Max-Cut Under Graph Constraints

An instance of the graph-constrained max-cut (GCMC) problem consists of (i) an undirected graph G = (V,E) and (ii) edge-weights c : ( V 2 ) → R+ on a complete undirected graph. The objective is to find a subset S ⊆ V of vertices satisfying some graph-based constraint in G that maximizes the weight ∑ u∈S,v 6∈S cuv of edges in the cut (S, V \ S). The types of graph constraints we can handle include independent set, vertex cover, dominating set and connectivity. Our main results are for the case when G is a graph with bounded treewidth, where we obtain a 1 2 -approximation algorithm. Our algorithm uses an LP relaxation based on the Sherali-Adams hierarchy. It can handle any graph constraint for which there is a (certain type of) dynamic program that exactly optimizes linear objectives. Using known decomposition results, these imply essentially the same approximation ratio for GCMC under constraints such as independent set, dominating set and connectivity on a planar graph G (more generally for bounded-genus or excluded-minor graphs).