a 9 k kernel for nonseparating independent set in planar
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A 9 k kernel for nonseparating independent set in planar graphs Lukasz Kowalik (speaker) and Marcin Mucha Institute of Informatics, University of Warsaw Jerusalem, 27.06.2012 Lukasz Kowalik (Warsaw) A kernel for nonseparating

  1. A 9 k kernel for nonseparating independent set in planar graphs � Lukasz Kowalik (speaker) and Marcin Mucha Institute of Informatics, University of Warsaw Jerusalem, 27.06.2012 � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 1 / 18

  2. Kernelization (of graph problems) Let ( G , k ) be an instance of a decision problem ( k is a parameter). ( G ′ , k ′ ) Graph G poly-time parameter k | V ( G ′ ) | ≤ f ( k ) Kernel Input instance ( G , k ) is a YES-instance iff ( G ′ , k ′ ) is a YES-instance. k ′ ≤ k , | V ( G ′ ) | ≤ f ( k ). � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 2 / 18

  3. Some examples of kernels General graphs: Vertex Cover 2 k , Feedback Vertex Set O ( k 2 ), Odd Cycle Transversal k O (1) , ... Planar graphs: Dominating Set 67 k , Feedback Vertex Set 112 k , Induced Matching 28 k , Connected Vertex Cover 11 3 k , ... � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 3 / 18

  4. Vertex Cover and Independent Set Let G = ( V , E ) be a graph. � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 4 / 18

  5. Vertex Cover and Independent Set Let G = ( V , E ) be a graph. C is a vertex cover ( ) when every edge has at least one endpoint in C . � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 4 / 18

  6. Vertex Cover and Independent Set Let G = ( V , E ) be a graph. C is a vertex cover ( ) when every edge has at least one endpoint in C . S is an independent set ( ) when every edge has at most one endpoint in S . � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 4 / 18

  7. Vertex Cover and Independent Set Let G = ( V , E ) be a graph. C is a vertex cover ( ) when every edge has at least one endpoint in C . S is an independent set ( ) when every edge has at most one endpoint in S . Observation C is a vertex cover iff V \ C is an independent set. G has a vertex cover of size k iff G has independent set of size | V | − k . � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 4 / 18

  8. Parametric Duality Corollary G has independent set of size k iff G has a vertex cover of size | V | − k . Vertex Cover Instance: Graph G = ( V , E ), k ∈ N Question: Does G contain a vertex cover of size k ? Independent Set Instance: Graph G = ( V , E ), k ∈ N Question: Does G contain an independent set of size k ? We can treat Independent Set as Vertex Cover with | V | − k as a parameter. Then, Vertex Cover is a parametric dual of Independent Set . But a small kernel for one problem does not give a small kernel for another. � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 5 / 18

  9. Connected Vertex Cover & Nonseparating Independent Set Let G = ( V , E ) be a connected graph. � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 6 / 18

  10. Connected Vertex Cover & Nonseparating Independent Set Let G = ( V , E ) be a connected graph. C is a connected vertex cover ( ) when C is a vertex cover and C induces a connected subgraph of G . � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 6 / 18

  11. Connected Vertex Cover & Nonseparating Independent Set Let G = ( V , E ) be a connected graph. C is a connected vertex cover ( ) when C is a vertex cover and C induces a connected subgraph of G . S is a nonseparating independent set ( ) when S is an independent set and G − S is connected. � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 6 / 18

  12. Connected Vertex Cover & Nonseparating Independent Set Let G = ( V , E ) be a connected graph. C is a connected vertex cover ( ) when C is a vertex cover and C induces a connected subgraph of G . S is a nonseparating independent set ( ) when S is an independent set and G − S is connected. Observation C is a connected vertex cover iff V \ C is a nonseparating independent set. G has a connected vertex cover of size k iff G has a nonseparating independent set of size | V | − k . � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 6 / 18

  13. Parametric Duality Connected Vertex Cover (CVC) Instance: Graph G = ( V , E ), k ∈ N Question: Does G contain a vertex cover of size k ? Nonseparating Independent Set (NSIS) Instance: Graph G = ( V , E ), k ∈ N Question: Does G contain an independent set of size k ? Connected Vertex Cover is a parametric dual of Nonseparating Independent Set . � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 7 / 18

