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Iterative Compression

The document describes the iterative compression technique for designing fixed-parameter tractable algorithms. It shows how to use a vertex/edge cover, feedback vertex set, or odd cycle transversal of size k+1 as "compression" to design a branching algorithm with running time of O*(ck) for some constant c. This technique involves guessing how an optimal solution of size k interacts with the given size k+1 solution, and branching based on including or excluding vertices in the optimal solution.

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Iterative Compression
Input A graph G = (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Vertex Cover Question
Input A graph G = (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Vertex Cover Question
Input A graph G = (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Vertex Cover Question
Input A graph G with a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Vertex Cover Question
Input A graph G with a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Vertex Cover Question
Input A graph G with a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Let us “guess” how a vertex cover of size at most k interacts with this one. Vertex Cover Question
Input A graph G with a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Let us “guess” how a vertex cover of size at most k interacts with this one. Vertex Cover Question
Input A graph G with a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Let us “guess” how a vertex cover of size at most k interacts with this one. Vertex Cover Question
Input A graph G with a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Vertex Cover Question
Input A graph G with a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Can there be an edge between two red vertices? Vertex Cover Question
Input A graph G with a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Vertex Cover Question
Input A graph G with a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Now we need to make up for the work that the red vertices were doing. Vertex Cover Question
Input A graph G with a vertex cover of size k+1. Is there a vertex cover of size at most k? Vertex Cover Question
Input A graph G with a vertex cover of size k+1. Is there a vertex cover of size at most k? Vertex Cover Question
Input A graph G = (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Vertex Cover Question
Input A graph G = (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Vertex Cover Question
Input A graph G = (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Vertex Cover Question
Input A graph G = (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Vertex Cover Question
Input A graph G = (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? The first k+2 vertices in G. Vertex Cover Question
Input A graph G = (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Question The first k+2 vertices in G. It has an easy vertex cover of size k+1. Vertex Cover
Input A graph G = (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Question The first k+2 vertices in G. It has an easy vertex cover of size k+1. Vertex Cover
Input A graph G = (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Question The first k+2 vertices in G. It has an easy vertex cover of size k+1. Vertex Cover
Input A graph G = (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Question The first k+2 vertices in G. It has an easy vertex cover of size k+1. Vertex Cover
Input A graph G = (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Question The first k+2 vertices in G. It has an easy vertex cover of size k+1. Vertex Cover
Input A graph G = (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Question The first k+2 vertices in G. It has an easy vertex cover of size k+1. Compress again… rinse, repeat. Vertex Cover
Input A graph G = (V,E) with n vertices, m edges, and k. Is there a subset of vertices S of size at most k that intersects all the edges? Question We have a O*(2k) algorithm for Vertex Cover. Vertex Cover
Input A graph G = (V,E) with n vertices, m edges, and k. Is there a subset of vertices S of size at most k that intersects all the edges? Question We have a O*(2k) algorithm for Vertex Cover. Vertex Cover
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? A Feedback Vertex Set of size (k+1). Let us “guess” how a FVS of size at most k interacts with this one. Feedback Vertex Set Question
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question A vertex with two neighbors in the same component is forced. Feedback Vertex Set
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question A leaf that has exactly one neighbor above can be preprocessed. Feedback Vertex Set
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question …a leaf with at least two neighbors in different components. Feedback Vertex Set
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question The leaf merges two components when we don’t include it. Feedback Vertex Set
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question The number of components “on top” decreases. Feedback Vertex Set
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question Start with a leaf. ! Two neighbors in one component: forced. At most one neighbor above: preprocess. At least two neighbors, all in different components above: branch. Feedback Vertex Set
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question Let t denote the number of components among the red vertices. Let w = (k+t). Feedback Vertex Set
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question Let t denote the number of components among the red vertices. Let w = (k+t). Include v… k drops by 1 Exclude v… t drops by at least 1 Feedback Vertex Set
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question Let t denote the number of components among the red vertices. Let w = (k+t). Include v… k drops by 1 Exclude v… t drops by at least 1 Either way, w drops by at least one. Feedback Vertex Set
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Either way, w drops by at least one. Feedback Vertex Set Question
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Either way, w drops by at least one. Running time: 2w = 2(k+t) Feedback Vertex Set Question
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Either way, w drops by at least one. Running time: 2w = 2(k+t) ≤ 2k+k ≤ 4k Feedback Vertex Set Question
