Monadic second order logic on graphs with local cardinality constraints

Abstract

We introduce the class of MSO-LCC problems, which are problems of the following form. Given a graph G and for each vertex v of G a set α( v ) of non-negative integers. Is there a set S of vertices or edges of G such that, (1) S satisfies a fixed property expressible in monadic second order logic, and (2) for each vertex v of G the number of vertices/edges in S adjacent/incident with v belongs to the set α( v )? We demonstrate that several hard combinatorial problems such as Lovász's General Factor Problem can be naturally formulated as MSO-LCC problems. Our main result is the polynomial-time tractability of MSO-LCC problems for graphs of bounded treewidth. We obtain this result by means of a tree-automata approach. By way of contrast we show that a more general class of MSO-LCC problems, where cardinality constraints are applied to second-order variables that are arbitrarily quantified, does not admit polynomial-time tractability for graphs of bounded treewidth unless P=NP.