A single-exponential FPT algorithm for Distance-Hereditary Vertex Deletion
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1 A single-exponential FPT algorithm for Distance-Hereditary Vertex Deletion O-joung Kwon Institute for Computer Science and Control, Hungarian Academy of Sciences in Budapest, Hungary Joint work with Eduard Eiben (TU Wien, Austria) Robert Ganian (TU Wien, Austria) Seminar at MTA SZTAKI 5th, May, / 24
2 Distance-hereditary graphs A graph G is distance-hereditary if for every connected induced subgraph H of G and u, v P V phq, the distance between u and v in H is the same as the distance in G. (First introduced by Howorka, 77) 2 / 24
3 Distance-hereditary graphs A graph G is distance-hereditary if for every connected induced subgraph H of G and u, v P V phq, the distance between u and v in H is the same as the distance in G. (First introduced by Howorka, 77) Equivalent characterizations (Bandelt, Mulder, 82): Every induced path is a shortest path. Every cycle of length at least 5 contains a pair of crossing chords. Graphs can be constructed from a vertex by a sequence of adding twins or adding leaf vertices. Having no obstructions : Induced cycles of length at least 5 + house gem domino 2 / 24
4 Distance-hereditary graphs A graph G is distance-hereditary if for every connected induced subgraph H of G and u, v P V phq, the distance between u and v in H is the same as the distance in G. (First introduced by Howorka, 77) Equivalent characterizations (Bandelt, Mulder, 82): Every induced path is a shortest path. Every cycle of length at least 5 contains a pair of crossing chords. Graphs can be constructed from a vertex by a sequence of adding twins or adding leaf vertices. Having no obstructions : Induced cycles of length at least 5 + house gem domino Rank-width ď 1 (Oum, 05). 2 / 24
5 Distance-Hereditary Vertex Deletion Input : A graph G, an integer k Parameter : k Question : D? S Ď V pgq with S ď k such that G S is distance-hereditary? This can be seen as a counterpart of Feedback Vertex Set. Rank-width w Vertex Deletion Input : A graph G, an integer k Parameter : k Question : D? S Ď V pgq with S ď k such that G S has rank-width at most w? 3 / 24
6 Distance-Hereditary Vertex Deletion Input : A graph G, an integer k Parameter : k Question : D? S Ď V pgq with S ď k such that G S is distance-hereditary? This can be seen as a counterpart of Feedback Vertex Set. Rank-width w Vertex Deletion Input : A graph G, an integer k Parameter : k Question : D? S Ď V pgq with S ď k such that G S has rank-width at most w? So, what is rank-width? 3 / 24
7 What is rank-width? A rank-decomposition pt, Lq of G consists of a subcubic tree T, and a bijective function L from V pgq to the leaves of T. a 2 a 1 a 5 a 4 a 3 a 7 a 6 4 / 24
8 What is rank-width? A rank-decomposition pt, Lq of G consists of a subcubic tree T, and a bijective function L from V pgq to the leaves of T. a 2 a 1 a 5 a 4 a 3 a 7 a 6 a 4 a 5 a 3 a 6 a 1 a 2 a 7 4 / 24
9 What is rank-width? A rank-decomposition pt, Lq of G consists of a subcubic tree T, and a bijective function L from V pgq to the leaves of T. a 2 a 1 a 5 a 4 a 3 a 7 a 6 a 4 a 5 a 3 a 6 e a 1 a 2 a 7 Width of an edge of T : the rank of the matrix with the partition induced by the edge. a 1 a 2 a 3 a Width of e rank a a =3 a / 24
10 What is rank-width? A rank-decomposition pt, Lq of G consists of a subcubic tree T, and a bijective function L from V pgq to the leaves of T. a 2 a 1 a 5 a 4 a 3 a 7 a 6 a 4 a 5 a 3 a 6 e a 1 a 2 a 7 Width of an edge of T : the rank of the matrix with the partition induced by the edge. a 1 a 2 a 3 a Width of e rank a a =3 a / 24
11 a 2 a 1 a 4 a 3 a 5 a 7 a 6 a 1 a 2 a5 a a 3a4 6 e a 7 Width of pt, Lq: max width over all edges in T Rank-width of G: min width over all rank-decompositions of G (Oum and Seymour 04; introduced for approximating clique-width) 5 / 24
12 a 2 a 1 a 4 a 3 a 5 a 7 a 6 a 1 a 2 a5 a a 3a4 6 e a 7 Width of pt, Lq: max width over all edges in T Rank-width of G: min width over all rank-decompositions of G (Oum and Seymour 04; introduced for approximating clique-width) compare with clique-width : rw ď cw ď 2 rw 1 ` 1. Clique-width : easier to design a dynamic programming algorithm, no direct FPT approximation algorithm, decomposition is always rooted. Rank-width : (3k+1)-FPT approximation algorithm, decomposition is not rooted, fit with vertex-minor (every bipartite graph with large rank-width contain a grid-like vertex-minor...) 5 / 24