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Either way, w drops by at least one. Running time: 2w = 2(k+t) ≤ 2k+k ≤ 4k Overall Running Time… Feedback Vertex Set Question
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Either way, w drops by at least one. Running time: 2w = 2(k+t) ≤ 2k+k ≤ 4k Overall Running Time… k i=1 k k k 4 =5 i Feedback Vertex Set Question
Input A tournament T on n vertices and an integer k. Is there a subset of k arcs that can be reversed to make the tournament acyclic? This implies an O(5k) algorithm for FVS. Feedback Vertex Set Question
Input A tournament T on n vertices and an integer k. Is there a subset of k arcs that can be reversed to make the tournament acyclic? This implies an O(5k) algorithm for FVS. Feedback Vertex Set Question
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question A Odd Cycle Transversal of size (k+1). Let us “guess” how a OCT of size at most k interacts with this one. Odd Cycle Transversal (OCT)
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Can we remove at most b vertices so that the resulting graph has a bipartition extending this pre-coloring? Odd Cycle Transversal (OCT)
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? GREEN Question YELLOW Can we remove at most b vertices so that the resulting graph has a bipartition extending this pre-coloring? Odd Cycle Transversal (OCT)
Input A graph on n vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? GREEN Question YELLOW Can we remove at most b vertices so that the resulting graph has a bipartition extending this pre-coloring? Odd Cycle Transversal (OCT)
Input A tournament T on n vertices and an integer k. Is there a subset of k arcs that can be reversed to make the tournament acyclic? Question This implies an O*(3k) algorithm for OCT. Odd Cycle Transversal (OCT)
Input A tournament T on n vertices and an integer k. Is there a subset of k arcs that can be reversed to make the tournament acyclic? Question This implies an O*(3k) algorithm for OCT. Odd Cycle Transversal (OCT)
Take Away…
Show that the problem has a hereditary property Solve the “compression” question, or even a “Disjoint” version.
Show that the problem has a hereditary property Solve the “compression” question, or even a “Disjoint” version. Is a framework that can be setup quite easily.
Show that the problem has a hereditary property Solve the “compression” question, or even a “Disjoint” version. Is a framework that can be setup quite easily. The compression algorithm can be fairly non-trivial.

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Iterative Compression

  • 1.
  • 2.
    Input A graph G= (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Vertex Cover Question
  • 3.
    Input A graph G= (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Vertex Cover Question
  • 4.
    Input A graph G= (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Vertex Cover Question
  • 5.
    Input A graph Gwith a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Vertex Cover Question
  • 6.
    Input A graph Gwith a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Vertex Cover Question
  • 7.
    Input A graph Gwith a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Let us “guess” how a vertex cover of size at most k interacts with this one. Vertex Cover Question
  • 8.
    Input A graph Gwith a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Let us “guess” how a vertex cover of size at most k interacts with this one. Vertex Cover Question
  • 9.
    Input A graph Gwith a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Let us “guess” how a vertex cover of size at most k interacts with this one. Vertex Cover Question
  • 10.
    Input A graph Gwith a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Vertex Cover Question
  • 11.
    Input A graph Gwith a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Can there be an edge between two red vertices? Vertex Cover Question
  • 12.
    Input A graph Gwith a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Vertex Cover Question
  • 13.
    Input A graph Gwith a vertex cover of size k+1. Is there a vertex cover of size at most k? A vertex cover of size (k+1). Now we need to make up for the work that the red vertices were doing. Vertex Cover Question
  • 14.
    Input A graph Gwith a vertex cover of size k+1. Is there a vertex cover of size at most k? Vertex Cover Question
  • 15.
    Input A graph Gwith a vertex cover of size k+1. Is there a vertex cover of size at most k? Vertex Cover Question
  • 16.
    Input A graph G= (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Vertex Cover Question
  • 17.
    Input A graph G= (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Vertex Cover Question
  • 18.
    Input A graph G= (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Vertex Cover Question
  • 19.
    Input A graph G= (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Vertex Cover Question
  • 20.
    Input A graph G= (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? The first k+2 vertices in G. Vertex Cover Question
  • 21.
    Input A graph G= (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Question The first k+2 vertices in G. It has an easy vertex cover of size k+1. Vertex Cover
  • 22.
    Input A graph G= (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Question The first k+2 vertices in G. It has an easy vertex cover of size k+1. Vertex Cover
  • 23.
    Input A graph G= (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Question The first k+2 vertices in G. It has an easy vertex cover of size k+1. Vertex Cover
  • 24.
    Input A graph G= (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Question The first k+2 vertices in G. It has an easy vertex cover of size k+1. Vertex Cover
  • 25.
    Input A graph G= (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Question The first k+2 vertices in G. It has an easy vertex cover of size k+1. Vertex Cover
  • 26.
    Input A graph G= (V,E) with n vertices, m edges, and k. Is there a vertex cover of size at most k? Question The first k+2 vertices in G. It has an easy vertex cover of size k+1. Compress again… rinse, repeat. Vertex Cover
  • 27.
    Input A graph G= (V,E) with n vertices, m edges, and k. Is there a subset of vertices S of size at most k that intersects all the edges? Question We have a O*(2k) algorithm for Vertex Cover. Vertex Cover
  • 28.
