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- W2065609263 abstract "In the Discrepancy problem, we are given M sets {S1,..., SM} on N elements. Our goal is to find an assignment χ of {−1, + 1} values to elements, so as to minimize the maximum discrepancy maxj | ΣieSjχ(i)|. Recently, Bansal gave an efficient algorithm for achieving O(√N) discrepancy for any set system where M = O(N) [Ban10], giving a constructive version of Spencer's proof that the discrepancy of any set system is at most O(√N) for this range of M [Spe85].We show that from the perspective of computational efficiency, these results are tight for general set systems where M = O(N). Specifically, we show that it is NP-hard to distinguish between such set systems with discrepancy zero and those with discrepancy Ω(√N). This means that even if the optimal solution has discrepancy zero, we cannot hope to efficiently find a coloring with discrepancy o(√N). We also consider the hardness of the Discrepancy problem on sets with bounded shatter function, and show that the upper bounds due to Matousek [Mat95] are tight for these sets systems as well.The hardness results in both settings are obtained from a common framework: we compose a family of high discrepancy set systems with set systems for which it is NP-hard to distinguish instances with discrepancy zero from instances in which a large number of the sets (i.e. constant fraction of the sets) have non-zero discrepancy. Our composition amplifies this zero versus non-zero gap." @default.
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- W2065609263 date "2011-01-23" @default.
- W2065609263 modified "2023-09-26" @default.
- W2065609263 title "Tight hardness results for minimizing discrepancy" @default.
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