13 Tree (tree-width ď 1) Distance-hereditary graph (rank-width ď 1) 6 / 24
14 Tree (tree-width ď 1) Distance-hereditary graph (rank-width ď 1) Graphs of bounded rank-width contain dense graphs but also extend useful properties of graphs of bounded tree-width : Courcelle, Makowsky, Rotics, 00 Every MSO 1 properties can be checked in cubic time on graphs of bounded rank-width. 6 / 24
15 Distance to graphs of bounded tree-width graph class C problem Edgeless graphs Vertex Cover FPT O p2 k q Kernel 2k vertices Forests Feedback Vertex Set (graphs of tree-width ď 1) FPT O p3.618 k q [Kociumaka et al. 14] Kernel 4k 2 vertices [Thomassé 10] Series-parallel graphs K 4 -minor Cover (graphs of tree-width ď 2) FPT O pc k q [Kim et al. 15] Graphs of tree-width ď w Tree-width w-vertex Deletion FPT O pc k q (Non-uniform) Polynomial kernel [Fomin et al.12 /Kim et al. 13] 7 / 24
16 Distance to graphs of bounded rank-width? graph class C problem Disjoint union of Cluster Vertex Deletion Complete graphs easy FPT O p3 k q (obstruction is P 3 ) Block graphs Block Graph Vertex Deletion Thread graphs (linear rank-width ď 1) Linear rank-width-1 Vertex Deletion Distance-hereditary graphs (graphs of rank-width ď 1) Graphs of rank-width ď w Distance-hereditary Vertex Deletion Rank-width w-vertex Deletion 8 / 24
17 Distance to graphs of bounded rank-width? graph class C problem Disjoint union of Cluster Vertex Deletion Complete graphs easy FPT O p3 k q (obstruction is P 3 ) Block graphs Block Graph Vertex Deletion FPT O p4 k q Kernel Opk 4 q vertices [Kim, K 15 / improved by Agrawal et al. 16] Thread graphs Linear rank-width-1 Vertex Deletion (linear rank-width ď 1) Distance-hereditary graphs (graphs of rank-width ď 1) Graphs of rank-width ď w Distance-hereditary Vertex Deletion Rank-width w-vertex Deletion 8 / 24
18 Distance to graphs of bounded rank-width? graph class C problem Disjoint union of Cluster Vertex Deletion Complete graphs easy FPT O p3 k q (obstruction is P 3 ) Block graphs Block Graph Vertex Deletion FPT O p4 k q Kernel Opk 4 q vertices [Kim, K 15 / improved by Agrawal et al. 16] Thread graphs Linear rank-width-1 Vertex Deletion (linear rank-width ď 1) FPT O p8 k q Kernel Opk 33 q vertices [Kanté, Kim, K, Paul 15] Distance-hereditary graphs Distance-hereditary Vertex Deletion (graphs of rank-width ď 1) Graphs of rank-width ď w Rank-width w-vertex Deletion 8 / 24
19 Distance to graphs of bounded rank-width? graph class C problem Disjoint union of Cluster Vertex Deletion Complete graphs easy FPT O p3 k q (obstruction is P 3 ) Block graphs Block Graph Vertex Deletion FPT O p4 k q Kernel Opk 4 q vertices [Kim, K 15 / improved by Agrawal et al. 16] Thread graphs Linear rank-width-1 Vertex Deletion (linear rank-width ď 1) FPT O p8 k q Kernel Opk 33 q vertices [Kanté, Kim, K, Paul 15] Distance-hereditary graphs Distance-hereditary Vertex Deletion (graphs of rank-width ď 1) FPT O p2 Opk log kq q [Kim, K, manuscript] Graphs of rank-width ď w Rank-width w-vertex Deletion FPT by meta-theorem OPEN: Rank-width w-vertex Deletion can be solved in O pc k q-time for some constant c? 8 / 24
20 Eiben, Ganian, K, 16 Distance-Hereditary Vertex Deletion can be solved in time 37 k n Op1q. 9 / 24
21 Eiben, Ganian, K, 16 Distance-Hereditary Vertex Deletion can be solved in time 37 k n Op1q. Recall : having no obstructions : Induced cycles of length at least 5 + house gem domino 9 / 24
22 Eiben, Ganian, K, 16 Distance-Hereditary Vertex Deletion can be solved in time 37 k n Op1q. Recall : having no obstructions : Induced cycles of length at least 5 + house gem domino Hitting induced cycles of length at least 5 is W[2]-hard. (Heggernes et al. 13) 9 / 24
23 Eiben, Ganian, K, 16 Distance-Hereditary Vertex Deletion can be solved in time 37 k n Op1q. Recall : having no obstructions : Induced cycles of length at least 5 + house gem domino Hitting induced cycles of length at least 5 is W[2]-hard. (Heggernes et al. 13) Important property Let G be a graph obtained from an induced path of length at least 3 by adding a vertex v adjacent to its end vertices. Then G has a DH obstruction containing v. 9 / 24
