Parameterized domination in circle graphs Ignasi Sau CNRS, LIRMM, Montpellier, France Joint work with: Nicolas Bousquet, Daniel Gonc¸alves, George B. Mertzios, Christophe Paul, Stephan´ Thomasse´ [an extended abstract has been presented at WG’12] Algorithms Research Group, University of Bergen October 5, 2012 1/28 Outline 1 Motivation 2 Hardness results Independent dominating set Acyclic dominating set Dominating set Tree dominating set 3 Sketch of the algorithms 4 Conclusions 2/28 Next section is... 1 Motivation 2 Hardness results Independent dominating set Acyclic dominating set Dominating set Tree dominating set 3 Sketch of the algorithms 4 Conclusions 3/28 Circle graphs and domination G 6 2 5 1 4 3 4 7 8 3 7 1 8 5 2 6 Circle graph: intersection graph of chords in a circle. 4/28 Circle graphs and domination G 6 2 5 1 4 3 4 7 8 3 7 1 8 5 2 6 S DOMINATING SET: S ⊆ V (G) s.t. each v 2 V (G) n S has a neighbor in S. 4/28 Circle graphs and domination G 6 2 5 1 4 3 4 7 8 3 7 1 8 5 2 6 G[S] We will impose extra conditions that G[S] must satisfy. 4/28 Circle graphs and domination G 6 2 5 1 4 3 4 7 8 3 7 1 8 5 2 6 G[S] CONNECTED DOMINATING SET: G[S] is connected. 4/28 Circle graphs and domination G 6 2 5 1 4 3 4 7 8 3 7 1 8 5 2 6 G[S] TOTAL DOMINATING SET: G[S] has no isolated vertices. 4/28 Circle graphs and domination G 6 2 5 1 4 3 4 7 8 3 7 1 8 5 2 6 G[S] ACYCLIC DOMINATING SET: G[S] has no cycles. 4/28 Circle graphs and domination G 6 2 5 1 4 3 4 7 8 3 7 1 8 5 2 6 G[S] ACYCLIC DOMINATING SET: G[S] has no cycles. 4/28 Circle graphs and domination G 6 2 5 1 4 3 4 7 8 3 7 1 8 5 2 6 G[S] INDEPENDENT DOMINATING SET: G[S] has no edges. 4/28 Circle graphs and domination G 6 2 5 1 4 3 4 7 8 3 7 1 8 5 2 6 G[S] INDEPENDENT DOMINATING SET: G[S] has no edges. 4/28 The complexity of some problems in circle graphs 2 I Circle graphs can be recognized in O(n ) time. [Spinrad. 1994] I MAXIMUM CLIQUE and MAXIMUM INDEPENDENT SET can be 3 solved in O(n ) time. [Gavril. 1973] 3 I TREEWIDTH can be solved in O(n ) time. [Kloks. 1996] I 3-COLORABILITY can be solved in O(n log n) time. [Unger. 1988] I k-COLORABILITY for k ≥ 4 is NP-complete. [Unger. 1992] 5/28 The complexity of some problems in circle graphs 2 I Circle graphs can be recognized in O(n ) time. [Spinrad. 1994] I MAXIMUM CLIQUE and MAXIMUM INDEPENDENT SET can be 3 solved in O(n ) time. [Gavril. 1973] 3 I TREEWIDTH can be solved in O(n ) time. [Kloks. 1996] I 3-COLORABILITY can be solved in O(n log n) time. [Unger. 1988] I k-COLORABILITY for k ≥ 4 is NP-complete. [Unger. 1992] 5/28 The complexity of some problems in circle graphs 2 I Circle graphs can be recognized in O(n ) time. [Spinrad. 1994] I MAXIMUM CLIQUE and MAXIMUM INDEPENDENT SET can be 3 solved in O(n ) time. [Gavril. 1973] 3 I TREEWIDTH can be solved in O(n ) time. [Kloks. 1996] I 3-COLORABILITY can be solved in O(n log n) time. [Unger. 1988] I k-COLORABILITY for k ≥ 4 is NP-complete. [Unger. 1992] 5/28 Domination in circle graphs I DOMINATING SET,CONNECTED DOMINATING SET, and TOTAL DOMINATING SET are NP-complete. [Keil. 1993] I INDEPENDENT DOMINATING SET is NP-complete. [Damian, Pemmaraju. 1999] I ACYCLIC DOMINATING SET is in P in interval and proper circular-arc graphs. [Hedetniemi, Hedetniemi, Rall. 2000] I ACYCLIC DOMINATING SET is in P in bipartite permutation graphs. [Xu, Kang, Shan. 2006] What about the parameterized complexity of these domination problems, when parameterized by the solution size? 6/28 Domination in circle graphs I DOMINATING SET,CONNECTED DOMINATING SET, and TOTAL DOMINATING SET are NP-complete. [Keil. 1993] I INDEPENDENT DOMINATING SET is NP-complete. [Damian, Pemmaraju. 1999] I ACYCLIC DOMINATING SET is in P in interval and proper circular-arc graphs. [Hedetniemi, Hedetniemi, Rall. 2000] I ACYCLIC DOMINATING SET is in P in bipartite permutation graphs. [Xu, Kang, Shan. 2006] What about the parameterized complexity of these domination problems, when parameterized by the solution size? 6/28 Domination in circle graphs I DOMINATING SET,CONNECTED DOMINATING SET, and TOTAL DOMINATING SET are NP-complete. [Keil. 1993] I INDEPENDENT DOMINATING SET is NP-complete. [Damian, Pemmaraju. 