280 research outputs found

    Approximate Hypergraph Coloring under Low-discrepancy and Related Promises

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    A hypergraph is said to be Ο‡\chi-colorable if its vertices can be colored with Ο‡\chi colors so that no hyperedge is monochromatic. 22-colorability is a fundamental property (called Property B) of hypergraphs and is extensively studied in combinatorics. Algorithmically, however, given a 22-colorable kk-uniform hypergraph, it is NP-hard to find a 22-coloring miscoloring fewer than a fraction 2βˆ’k+12^{-k+1} of hyperedges (which is achieved by a random 22-coloring), and the best algorithms to color the hypergraph properly require β‰ˆn1βˆ’1/k\approx n^{1-1/k} colors, approaching the trivial bound of nn as kk increases. In this work, we study the complexity of approximate hypergraph coloring, for both the maximization (finding a 22-coloring with fewest miscolored edges) and minimization (finding a proper coloring using fewest number of colors) versions, when the input hypergraph is promised to have the following stronger properties than 22-colorability: (A) Low-discrepancy: If the hypergraph has discrepancy β„“β‰ͺk\ell \ll \sqrt{k}, we give an algorithm to color the it with β‰ˆnO(β„“2/k)\approx n^{O(\ell^2/k)} colors. However, for the maximization version, we prove NP-hardness of finding a 22-coloring miscoloring a smaller than 2βˆ’O(k)2^{-O(k)} (resp. kβˆ’O(k)k^{-O(k)}) fraction of the hyperedges when β„“=O(log⁑k)\ell = O(\log k) (resp. β„“=2\ell=2). Assuming the UGC, we improve the latter hardness factor to 2βˆ’O(k)2^{-O(k)} for almost discrepancy-11 hypergraphs. (B) Rainbow colorability: If the hypergraph has a (kβˆ’β„“)(k-\ell)-coloring such that each hyperedge is polychromatic with all these colors, we give a 22-coloring algorithm that miscolors at most kβˆ’Ξ©(k)k^{-\Omega(k)} of the hyperedges when β„“β‰ͺk\ell \ll \sqrt{k}, and complement this with a matching UG hardness result showing that when β„“=k\ell =\sqrt{k}, it is hard to even beat the 2βˆ’k+12^{-k+1} bound achieved by a random coloring.Comment: Approx 201
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