Vertex cover in graphs with locally few colors

Abstract

Erdős 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 (UGC), we provide a tight characterization. We prove that it is UGC-hard to improve the approximation ratio of 2−2/(Δ+1) on (Δ+1)-locally (but not necessarily biclique) colorable graphs. 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)lnlnΔlnΔ. This matches known inapproximability results for the special case of bounded degree graphs.Moreover, we show that when both the above two properties (i) and (ii) hold, 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, denoted as 1|prec|∑wjCj in standard scheduling notation. 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.