees bounded by a constant d then for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. The main fact we use here is that any bounded degree graphs always contains a linear size matching Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 48.
    Introduction Matching Our Contribution Results Theorem Let Gbe a graph with degrees bounded by a constant d then for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. The main fact we use here is that any bounded degree graphs always contains a linear size matching Many planar graph classes, such as 3-connected planar graphs, are known to be containing a large matching Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 49.
    Introduction Matching Our Contribution Results Theorem Let Gbe a graph with degrees bounded by a constant d then for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in Logspace. The main fact we use here is that any bounded degree graphs always contains a linear size matching Many planar graph classes, such as 3-connected planar graphs, are known to be containing a large matching In fact our algorithm works for any recursively sparse graph containing a large matching. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 50.
    Introduction Matching Our Contribution A BriefIdea 1 Consider short augmenting paths. In a bounded degree graph, there exist linearly many short augmenting paths Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 51.
    Introduction Matching Our Contribution A BriefIdea 1 Consider short augmenting paths. In a bounded degree graph, there exist linearly many short augmenting paths 2 Pick a large subset of non-intersecting augmenting paths i.e find a large independent set of in Logspace Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 52.
    Introduction Matching Our Contribution A BriefIdea 1 Consider short augmenting paths. In a bounded degree graph, there exist linearly many short augmenting paths 2 Pick a large subset of non-intersecting augmenting paths i.e find a large independent set of in Logspace 3 To convert a planar graph to a bounded degree graph we delete high degree vertices Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 53.
    Introduction Matching Our Contribution A BriefIdea 1 Consider short augmenting paths. In a bounded degree graph, there exist linearly many short augmenting paths 2 Pick a large subset of non-intersecting augmenting paths i.e find a large independent set of in Logspace 3 To convert a planar graph to a bounded degree graph we delete high degree vertices 4 The number of such vertices is small though possibly still linear in the graph size Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 54.
    Introduction Matching Our Contribution A BriefIdea 1 Consider short augmenting paths. In a bounded degree graph, there exist linearly many short augmenting paths 2 Pick a large subset of non-intersecting augmenting paths i.e find a large independent set of in Logspace 3 To convert a planar graph to a bounded degree graph we delete high degree vertices 4 The number of such vertices is small though possibly still linear in the graph size 5 Remove small number of vertices and edges to transform the graph down to one containing a linear sized matching. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 55.
    Introduction Matching Our Contribution Bounded degreegraphs I We deal with augmenting paths of length at most 2k + 1 Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 56.
    Introduction Matching Our Contribution Bounded degreegraphs I We deal with augmenting paths of length at most 2k + 1 Such paths can be found in Logspace by say exhaustively listing all (2k + 1)-tuples of vertices using L-transducers Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 57.
    Introduction Matching Our Contribution Bounded degreegraphs I We deal with augmenting paths of length at most 2k + 1 Such paths can be found in Logspace by say exhaustively listing all (2k + 1)-tuples of vertices using L-transducers If |M| differs significantly from |Mopt| then we show that there are many short augmenting paths Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 58.
    Introduction Matching Our Contribution Bounded degreegraphs I We deal with augmenting paths of length at most 2k + 1 Such paths can be found in Logspace by say exhaustively listing all (2k + 1)-tuples of vertices using L-transducers If |M| differs significantly from |Mopt| then we show that there are many short augmenting paths Lemma If |M| < (1 − 3 k )|Mopt| for some k then there are at least 3|Mopt|/2k augmenting paths consisting of at most 2k + 1 edges. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 59.
    Introduction Matching Our Contribution Bounded degreegraphs I We deal with augmenting paths of length at most 2k + 1 Such paths can be found in Logspace by say exhaustively listing all (2k + 1)-tuples of vertices using L-transducers If |M| differs significantly from |Mopt| then we show that there are many short augmenting paths Lemma If |M| < (1 − 3 k )|Mopt| for some k then there are at least 3|Mopt|/2k augmenting paths consisting of at most 2k + 1 edges. Form an intersection graph of these short augmenting paths by making two paths adjacent if they have a vertex in common Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 60.
    Introduction Matching Our Contribution Maximum matchingin bounded degree graphs II Lemma A β-factor approximation to the maximum independent set can be computed in Logspace Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 61.
    Introduction Matching Our Contribution Maximum matchingin bounded degree graphs II Lemma A β-factor approximation to the maximum independent set can be computed in Logspace Colour the paths and the largest colour class works Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 62.
    Introduction Matching Our Contribution Maximum matchingin bounded degree graphs II Lemma A β-factor approximation to the maximum independent set can be computed in Logspace Colour the paths and the largest colour class works As the degree is bounded by some D, find at most D disjoint forests that partition the edge set Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 63.
    Introduction Matching Our Contribution Maximum matchingin bounded degree graphs II Lemma A β-factor approximation to the maximum independent set can be computed in Logspace Colour the paths and the largest colour class works As the degree is bounded by some D, find at most D disjoint forests that partition the edge set Can be done using Reingold’s algorithm for connectivity Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 64.
