Subexponential parameterized complexity of completion problems Survey of the upper bounds Marcin Pilipczuk University of Warsaw 4th Nov 2015 Marcin Pilipczuk Subexponential parameterized complexity of completion problems 1/22 Theory: result should have some property. Modification: error in our measurments. A few errors ) #modifications as a parameter. Graph modification problems An input graph: a result of an experiment. Marcin Pilipczuk Subexponential parameterized complexity of completion problems 2/22 Modification: error in our measurments. A few errors ) #modifications as a parameter. Graph modification problems An input graph: a result of an experiment. Theory: result should have some property. Marcin Pilipczuk Subexponential parameterized complexity of completion problems 2/22 A few errors ) #modifications as a parameter. Graph modification problems An input graph: a result of an experiment. Theory: result should have some property. Modification: error in our measurments. Marcin Pilipczuk Subexponential parameterized complexity of completion problems 2/22 Graph modification problems An input graph: a result of an experiment. Theory: result should have some property. Modification: error in our measurments. A few errors ) #modifications as a parameter. Marcin Pilipczuk Subexponential parameterized complexity of completion problems 2/22 G Completion F-Completion obtain a graph 2 G kill all induced subgraphs 2 F Side note: no known P vs NP dichotomy for completion problems. Completions completion = only edge insertions are allowed Marcin Pilipczuk Subexponential parameterized complexity of completion problems 3/22 F-Completion kill all induced subgraphs 2 F Side note: no known P vs NP dichotomy for completion problems. Completions completion = only edge insertions are allowed G Completion obtain a graph 2 G Marcin Pilipczuk Subexponential parameterized complexity of completion problems 3/22 Side note: no known P vs NP dichotomy for completion problems. Completions completion = only edge insertions are allowed G Completion F-Completion obtain a graph 2 G kill all induced subgraphs 2 F Marcin Pilipczuk Subexponential parameterized complexity of completion problems 3/22 Completions completion = only edge insertions are allowed G Completion F-Completion obtain a graph 2 G kill all induced subgraphs 2 F Side note: no known P vs NP dichotomy for completion problems. Marcin Pilipczuk Subexponential parameterized complexity of completion problems 3/22 Interesting classes for completion threshold vertex cover ⊆ ≥ proper trivially perfect treedepth bandwidth interval ≥ ≤ ⊆ ⊇ pathwidth interval ≥ ⊆ chordal treewidth measure(G) ' minf!(H): H 2 G and H is a completion of Gg: Marcin Pilipczuk Subexponential parameterized complexity of completion problems 4/22 Disclaimer In this talk we mostly focus on f (k) in the FPT running time f (k)nO(1). We denote O∗(f (k)) = f (k)nO(1). Marcin Pilipczuk Subexponential parameterized complexity of completion problems 5/22 Theorem (Cai, IPL'96) A simple branching strategy )O∗(ck ) FPT algorithm Works also for: Co-cluster, Cograph, Threshold, Pseudosplit,... Finite set F Split Completion f2K2; C4; C5g-Completion Marcin Pilipczuk Subexponential parameterized complexity of completion problems 6/22 Works also for: Co-cluster, Cograph, Threshold, Pseudosplit,... Finite set F Split Completion f2K2; C4; C5g-Completion Theorem (Cai, IPL'96) A simple branching strategy )O∗(ck ) FPT algorithm Marcin Pilipczuk Subexponential parameterized complexity of completion problems 6/22 Finite set F Split Completion f2K2; C4; C5g-Completion Theorem (Cai, IPL'96) A simple branching strategy )O∗(ck ) FPT algorithm Works also for: Co-cluster, Cograph, Threshold, Pseudosplit,... Marcin Pilipczuk Subexponential parameterized complexity of completion problems 6/22 Theorem (Kaplan, Shamir, Tarjan, SICOMP'99) Large hole ) many options but big cost (Catalan number C`−2 for ` − 3 edges) )O∗(4k ) branching. Gives also O∗(ck ) FPT algorithm for Proper Interval Completion, as Chordal −! Proper Interval means killing fS3; claw; netg. Chordal completion Chordal Completion fC4; C5; C6;:::g-Completion Marcin Pilipczuk Subexponential parameterized complexity of completion problems 7/22 Theorem (Kaplan, Shamir, Tarjan, SICOMP'99) Large hole ) many options but big cost (Catalan number C`−2 for ` − 3 edges) )O∗(4k ) branching. Gives also O∗(ck ) FPT algorithm for Proper Interval Completion, as Chordal −! Proper Interval means killing fS3; claw; netg. Chordal completion Chordal Completion fC4; C5; C6;:::g-Completion Marcin Pilipczuk Subexponential parameterized complexity of completion problems 7/22 Theorem (Kaplan, Shamir, Tarjan, SICOMP'99) Large hole ) many options but big cost (Catalan number C`−2 for ` − 3 edges) )O∗(4k ) branching. Gives