Max-Cut under Graph Constraints

TL;DR

This paper introduces a 0.5-approximation algorithm for the graph-constrained max-cut problem on graphs with bounded treewidth, using LP relaxations and dynamic programming techniques.

Contribution

It provides the first approximation algorithm for GCMC with various constraints on bounded-treewidth graphs, leveraging Sherali-Adams LP relaxations and dynamic programming.

Findings

Achieves a 0.5-approximation ratio for GCMC with certain constraints.

Extends results to planar, bounded-genus, and minor-excluded graphs.

Uses LP relaxations based on Sherali-Adams hierarchy and dynamic programming.

Abstract

An instance of the graph-constrained max-cut (GCMC) problem consists of (i) an undirected graph G and (ii) edge-weights on a complete undirected graph on the same vertex set. The objective is to find a subset of vertices satisfying some graph-based constraint in G that maximizes the total weight of edges in the cut. 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 0.5-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).