24 Eiben, Ganian, K, 16 Distance-Hereditary Vertex Deletion can be solved in time 37 k n Op1q. 10 / 24
25 Eiben, Ganian, K, 16 Distance-Hereditary Vertex Deletion can be solved in time 37 k n Op1q. (1) Iterative Compression. Reduce to Disjoint Distance-Hereditary Vertex Deletion. Disjoint Distance-Hereditary Vertex Deletion Input : A graph G, an integer k, and S Ď V pgq with S ď k ` 1 such that G S is distance-hereditary. Parameter : k Question : Is there Q Ď V pgqzs with Q ď k such that G Q is distancehereditary? 10 / 24
26 Eiben, Ganian, K, 16 Distance-Hereditary Vertex Deletion can be solved in time 37 k n Op1q. (1) Iterative Compression. Reduce to Disjoint Distance-Hereditary Vertex Deletion. Disjoint Distance-Hereditary Vertex Deletion Input : A graph G, an integer k, and S Ď V pgq with S ď k ` 1 such that G S is distance-hereditary. Parameter : k Question : Is there Q Ď V pgqzs with Q ď k such that G Q is distancehereditary? (2) Branching Rules. Apply two branching rules Ñ G has no small obstructions and furthermore that certain connectivity conditions hold on GrSs. (3) Simplification of Split Decomposition. We compute the split decomposition of G S and exploit the properties of our instance G guaranteed by branching to prune the decomposition. 10 / 24
27 Split Decompositions B 3 G B 1 B 2 B 4 B 5 Two types of vertices : original vertices, marked vertices. Two types of edges : marked edges, unmarked edges. Two original vertices are adjacent if and only if they are linked by an alternating path. (unmarked marked unmarked unmarked) 11 / 24
28 Split Decompositions B 3 G B 1 B 2 B 4 B 5 Cunningham, 82 This decomposition is unique if every bag is a complete graph or a star or a prime graph, we cannot obtain a decomposition with same property by reversing a marked edge. It is called the canonical split decomposition of a graph. 12 / 24
29 Split Decompositions B 3 G B 1 B 2 B 4 B 5 Cunningham, 82 This decomposition is unique if every bag is a complete graph or a star or a prime graph, we cannot obtain a decomposition with same property by reversing a marked edge. It is called the canonical split decomposition of a graph. Bouchet, 88 A graph is distance-hereditary if and only if every bag of its canonical split decomposition is either a star or a complete graph. 12 / 24
30 Now, we are ready.. 13 / 24
31 Now, we are ready.. There is a given solution S and the canonical split decomposition of G S. S 13 / 24
32 Now, we are ready.. There is a given solution S and the canonical split decomposition of G S. S Branching Rule Ď V pg Sq with X ď 5, if GrS Y Xs is not distance-hereditary, then we remove one of the vertices in X. Branching Rule Ď V pg Sq with X ď 5 such that GrXs is connected and adding X to S decreases the number of components in GrSs, then we either remove one of the vertices in X, or put all of them into S. 13 / 24
33 And nine reduction rules.. what? 14 / 24
34 And nine reduction rules.. what? Main ingredient : bad vertex for an induced cycle of length at least 5. For every induced cycle C of length at least 5 and v P V pcq, a vertex w P S is a bad vertex for C and v if it is adjacent to the two neighbors of v in C. 14 / 24
35 And nine reduction rules.. what? Main ingredient : bad vertex for an induced cycle of length at least 5. For every induced cycle C of length at least 5 and v P V pcq, a vertex w P S is a bad vertex for C and v if it is adjacent to the two neighbors of v in C. It is easy to see that v P V pg Sq can be safely removed if for every induced cycle of length at least 5 containing v, there is a bad vertex for C and v. Is it easy to find a bad vertex? 14 / 24
36 And nine reduction rules.. what? Main ingredient : bad vertex for an induced cycle of length at least 5. For every induced cycle C of length at least 5 and v P V pcq, a vertex w P S is a bad vertex for C and v if it is adjacent to the two neighbors of v in C. It is easy to see that v P V pg Sq can be safely removed if for every induced cycle of length at least 5 containing v, there is a bad vertex for C and v. Is it easy to find a bad vertex? S 14 / 24
