Vertex cover in graphs with locally few colors

• Authors: Fabian Kuhn;Monaldo Mastrolilli
• Venue: ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
• Year: 2011

In [13], Erdös et al. defined the local chromatic number of a graph as the minimum number of colors that must appear within distance 1 of a vertex. For any Δ ≥ 2, there are graphs with arbitrarily large chromatic number that can be colored so that (i) no vertex neighborhood contains more than δ different colors (bounded local colorability), and (ii) adjacent vertices from two color classes induce a complete bipartite graph (biclique coloring). We investigate the weighted vertex cover problem in graphs when a locally bounded coloring is given. This generalizes the vertex cover problem in bounded degree graphs to a class of graphs with arbitrarily large chromatic number. Assuming the Unique Game Conjecture, we provide a tight characterization. We prove that it is UGC-hard to improve the approximation ratio of 2 - 2/(Δ + 1) if the given local coloring is not a biclique coloring. A matching upper bound is also provided. Vice versa, when properties (i) and (ii) hold, we present a randomized algorithm with approximation ratio of 2 - Ω(1) ln lnΔ/lnΔ . This matches known inapproximability results for the special case of bounded degree graphs. Moreover, we show that the obtained result finds a natural application in a classical scheduling problem, namely the precedence constrained single machine scheduling problem to minimize the total weighted completion time. In a series of recent papers it was established that this scheduling problem is a special case of the minimum weighted vertex cover in graphs GP of incomparable pairs defined in the dimension theory of partial orders. We show that GP satisfies properties (i) and (ii) where δ - 1 is the maximum number of predecessors (or successors) of each job.