    Input A graph G= (V,E) with n vertices, m edges, and k. Is there a subset of vertices S of size at most k that intersects all the edges? Question We have a O*(2k) algorithm for Vertex Cover. Vertex Cover
  • 29.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? A Feedback Vertex Set of size (k+1). Let us “guess” how a FVS of size at most k interacts with this one. Feedback Vertex Set Question
  • 30.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
  • 31.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
  • 32.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
  • 33.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
  • 34.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question A vertex with two neighbors in the same component is forced. Feedback Vertex Set
  • 35.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
  • 36.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question A leaf that has exactly one neighbor above can be preprocessed. Feedback Vertex Set
  • 37.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
  • 38.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question …a leaf with at least two neighbors in different components. Feedback Vertex Set
  • 39.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
  • 40.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question The leaf merges two components when we don’t include it. Feedback Vertex Set
  • 41.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
  • 42.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question The number of components “on top” decreases. Feedback Vertex Set
  • 43.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question Start with a leaf. ! Two neighbors in one component: forced. At most one neighbor above: preprocess. At least two neighbors, all in different components above: branch. Feedback Vertex Set
  • 44.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Feedback Vertex Set Question
  • 45.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question Let t denote the number of components among the red vertices. Let w = (k+t). Feedback Vertex Set
  • 46.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question Let t denote the number of components among the red vertices. Let w = (k+t). Include v… k drops by 1 Exclude v… t drops by at least 1 Feedback Vertex Set
  • 47.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Question Let t denote the number of components among the red vertices. Let w = (k+t). Include v… k drops by 1 Exclude v… t drops by at least 1 Either way, w drops by at least one. Feedback Vertex Set
  • 48.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Either way, w drops by at least one. Feedback Vertex Set Question
  • 49.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Either way, w drops by at least one. Running time: 2w = 2(k+t) Feedback Vertex Set Question
  • 50.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Either way, w drops by at least one. Running time: 2w = 2(k+t) ≤ 2k+k ≤ 4k Feedback Vertex Set Question
  • 51.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Either way, w drops by at least one. Running time: 2w = 2(k+t) ≤ 2k+k ≤ 4k Overall Running Time… Feedback Vertex Set Question
  • 52.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph acyclic? Either way, w drops by at least one. Running time: 2w = 2(k+t) ≤ 2k+k ≤ 4k Overall Running Time… k i=1 k k k 4 =5 i Feedback Vertex Set Question
  • 53.
    Input A tournament Ton n vertices and an integer k. Is there a subset of k arcs that can be reversed to make the tournament acyclic? This implies an O(5k) algorithm for FVS. Feedback Vertex Set Question
  • 54.
    Input A tournament Ton n vertices and an integer k. Is there a subset of k arcs that can be reversed to make the tournament acyclic? This implies an O(5k) algorithm for FVS. Feedback Vertex Set Question
  • 55.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question A Odd Cycle Transversal of size (k+1). Let us “guess” how a OCT of size at most k interacts with this one. Odd Cycle Transversal (OCT)
  • 56.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
  • 57.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
  • 58.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
  • 59.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
  • 60.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
  • 61.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
  • 62.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
  • 63.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
  • 64.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
  • 65.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Odd Cycle Transversal (OCT)
  • 66.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? Question Can we remove at most b vertices so that the resulting graph has a bipartition extending this pre-coloring? Odd Cycle Transversal (OCT)
  • 67.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? GREEN Question YELLOW Can we remove at most b vertices so that the resulting graph has a bipartition extending this pre-coloring? Odd Cycle Transversal (OCT)
  • 68.
    Input A graph onn vertices, m edges and an integer k. Is there a subset of at most k vertices whose removal makes the graph bipartite? GREEN Question YELLOW Can we remove at most b vertices so that the resulting graph has a bipartition extending this pre-coloring? Odd Cycle Transversal (OCT)
  • 69.
    Input A tournament Ton n vertices and an integer k. Is there a subset of k arcs that can be reversed to make the tournament acyclic? Question This implies an O*(3k) algorithm for OCT. Odd Cycle Transversal (OCT)
  • 70.
    Input A tournament Ton n vertices and an integer k. Is there a subset of k arcs that can be reversed to make the tournament acyclic? Question This implies an O*(3k) algorithm for OCT. Odd Cycle Transversal (OCT)
  • 71.
  • 73.
    Show that the problemhas a hereditary property Solve the “compression” question, or even a “Disjoint” version.
  • 74.
    Show that the problemhas a hereditary property Solve the “compression” question, or even a “Disjoint” version. Is a framework that can be setup quite easily.
  • 75.
    Show that the problemhas a hereditary property Solve the “compression” question, or even a “Disjoint” version. Is a framework that can be setup quite easily. The compression algorithm can be fairly non-trivial.