37 Twin Reduction Rule If there exist two non-adjacent twins v, w in G S having at least one common neighbor in G S such that pn G pvq X SqzpN G pwq X Sq H, then we can safely remove v. 15 / 24
38 Twin Reduction Rule If there exist two non-adjacent twins v, w in G S having at least one common neighbor in G S such that pn G pvq X SqzpN G pwq X Sq H, then we can safely remove v. Claim 1. pn G pvq X Sq X pn G pwq X Sq H. v 15 / 24
39 Twin Reduction Rule If there exist two non-adjacent twins v, w in G S having at least one common neighbor in G S such that pn G pvq X SqzpN G pwq X Sq H, then we can safely remove v. Claim 1. pn G pvq X Sq X pn G pwq X Sq H. Claim 2. Let C be an induced cycle of length at least 5 containing v. 3 types: 1) S v S 2) S v V pg Sq 3) V pg Sq v V pg Sq There is a bad vertex for C and v. v v 16 / 24
40 Twin Reduction Rule If there exist two non-adjacent twins v, w in G S having at least one common neighbor in G S such that pn G pvq X SqzpN G pwq X Sq H, then we can safely remove v. Claim 1. pn G pvq X Sq X pn G pwq X Sq H. Claim 2. Let C be an induced cycle of length at least 5 containing v. 3 types: 1) S v S 2) S v V pg Sq 3) V pg Sq v V pg Sq There is a bad vertex for C and v. v v 17 / 24
41 Twin Reduction Rule If there exist two non-adjacent twins v, w in G S having at least one common neighbor in G S such that pn G pvq X SqzpN G pwq X Sq H, then we can safely remove v. Claim 1. pn G pvq X Sq X pn G pwq X Sq H. Claim 2. Let C be an induced cycle of length at least 5 containing v. 3 types: 1) S v S 2) S v V pg Sq 3) V pg Sq v V pg Sq There is a bad vertex for C and v. v v 18 / 24
42 Rule 1. D contains at most one class having neighbors on S, then we remove it. Rule 2. Deleting a vertex of degree 1. Rule 3. Twin Reduction Rule. 19 / 24
43 Rule 1. D contains at most one class having neighbors on S, then we remove it. Rule 2. Deleting a vertex of degree 1. Rule 3. Twin Reduction Rule. Rule 4. B1 B1 B2 B2 19 / 24
44 Rule 1. D contains at most one class having neighbors on S, then we remove it. Rule 2. Deleting a vertex of degree 1. Rule 3. Twin Reduction Rule. Rule 4. B1 B1 Rule 5. B2 B2 B1 B1 B2 B2 19 / 24
45 Rule 1. D contains at most one class having neighbors on S, then we remove it. Rule 2. Deleting a vertex of degree 1. Rule 3. Twin Reduction Rule. Rule 4. B1 B1 Rule 5. B2 B2 B1 B1 Rule 6. Swapping the type of bags. B2 B2 19 / 24
46 Rule 7. C1 C3 C2 g C We obtain a property in a reduced instance: (1) Every non-trivial branch should have a neighbor in S, and (2) if there is a bag branching two branches, then two classes having neighbors in S in these branches have distance at most 4 in G S. 20 / 24
47 Rule 7. C1 C3 C2 g C We obtain a property in a reduced instance: (1) Every non-trivial branch should have a neighbor in S, and (2) if there is a bag branching two branches, then two classes having neighbors in S in these branches have distance at most 4 in G S. We can apply some special rule to remove one of them (an extension of Twin Reduction Rule). 20 / 24
48 Rule 7. C1 C3 C2 g C We obtain a property in a reduced instance: (1) Every non-trivial branch should have a neighbor in S, and (2) if there is a bag branching two branches, then two classes having neighbors in S in these branches have distance at most 4 in G S. We can apply some special rule to remove one of them (an extension of Twin Reduction Rule). Eventually, we will arrive at a trivial instance. 20 / 24
49 Conclusion Distance-Hereditary Vertex Deletion can be solved in time O p37 k q. Can we improve the running time? Does Distance-Hereditary Vertex Deletion admit a polynomial kernel? Twin Reduction Rule is not possible, because we cannot kill the small obstructions by branching. For w ě 2, Rank-width w Vertex Deletion can be solved in time O pc k q for some constant c? 21 / 24
50 Conclusion Distance-Hereditary Vertex Deletion can be solved in time O p37 k q. Can we improve the running time? Does Distance-Hereditary Vertex Deletion admit a polynomial kernel? Twin Reduction Rule is not possible, because we cannot kill the small obstructions by branching. For w ě 2, Rank-width w Vertex Deletion can be solved in time O pc k q for some constant c? Thank for listening! 21 / 24
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