also O∗(ck ) FPT algorithm for Proper Interval Completion, as Chordal −! Proper Interval means killing fS3; claw; netg. Chordal completion Chordal Completion fC4; C5; C6;:::g-Completion Marcin Pilipczuk Subexponential parameterized complexity of completion problems 7/22 Theorem (Kaplan, Shamir, Tarjan, SICOMP'99) Large hole ) many options but big cost (Catalan number C`−2 for ` − 3 edges) )O∗(4k ) branching. Gives also O∗(ck ) FPT algorithm for Proper Interval Completion, as Chordal −! Proper Interval means killing fS3; claw; netg. Chordal completion Chordal Completion fC4; C5; C6;:::g-Completion Marcin Pilipczuk Subexponential parameterized complexity of completion problems 7/22 Gives also O∗(ck ) FPT algorithm for Proper Interval Completion, as Chordal −! Proper Interval means killing fS3; claw; netg. Chordal completion Chordal Completion fC4; C5; C6;:::g-Completion Theorem (Kaplan, Shamir, Tarjan, SICOMP'99) Large hole ) many options but big cost (Catalan number C`−2 for ` − 3 edges) )O∗(4k ) branching. Marcin Pilipczuk Subexponential parameterized complexity of completion problems 7/22 Chordal completion Chordal Completion fC4; C5; C6;:::g-Completion Theorem (Kaplan, Shamir, Tarjan, SICOMP'99) Large hole ) many options but big cost (Catalan number C`−2 for ` − 3 edges) )O∗(4k ) branching. Gives also O∗(ck ) FPT algorithm for Proper Interval Completion, as Chordal −! Proper Interval means killing fS3; claw; netg. Marcin Pilipczuk Subexponential parameterized complexity of completion problems 7/22 Long-standing open problem for more than a decade. Theorem (Villanger, Heggernes, Paul, Telle, SICOMP'09) Branching still doable! An O∗(k2k ) FPT algorithm. Theorem (Cao, SODA'16) Can be solved in O(ck (n + m)) time. Interval completion Interval Completion fholes; ATsg-Completion Marcin Pilipczuk Subexponential parameterized complexity of completion problems 8/22 Long-standing open problem for more than a decade. Theorem (Villanger, Heggernes, Paul, Telle, SICOMP'09) Branching still doable! An O∗(k2k ) FPT algorithm. Theorem (Cao, SODA'16) Can be solved in O(ck (n + m)) time. Interval completion Interval Completion fholes; ATsg-Completion Marcin Pilipczuk Subexponential parameterized complexity of completion problems 8/22 Theorem (Villanger, Heggernes, Paul, Telle, SICOMP'09) Branching still doable! An O∗(k2k ) FPT algorithm. Theorem (Cao, SODA'16) Can be solved in O(ck (n + m)) time. Interval completion Interval Completion fholes; ATsg-Completion Long-standing open problem for more than a decade. Marcin Pilipczuk Subexponential parameterized complexity of completion problems 8/22 Interval completion Interval Completion fholes; ATsg-Completion Long-standing open problem for more than a decade. Theorem (Villanger, Heggernes, Paul, Telle, SICOMP'09) Branching still doable! An O∗(k2k ) FPT algorithm. Theorem (Cao, SODA'16) Can be solved in O(ck (n + m)) time. Marcin Pilipczuk Subexponential parameterized complexity of completion problems 8/22 Theorem (Guo, ISAAC'07) Chain, Split, Threshold and Trivially Perfect Completion admit polynomial kernels. Theorem (Kratsch, Wahlstr¨om,IWPEC'09) There is a finite F such that F-Completion does not admit a polynomial kernel unless NP ⊆ coNP=poly. Theorem (Guillemot, Havet, Paul, Perez, Algorithmica'13) Cograph Completion admits a polynomial kernel. Theorem (Bessy, Perez, Inf. Comp.'13) Proper Interval Completion admits a polynomial kernel. Polynomial kernels Theorem (Kaplan, Shamir, Tarjan, SICOMP'99) Chordal Completion admits a polynomial kernel. Marcin Pilipczuk Subexponential parameterized complexity of completion problems 9/22 Theorem (Kratsch, Wahlstr¨om,IWPEC'09) There is a finite F such that F-Completion does not admit a polynomial kernel unless NP ⊆ coNP=poly. Theorem (Guillemot, Havet, Paul, Perez, Algorithmica'13) Cograph Completion admits a polynomial kernel. Theorem (Bessy, Perez, Inf. Comp.'13) Proper Interval Completion admits a polynomial kernel. Polynomial kernels Theorem (Kaplan, Shamir, Tarjan, SICOMP'99) Chordal Completion admits a polynomial kernel. Theorem (Guo, ISAAC'07) Chain, Split, Threshold and Trivially Perfect Completion admit polynomial kernels. Marcin Pilipczuk Subexponential parameterized complexity of completion problems 9/22 Theorem (Guillemot, Havet, Paul, Perez, Algorithmica'13) Cograph Completion admits a polynomial kernel. Theorem (Bessy, Perez, Inf. Comp.'13) Proper Interval Completion admits a polynomial kernel. Polynomial kernels Theorem (Kaplan, Shamir, Tarjan, SICOMP'99) Chordal Completion admits a polynomial kernel. Theorem (Guo, ISAAC'07) Chain, Split, Threshold and Trivially Perfect Completion admit polynomial kernels. Theorem (Kratsch, Wahlstr¨om,IWPEC'09) There is a finite F such that F-Completion does not admit a polynomial kernel unless