Low-degree graph partitioning via local search with applications to constraint satisfaction, max cut, and coloring

Abstract

We present practical algorithms for constructing partitions of graphs into a fixed number of vertex-disjoint subgraphs that satisfy particular degree constraints. We use this in particular to find k-cuts of graphs of maximum degree △ that cut at least a k-1/k (1 + 1/2△+k-1) fraction of the edges, improving previous bounds known. The partitions also apply to constraint networks, for which we give a tight analysis of natural local search heuristics for the maximum constraint satisfaction problem. These partitions also imply efficient approximations for several problems on weighted bounded-degree graphs. In particular, we improve the best performance ratio for the weighted independent set problem to 3/△+2, and obtain an efficient algorithm for coloring 3-colorable graphs with at most 3△+2/4 colors.