Monadic second order logic on graphs with local cardinality constraints

We introduce the class of MSO-LCC problems, which are problems of the following form. Given a graph <i>G</i> and for each vertex <i>v</i> of <i>G</i> a set α(<i>v</i>) of non-negative integers. Is there a set <i>S</i> of vertices or edges of <i>G</i> such that, (1) <i>S</i> satisfies a fixed property expressible in monadic second order logic, and (2) for each vertex <i>v</i> of <i>G</i> the number of vertices/edges in <i>S</i> adjacent/incident with <i>v</i> belongs to the set α(<i>v</i>)? 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.