Tree Decompositions and Tree-Width

Transcription

1 Tree Decompositions and Tree-Width CS 511 Iowa State University December 6, 2010 CS 511 (Iowa State University) Tree Decompositions and Tree-Width December 6, / 15

2 Tree Decompositions Definition A tree decomposition of a graph G = (V, E) consists of a tree T and a subset V t V for every node t T, such that the collection {V t : t T } satisfies: (Node coverage) For every v V, there is some node t in T such that v V t. (Edge coverage) For every e E, there is some node t in T such that V t contains both endpoints of e. (Coherence) Let t 1, t 2, t 3 be three nodes in T such that t 2 lies on the path between t 1 and t 3 in T. Then, if v V belongs to both V t1 and V t3, v must also belong to V t2. CS 511 (Iowa State University) Tree Decompositions and Tree-Width December 6, / 15

3 Author: David Eppstein. Source: Wikipedia. CS 511 (Iowa State University) Tree Decompositions and Tree-Width December 6, / 15

4 Tree-Width Definition The width of tree decomposition (T, {V t : t T }) is width(t, {V t : t T }) = max t T V t 1. Definition The tree-width of G, denoted tw(g), is the minimum width of a tree decomposition of G. CS 511 (Iowa State University) Tree Decompositions and Tree-Width December 6, / 15

5 Complexity of Tree-Width Let TW(k) be the class of graphs G such that tw(g) k. Tree-Width (Decision Version) Input: An undirected graph G and an integer k. Question: Is G TW(k)? Theorem Tree-width (decision version) is NP-complete. CS 511 (Iowa State University) Tree Decompositions and Tree-Width December 6, / 15

6 Complexity of Tree-Width Lemma For every positive integer k, TW(k) is minor closed. Corollary (Tree-width is fixed-parameter tractable) For every fixed k, the problem of determining whether or not G TW(k) can be solved in O(f (k) n O(1) ) time. Corollary follows from Robertson & Seymour s graph minor results. f (k) is superpolynomial, but depends only on k. Running time can be improved to O(n) for each fixed k. Simple O(n) algorithms exist for k 4. CS 511 (Iowa State University) Tree Decompositions and Tree-Width December 6, / 15

7 Notation Let (T, {V t : t T }) be a tree decomposition of G. Then, if T is a subgraph of T, G T denotes the subgraph induced by the set t T V t. CS 511 (Iowa State University) Tree Decompositions and Tree-Width December 6, / 15

8 Theorem (Node Separation Property) Suppose T t has components T 1,..., T d. Then, the subgraphs G T1 V t, G T2 V t,..., G Td V t have no nodes in common, and there are no edges between them. t1 t2 t4 t t3 CS 511 (Iowa State University) Tree Decompositions and Tree-Width December 6, / 15

9 Theorem (Edge Separation Property) Let X and Y be the two components of T after the deletion of edge (x, y). Then, deleting V x V y disconnects G into two subgraphs H X = G X (V x V y ) and H Y = G Y (V x V y ). That is, H X and H Y share no nodes and there is no edge in G with one endpoint in H X and the other in H Y. y x Vx Vy CS 511 (Iowa State University) Tree Decompositions and Tree-Width December 6, / 15

10 Definition A tree decomposition (T, {V t : t T }) of G is nonredundant if there is no edge (x, y) in T such that V x V y. Lemma Any graph has a nonredundant tree decomposition. Lemma Any non-redundant tree decomposition of an n-node graph has at most n pieces. CS 511 (Iowa State University) Tree Decompositions and Tree-Width December 6, / 15

11 Rooted tree decomposition Definition A rooted tree decomposition of G is a tree decomposition (T, {V t : t T }) of G where some node r in T is declared to be the root. Let t be a node in a rooted tree decomposition. Then, T t is the subtree of T rooted at t, G t is the subgraph of G induced by the vertices in x T t V x. CS 511 (Iowa State University) Tree Decompositions and Tree-Width December 6, / 15

12 Subproblems Definition For each node t in a rooted tree decomposition of G and each independent set U V t, opt U (t) is the maximum weight of an independent set S of G t such that S V t = U. CS 511 (Iowa State University) Tree Decompositions and Tree-Width December 6, / 15

13 Optimal Substructure Let t be a node in T with children t 1,..., t d, U be an independent set of V t, S be a maximum independent set in G t subject to S V t = U (i.e., w(s) = opt U (t)), S i be the intersection of S with the nodes of G Ti. Lemma (Optimal Substructure) S i is a maximum-weight independent set of G ti, subject to the constraint that S i V t = U V ti. CS 511 (Iowa State University) Tree Decompositions and Tree-Width December 6, / 15

14 A Recurrence for Maximum Weight Independent Set Theorem (Dynamic Programming Recurrence Relation for MWIS) d opt U (t) = w(u) + max{opt Ui (t i ) w(u i U) : i=1 U i V ti is independent and U i V t = U V ti }. CS 511 (Iowa State University) Tree Decompositions and Tree-Width December 6, / 15

15 Theorem (Running time analysis) Suppose we are given a vertex-weighted graph G TW(k) with n nodes along with a tree decomposition of width k for G. Then, we can find a maximum weight independent set in G in O(4 k+1 kn) time. Proof. Traverse the tree decomposition bottom-up. For a leaf, use exhaustive enumeration O(2 k+1 ) time. For an internal node t, apply the recurrence relation. Enumerate each of the O(2 k+1 ) subsets U of V t. For each child ti of t, enumerate each of the O(2 k+1 ) subsets U i of V ti, checking that U i V t = U V ti. Time = O(2 k+1 }{{} #U s d }{{} #children 2 }{{} k+1 #U i s Total time = O(4 k+1 k t T degree(t)) = O(4k+1 kn). k }{{} checking U i ) = O(4 k+1 kd). CS 511 (Iowa State University) Tree Decompositions and Tree-Width December 6, / 15