Vertex Cover in Graphs with Locally Few Colors

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Vertex Cover in Graphs with Locally Few Colors
In [13], Erd˝os 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 ∆ ≥ 2, there are graphs with arbitrarily large chromatic number that can be colored so that (i) no vertex neighborhood contains more than ∆ 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/(∆ + 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 presen...
Fabian Kuhn, Monaldo Mastrolilli
Added 29 Aug 2011
Updated 29 Aug 2011
Type Journal
Year 2011
Authors Fabian Kuhn, Monaldo Mastrolilli
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