1999] I ACYCLIC DOMINATING SET is in P in interval and proper circular-arc graphs. [Hedetniemi, Hedetniemi, Rall. 2000] I ACYCLIC DOMINATING SET is in P in bipartite permutation graphs. [Xu, Kang, Shan. 2006] What about the parameterized complexity of these domination problems, when parameterized by the solution size? 6/28 Our results (in circle graphs) 1 DOMINATING SET, CONNECTED DOMINATING SET, TOTAL DOMINATING SET, INDEPENDENT DOMINATING SET, ACYCLIC DOMINATING SET are W [1]-hard, parameterized by the size of the solution. 2 Whereas bothC ONNECTED andA CYCLIC DOMINATING SET are W [1]-hard,C ONNECTED ACYCLIC (i.e.,T REE) DOM.SET is inP. 3 If T is a given fixed tree, the problem of deciding whether a circle graph has a dominating set isomorphic to T is I NP-complete, when T is part of the input. I FPT, when parameterized by jV (T )j. 7/28 Our results (in circle graphs) 1 DOMINATING SET, CONNECTED DOMINATING SET, TOTAL DOMINATING SET, INDEPENDENT DOMINATING SET, ACYCLIC DOMINATING SET are W [1]-hard, parameterized by the size of the solution. 2 Whereas bothC ONNECTED andA CYCLIC DOMINATING SET are W [1]-hard, CONNECTED ACYCLIC (i.e.,T REE) DOM.SET is inP. 3 If T is a given fixed tree, the problem of deciding whether a circle graph has a dominating set isomorphic to T is I NP-complete, when T is part of the input. I FPT, when parameterized by jV (T )j. 7/28 Our results (in circle graphs) 1 DOMINATING SET, CONNECTED DOMINATING SET, TOTAL DOMINATING SET, INDEPENDENT DOMINATING SET, ACYCLIC DOMINATING SET are W [1]-hard, parameterized by the size of the solution. 2 Whereas bothC ONNECTED andA CYCLIC DOMINATING SET are W [1]-hard, CONNECTED ACYCLIC (i.e.,T REE) DOM.SET is inP. 3 If T is a given fixed tree, the problem of deciding whether a circle graph has a dominating set isomorphic to T is I NP-complete, when T is part of the input. I FPT, when parameterized by jV (T )j. 7/28 Our results (in circle graphs) 1 DOMINATING SET, CONNECTED DOMINATING SET, TOTAL DOMINATING SET, INDEPENDENT DOMINATING SET, ACYCLIC DOMINATING SET are W [1]-hard, parameterized by the size of the solution. 2 Whereas bothC ONNECTED andA CYCLIC DOMINATING SET are W [1]-hard, CONNECTED ACYCLIC (i.e.,T REE) DOM.SET is inP. 3 If T is a given fixed tree, the problem of deciding whether a circle graph has a dominating set isomorphic to T is I NP-complete, when T is part of the input. I FPT, when parameterized by jV (T )j. 7/28 Our results (in circle graphs) 1 DOMINATING SET, CONNECTED DOMINATING SET, TOTAL DOMINATING SET, INDEPENDENT DOMINATING SET, ACYCLIC DOMINATING SET are W [1]-hard, parameterized by the size of the solution. 2 Whereas bothC ONNECTED andA CYCLIC DOMINATING SET are W [1]-hard, CONNECTED ACYCLIC (i.e.,T REE) DOM.SET is inP. 3 If T is a given fixed tree, the problem of deciding whether a circle graph has a dominating set isomorphic to T is I NP-complete, when T is part of the input. I FPT, when parameterized by jV (T )j. 7/28 Our results (in circle graphs) 1 DOMINATING SET, CONNECTED DOMINATING SET, TOTAL DOMINATING SET, INDEPENDENT DOMINATING SET, ACYCLIC DOMINATING SET are W [1]-hard, parameterized by the size of the solution. 2 Whereas bothC ONNECTED andA CYCLIC DOMINATING SET are W [1]-hard, CONNECTED ACYCLIC (i.e.,T REE) DOM.SET is inP. 3 If T is a given fixed tree, the problem of deciding whether a circle graph has a dominating set isomorphic to T is I NP-complete, when T is part of the input. I FPT, when parameterized by jV (T )j. 7/28 Parameterized complexity in one slide I Idea: given an NP-hard problem, fix one parameter of the input to see if the problem gets more “tractable”. Example: the size of a VERTEX COVER. I Given a (NP-hard) problem with input of size n and a parameter k, a fixed-parameter tractable(FPT) algorithm runs in f (k) · nO(1); for some function f . Examples: k-VERTEX COVER, k-LONGEST PATH. I Barometer of intractability: FPT ⊆ W [1] ⊆ W [2] ⊆ W [3] ⊆ · · · ⊆ XP I The higher a problem is located in the W -hierarchy, the more unlikely it is to be in FPT. 8/28 Next section is... 1 Motivation 2 Hardness results Independent dominating set Acyclic dominating set Dominating set Tree dominating set 3 Sketch of the algorithms 4 Conclusions 9/28 Multicolored Clique k-COLORED CLIQUE espai Instance: A graph G = (V ; E) and a coloring of V using k colors.