We present a Logspace Approximation Scheme (LSAS), i.e. an approximation algorithm for maximum matching in planar graphs (not necessarily bipartite) that achieves an approximation ratio arbitrarily close to one, in Logspace This deviates from the well known Baker’s approach for approximation in planar graphs by avoiding the use of distance computation - which is not known to be in Logspace. Our algorithm actually works for any “recursively sparse” graph class which contains a linear size matching.The scheme is based on an LSAS in bounded degree graphs which are not known to be amenable to Baker’s method. We solve the bounded degree case by parallel augmentation of short augmenting paths. Finding a large number of such disjoint paths can, in turn, be reduced to finding a large independent set in a bounded degree graph. This is joint work with Samir Datta and Raghav Kulkarni.

1. 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

2. 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

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
Basic Idea
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
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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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

46. 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

47. 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

48. 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

49. 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

50. 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

51. 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

52. 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

53. 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

54. 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

55. 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

56. 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

57. 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

58. 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

59. 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

60. 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

61. 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

62. 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

63. 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

64. 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

65. 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

66. 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

67. 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

68. 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

69. 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

70. 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

71. 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

72. 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

73. 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

74. 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

75. 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

76. 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

77. 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

78. 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

79. 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

80. 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

81. 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

82. 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

83. 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

84. 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

85. 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

86. 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

87. 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

88. 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

89. 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

90. 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

91. 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

92. 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

93. 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

94. 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

95. Introduction
Matching
Our Contribution
Thank You
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in

96. 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

97. 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

98. 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

99. 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

100. 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

101. 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