  14. Known complexity results for Connected Vertex Cover (CVC) and Nonseparating Independent Set (NSIS) Both problems NP-complete even for planar graphs, in P for graphs of maximum degree 3 (Ueno 1988). CVC: kernelization CVC has no kernel of polynomial size (Dom et al 2009), Planar CVC has a 11 3 k -kernel (Kowalik et al 2011). NSIS: kernelization NSIS is W [1]-hard, so no kernel at all (folklore), Planar NSIS : O ( k )-sized kernel (Fomin et al. 2010) � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 8 / 18

  15. Our results: kernel upper bounds Main Result There is a 9 k -kernel for Planar Nonseparating Independent Set . A Bonus Result (skipped in this presentation) There is a 5 k -kernel for Planar Max Leaf . � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 9 / 18

  16. Our results: kernel lower bounds Theorem (Chen et al. 2007) If a problem admits a kernel of size at most α k , then the dual problem has no kernel of size at most ( α − 1 − ǫ ) k , for any ǫ > 0, unless P=NP. α Two corollaries Planar CVC has no kernel of size at most ( 9 8 − ǫ ) k , unless P=NP, Planar Connected Dominating Set has no kernel of size at most ( 5 4 − ǫ ) k , unless P=NP, � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 10 / 18

  17. A simple 12 k kernel for Planar NSIS For a tree T let L ( T ) denote the set of leaves of T . Maximum Independent Leaf Instance: Graph G = ( V , E ), k ∈ N Question: Is there a spanning tree T such that L ( T ) contains a subset of size k that is independent in G ? Observation Connected graph G has a nonseparating independent set of size k iff G has a spanning tree T such that L ( T ) contains a subset S of size k that is independent in G . � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 11 / 18

  18. A simple 12 k kernel for Planar NSIS For a tree T let L ( T ) denote the set of leaves of T . Maximum Independent Leaf Instance: Graph G = ( V , E ), k ∈ N Question: Is there a spanning tree T such that L ( T ) contains a subset of size k that is independent in G ? Observation Connected graph G has a nonseparating independent set of size k iff G has a spanning tree T such that L ( T ) contains a subset S of size k that is independent in G . � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 11 / 18

  19. A simple 12 k kernel for Planar NSIS For a tree T let L ( T ) denote the set of leaves of T . Maximum Independent Leaf Instance: Graph G = ( V , E ), k ∈ N Question: Is there a spanning tree T such that L ( T ) contains a subset of size k that is independent in G ? Observation Connected graph G has a nonseparating independent set of size k iff G has a spanning tree T such that L ( T ) contains a subset S of size k that is independent in G . � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 11 / 18

  20. A simple 12 k kernel for Planar NSIS For a tree T let L ( T ) denote the set of leaves of T . Maximum Independent Leaf Instance: Graph G = ( V , E ), k ∈ N Question: Is there a spanning tree T such that L ( T ) contains a subset of size k that is independent in G ? Observation Connected graph G has a nonseparating independent set of size k iff G has a spanning tree T such In what follows... that L ( T ) contains a subset S of size k that is independent in G . we focus on Maximum Independent Leaf ! � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 11 / 18

  21. A 12 k -kernel for Planar NSIS for minimum degree 3 � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 12 / 18

  22. A 12 k -kernel for Planar NSIS for minimum degree 3 Theorem (Kleitman, West 1991) Every n -vertex graph of minimum degree 3 has a spanning tree of ≥ n / 4 leaves. = leaves. � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 12 / 18

  23. A 12 k -kernel for Planar NSIS for minimum degree 3 Theorem (Kleitman, West 1991) Every n -vertex graph of minimum degree 3 has a spanning tree of ≥ n / 4 leaves. Observation If G is planar and T is a spanning tree of G then G [ L ( T )] is outerplanar. = leaves. � Lukasz Kowalik (Warsaw) A kernel for nonseparating independent set... Jerusalem, 27.06.2012 12 / 18

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