Low-degree Graph Partitioning via Local Search with Applications to Constraint Satisfaction, Max Cut, and Coloring

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 \Delta that cut at least a k\Gamma1 k (1 + 1 2\Delta+k\Gamma1 ) 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 \Delta+2 , and obtain an efficient algorithm for coloring 3-colorable graphs with at most 3\Delta+2 4 colors