    Introduction Matching Our Contribution Maximum matchingin bounded degree graphs II Lemma A β-factor approximation to the maximum independent set can be computed in Logspace Colour the paths and the largest colour class works As the degree is bounded by some D, find at most D disjoint forests that partition the edge set Can be done using Reingold’s algorithm for connectivity Colour each forest with 2 colours and it gives D bit colours to every node Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 65.
    Introduction Matching Our Contribution Maximum matchingin bounded degree graphs II Lemma A β-factor approximation to the maximum independent set can be computed in Logspace Colour the paths and the largest colour class works As the degree is bounded by some D, find at most D disjoint forests that partition the edge set Can be done using Reingold’s algorithm for connectivity Colour each forest with 2 colours and it gives D bit colours to every node This yields a 2D i.e. constant colouring of the graph. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 66.
    Introduction Matching Our Contribution Theorem In abounded degree graph for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in L. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 67.
    Introduction Matching Our Contribution Theorem In abounded degree graph for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in L. Previous lemma yields large fraction of short paths, augmentable in parallel Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 68.
    Introduction Matching Our Contribution Theorem In abounded degree graph for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in L. Previous lemma yields large fraction of short paths, augmentable in parallel A L-transducer can do the augmentation and we chain (1 − 3/k)2k/β such transducers Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 69.
    Introduction Matching Our Contribution Theorem In abounded degree graph for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in L. Previous lemma yields large fraction of short paths, augmentable in parallel A L-transducer can do the augmentation and we chain (1 − 3/k)2k/β such transducers At each step we increase the matching size by an additive term of |Mopt|/(2k/β) Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 70.
    Introduction Matching Our Contribution Theorem In abounded degree graph for any fixed > 0, we can find a (1 − ) factor approximation to the maximum matching in L. Previous lemma yields large fraction of short paths, augmentable in parallel A L-transducer can do the augmentation and we chain (1 − 3/k)2k/β such transducers At each step we increase the matching size by an additive term of |Mopt|/(2k/β) After k rounds the ratio would be at least (1 − 3/k) ≥ 1 − . Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 71.
    Introduction Matching Our Contribution Algorithm 1 1Fix integer k = 3 . 2 Construct the intersection graph of augmenting paths of length at most 2k + 1 in G. 3 Let the graph be H with maximum degree ≤ D = (2k + 1)2d2k+1 4 Find at most D disjoint forests that partition the edge set. 5 Colour each forest with 2 colours, giving D bit colours to every node 6 Augment the vertex disjoint augmenting paths in parallel 7 Add the new matching to M 8 Return M Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 72.
    Introduction Matching Our Contribution Planar maximummatching Definition A graph is tame if all pairs of vertices (a, b) which are endpoints of a even length isolated path, support at most two of them. This can be ensured by deleting a set of edges E from G Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 73.
    Introduction Matching Our Contribution Planar maximummatching Definition A graph is tame if all pairs of vertices (a, b) which are endpoints of a even length isolated path, support at most two of them. This can be ensured by deleting a set of edges E from G Lemma The size of the maximum matching in G E is the same as in G. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 74.
    Introduction Matching Our Contribution Planar maximummatching Definition A graph is tame if all pairs of vertices (a, b) which are endpoints of a even length isolated path, support at most two of them. This can be ensured by deleting a set of edges E from G Lemma The size of the maximum matching in G E is the same as in G. Main Lemma A tame planar graph has a linear sized maximum matching. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 75.
    Introduction Matching Our Contribution Planar maximummatching: tame graphs One of the following is true : Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 76.
    Introduction Matching Our Contribution Planar maximummatching: tame graphs One of the following is true : Total length of long isolated paths in G is large enough Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 77.
    Introduction Matching Our Contribution Planar maximummatching: tame graphs One of the following is true : Total length of long isolated paths in G is large enough We can transform the graph by case analysis to a minimum degree 3 planar graph Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 78.
    Introduction Matching Our Contribution Planar maximummatching: tame graphs One of the following is true : Total length of long isolated paths in G is large enough We can transform the graph by case analysis to a minimum degree 3 planar graph Lemma A graph in which the total length of isolated paths is N has a matching of size at least N/4. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 79.
    Introduction Matching Our Contribution Planar maximummatching: tame graphs One of the following is true : Total length of long isolated paths in G is large enough We can transform the graph by case analysis to a minimum degree 3 planar graph Lemma A graph in which the total length of isolated paths is N has a matching of size at least N/4. Lemma A min degree 3 planar graph has a matching of size at least n/140. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 80.
    Introduction Matching Our Contribution Planar maximummatching III Theorem There is a LSAS for maximum matching in planar graphs. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 81.
    Introduction Matching Our Contribution Planar maximummatching III Theorem There is a LSAS for maximum matching in planar graphs. proof Tame the graph G to G preserving the maximum matching size. Suppose there are least αn matching edges in G Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 82.
    Introduction Matching Our Contribution Planar maximummatching III Theorem There is a LSAS for maximum matching in planar graphs. proof Tame the graph G to G preserving the maximum matching size. Suppose there are least αn matching edges in G Delete vertices of degree more than d from G which removes at most 6n/d many matching edges Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 83.
