On hard instances of approximate vertex cover
ACM Transactions on Algorithms (TALG)
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
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Karger, Motwani and Sudan (JACM 1998) introduced the notion of a vector coloring of a graph. In particular they show that every k-colorable graph is also vector k-colorable, and that for constant k, graphs that are vector k-colorable can be colored by roughly \Delta ^{1 - {2 \mathord{\left/ {\vphantom {2 k}} \right. \kern-\nulldelimiterspace} k}} colors. Here \Delta is the maximum degree in the graph. Their results play a major role in the best approximation algorithms for coloring and for maximal independent set.We showthat for every positive integer k there are graphs that are vector k-colorable but do not have independent sets significantly larger than {n \mathord{\left/ {\vphantom {n {\Delta ^{1 - 2k} }}} \right. \kern-\nulldelimiterspace} (and hence cannot be colored with significantly less that {\Delta ^{1 - 2k} }} colors). For k = 0({{\log n} \mathord{\left/ {\vphantom {{\log n} {\log n)}}} \right. \kern-\nulldelimiterspace} {\log n)}} we show vector k-colorable graphs that do not have independent sets of size(\log n)^c for some constant c. This shows that the vector chromatic number does not approximate the chromatic number within factors better than n/polylogn.As part of our proof, we analyze "property testing" algorithms that distinguish between graphs that have an independent set of size n/k, and graphs that are "far" from having such an independent set. Our bounds on the sample size improve previous bounds of Goldreich, Goldwasser and Ron (JACM 1998) for this problem.