Large induced subgraphs via triangulations and CMSO

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 ≤ ` copies of Fm-cycles, where for fixed nonnegative integers m and `, the task is to find a maximum induced subgraph of a given graph with at most ` 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 F containing a planar graph, the task is to find a maximum induced subgraph of a given graph containing no graph from F 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(|ΠG| · n · f(t, φ)), where ΠG 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. ∗Partially supported by the ANR project AGAPE. ar X iv :1 30 9. 15 59 v1 [ cs .D S] 6 S ep 2 01 3