    Introduction Matching Our Contribution Planar maximummatching III Theorem There is a LSAS for maximum matching in planar graphs. proof Tame the graph G to G preserving the maximum matching size. Suppose there are least αn matching edges in G Delete vertices of degree more than d from G which removes at most 6n/d many matching edges So we have a (α − 6/d)n sized matching remaining Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 84.
    Introduction Matching Our Contribution Planar maximummatching III Theorem There is a LSAS for maximum matching in planar graphs. proof Tame the graph G to G preserving the maximum matching size. Suppose there are least αn matching edges in G Delete vertices of degree more than d from G which removes at most 6n/d many matching edges So we have a (α − 6/d)n sized matching remaining Taking d = 12 2α− reduces the problem to find a (1 − /2) factor approximation algorithm for bounded degree graphs. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 85.
    Introduction Matching Our Contribution Conclusion We showedthat maximum matching can be approximated to any arbitrary constant factor for bounded degree graphs Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 86.
    Introduction Matching Our Contribution Conclusion We showedthat maximum matching can be approximated to any arbitrary constant factor for bounded degree graphs For planar graphs we require only the following properties: Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 87.
    Introduction Matching Our Contribution Conclusion We showedthat maximum matching can be approximated to any arbitrary constant factor for bounded degree graphs For planar graphs we require only the following properties: Sparsity: The average degree is bounded by 6. Bipartite sparsity: Even lower, i.e 4. Min-degree: The minimum degree is at least 3 Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 88.
    Introduction Matching Our Contribution Conclusion We showedthat maximum matching can be approximated to any arbitrary constant factor for bounded degree graphs For planar graphs we require only the following properties: Sparsity: The average degree is bounded by 6. Bipartite sparsity: Even lower, i.e 4. Min-degree: The minimum degree is at least 3 So can be extended many other classes of sparse graphs Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 89.
    Introduction Matching Our Contribution Conclusion We showedthat maximum matching can be approximated to any arbitrary constant factor for bounded degree graphs For planar graphs we require only the following properties: Sparsity: The average degree is bounded by 6. Bipartite sparsity: Even lower, i.e 4. Min-degree: The minimum degree is at least 3 So can be extended many other classes of sparse graphs bounded genus graphs, k-page graphs, 1-planar graphs, k-Apex graphs etc recursively sparse graph containing a linear size matching. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 90.
    Introduction Matching Our Contribution Open Problems Baker’sTheorem in Logspace ? Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 91.
    Introduction Matching Our Contribution Open Problems Baker’sTheorem in Logspace ? Devise an LSAS for maximum matching in general graphs Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 92.
    Introduction Matching Our Contribution Open Problems Baker’sTheorem in Logspace ? Devise an LSAS for maximum matching in general graphs or at least in arbitrary sparse graphs Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 93.
    Introduction Matching Our Contribution Open Problems Baker’sTheorem in Logspace ? Devise an LSAS for maximum matching in general graphs or at least in arbitrary sparse graphs Lower bounds in the context of approximation ? Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 94.
    Introduction Matching Our Contribution Open Problems Baker’sTheorem in Logspace ? Devise an LSAS for maximum matching in general graphs or at least in arbitrary sparse graphs Lower bounds in the context of approximation ? Currently we do not know of any non-trivial, even TC0 -hardness results for approximation to any factor. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 95.
    Introduction Matching Our Contribution Thank You SamirDatta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 96.
    Introduction Matching Our Contribution Packing complexpatterns ? H-Matching Pack disjoint copies of a fixed graph H Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 97.
    Introduction Matching Our Contribution Packing complexpatterns ? H-Matching Pack disjoint copies of a fixed graph H Maximum planar H-matching is NP-Complete for any H containing at least three nodes. Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 98.
    Introduction Matching Our Contribution Packing complexpatterns ? H-Matching Pack disjoint copies of a fixed graph H Maximum planar H-matching is NP-Complete for any H containing at least three nodes. Approximation and hardness is known for some restricted cases Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 99.
    Introduction Matching Our Contribution Packing complexpatterns ? H-Matching Pack disjoint copies of a fixed graph H Maximum planar H-matching is NP-Complete for any H containing at least three nodes. Approximation and hardness is known for some restricted cases We give LSAS for graphs with a small balanced separator, for packing any fixed graph H when degrees are bounded Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 100.
    Introduction Matching Our Contribution Packing complexpatterns ? H-Matching Pack disjoint copies of a fixed graph H Maximum planar H-matching is NP-Complete for any H containing at least three nodes. Approximation and hardness is known for some restricted cases We give LSAS for graphs with a small balanced separator, for packing any fixed graph H when degrees are bounded Otherwise, Packing some special class of patterns Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in
  • 101.
    Introduction Matching Our Contribution Packing complexpatterns ? H-Matching Pack disjoint copies of a fixed graph H Maximum planar H-matching is NP-Complete for any H containing at least three nodes. Approximation and hardness is known for some restricted cases We give LSAS for graphs with a small balanced separator, for packing any fixed graph H when degrees are bounded Otherwise, Packing some special class of patterns As before, the idea is to delete high degree vertices and tame the graph by removing some forbidden patterns Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in