A Logarithmic Additive Integrality Gap for Bin Packing
نویسندگان
چکیده
For bin packing, the input consists of n items with sizes s1, . . . , sn ∈ [0,1] which have to be assigned to aminimum number of bins of size 1. Recently, the second author gave an LP-based polynomial time algorithm that employed techniques from discrepancy theory to find a solution using at mostOPT +O(logOPT · log logOPT ) bins. In this paper, we present an approximation algorithm that has an additive gap of onlyO(logOPT ) bins, which matches certain combinatorial lower bounds. Any further improvement would have to use more algebraic structure. Our improvement is based on a combination of discrepancy theory techniques and a novel 2-stage packing: first we pack items into containers; then we pack containers into bins of size 1. Apart frombeing more effective, we believe our algorithm is much cleaner than the one of Rothvoss.
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