Large induced subgraphs via triangulations and CMSO

Abstract

We obtain an algorithmic meta-theorem for the following optimization problem. Let φ be a Counting Monadic Second Order Logic (CMSO) formula and t ≥ 0 be an integer. For a given graph G = (V,E), the task is to maximize |X| subject to the following: There is a set F ⊆ V such that X ⊆ F, the subgraph G[F] induced by F is of treewidth at most t, and structure (G[F], X) models φ, i.e. (G[F],X) |= φ. Special cases of this optimization problem are the following generic examples. Each of these special cases contains various problems as a special subcase: Maximum Induced Subgraph with ≤ ℓ copies of Fm-cycles, where for fixed nonnegative integers m and ℓ. the task is to find a maximum induced subgraph of a given graph with at most ℓ vertex-disjoint cycles of length 0 (mod m). For example, this encompasses the problems of finding a maximum induced forest or a maximum subgraph without even cycles. Minimum F-Deletion, where for a fixed finite set of graphs T containing a planar graph, the task is to find a maximum induced subgraph of a given graph containing no graph from T as a minor. Examples of Minimum F -Deletion are the problems of finding a minimum vertex cover or a minimum number of vertices required to delete from the graph to obtain an outerplanar graph. Independent H-packing, where for a fixed finite set of connected graphs H, the task is to find an induced subgraph F of a given graph with the maximum number of connected components, such that each connected component of F is isomorphic to some graph from H. For example, the problem of finding a maximum induced matching or packing into nonadjacent triangles, are the special cases of this problem. We give an algorithm solving the optimization problem on an n-vertex graph G in time O(|IIG| · n t+4 · f(t, φ)), where IIG is the set of all potential maximal cliques in G and f is a function of t and φ only. We also show how similar running time can be obtained for the weighted version of the problem. Pipelined with known bounds on the number of potential maximal cliques, we derive a plethora of algorithmic consequences extending and subsuming many known results on algorithms for special graph classes and exact exponential algorithms. Copyright © 2014 by the Society for Industrial and Applied Mathematics.