834 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

    Rainbow Coloring Hardness via Low Sensitivity Polymorphisms

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    A k-uniform hypergraph is said to be r-rainbow colorable if there is an r-coloring of its vertices such that every hyperedge intersects all r color classes. Given as input such a hypergraph, finding a r-rainbow coloring of it is NP-hard for all k >= 3 and r >= 2. Therefore, one settles for finding a rainbow coloring with fewer colors (which is an easier task). When r=k (the maximum possible value), i.e., the hypergraph is k-partite, one can efficiently 2-rainbow color the hypergraph, i.e., 2-color its vertices so that there are no monochromatic edges. In this work we consider the next smaller value of r=k-1, and prove that in this case it is NP-hard to rainbow color the hypergraph with q := ceil[(k-2)/2] colors. In particular, for k <=6, it is NP-hard to 2-color (k-1)-rainbow colorable k-uniform hypergraphs. Our proof follows the algebraic approach to promise constraint satisfaction problems. It proceeds by characterizing the polymorphisms associated with the approximate rainbow coloring problem, which are rainbow colorings of some product hypergraphs on vertex set [r]^n. We prove that any such polymorphism f: [r]^n -> [q] must be C-fixing, i.e., there is a small subset S of C coordinates and a setting a in [q]^S such that fixing x_{|S} = a determines the value of f(x). The key step in our proof is bounding the sensitivity of certain rainbow colorings, thereby arguing that they must be juntas. Armed with the C-fixing characterization, our NP-hardness is obtained via a reduction from smooth Label Cover

    Hardness of Finding Independent Sets in 2-Colorable Hypergraphs and of Satisfiable CSPs

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    This work revisits the PCP Verifiers used in the works of Hastad [Has01], Guruswami et al.[GHS02], Holmerin[Hol02] and Guruswami[Gur00] for satisfiable Max-E3-SAT and Max-Ek-Set-Splitting, and independent set in 2-colorable 4-uniform hypergraphs. We provide simpler and more efficient PCP Verifiers to prove the following improved hardness results: Assuming that NP\not\subseteq DTIME(N^{O(loglog N)}), There is no polynomial time algorithm that, given an n-vertex 2-colorable 4-uniform hypergraph, finds an independent set of n/(log n)^c vertices, for some constant c > 0. There is no polynomial time algorithm that satisfies 7/8 + 1/(log n)^c fraction of the clauses of a satisfiable Max-E3-SAT instance of size n, for some constant c > 0. For any fixed k >= 4, there is no polynomial time algorithm that finds a partition splitting (1 - 2^{-k+1}) + 1/(log n)^c fraction of the k-sets of a satisfiable Max-Ek-Set-Splitting instance of size n, for some constant c > 0. Our hardness factor for independent set in 2-colorable 4-uniform hypergraphs is an exponential improvement over the previous results of Guruswami et al.[GHS02] and Holmerin[Hol02]. Similarly, our inapproximability of (log n)^{-c} beyond the random assignment threshold for Max-E3-SAT and Max-Ek-Set-Splitting is an exponential improvement over the previous bounds proved in [Has01], [Hol02] and [Gur00]. The PCP Verifiers used in our results avoid the use of a variable bias parameter used in previous works, which leads to the improved hardness thresholds in addition to simplifying the analysis substantially. Apart from standard techniques from Fourier Analysis, for the first mentioned result we use a mixing estimate of Markov Chains based on uniform reverse hypercontractivity over general product spaces from the work of Mossel et al.[MOS13].Comment: 23 Page

    Super-polylogarithmic hypergraph coloring hardness via low-degree long codes

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    We prove improved inapproximability results for hypergraph coloring using the low-degree polynomial code (aka, the 'short code' of Barak et. al. [FOCS 2012]) and the techniques proposed by Dinur and Guruswami [FOCS 2013] to incorporate this code for inapproximability results. In particular, we prove quasi-NP-hardness of the following problems on nn-vertex hyper-graphs: * Coloring a 2-colorable 8-uniform hypergraph with 22Ω(log⁑log⁑n)2^{2^{\Omega(\sqrt{\log\log n})}} colors. * Coloring a 4-colorable 4-uniform hypergraph with 22Ω(log⁑log⁑n)2^{2^{\Omega(\sqrt{\log\log n})}} colors. * Coloring a 3-colorable 3-uniform hypergraph with (log⁑n)Ω(1/log⁑log⁑log⁑n)(\log n)^{\Omega(1/\log\log\log n)} colors. In each of these cases, the hardness results obtained are (at least) exponentially stronger than what was previously known for the respective cases. In fact, prior to this result, polylog n colors was the strongest quantitative bound on the number of colors ruled out by inapproximability results for O(1)-colorable hypergraphs. The fundamental bottleneck in obtaining coloring inapproximability results using the low- degree long code was a multipartite structural restriction in the PCP construction of Dinur-Guruswami. We are able to get around this restriction by simulating the multipartite structure implicitly by querying just one partition (albeit requiring 8 queries), which yields our result for 2-colorable 8-uniform hypergraphs. The result for 4-colorable 4-uniform hypergraphs is obtained via a 'query doubling' method. For 3-colorable 3-uniform hypergraphs, we exploit the ternary domain to design a test with an additive (as opposed to multiplicative) noise function, and analyze its efficacy in killing high weight Fourier coefficients via the pseudorandom properties of an associated quadratic form.Comment: 25 page
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