20,951 research outputs found

    A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover

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    Given a kk-uniform hyper-graph, the Ekk-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyper-edge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that Ekk-Vertex-Cover is NP-hard to approximate within factor (k1ϵ)(k-1-\epsilon) for any k3k \geq 3 and any ϵ>0\epsilon>0. The result is essentially tight as this problem can be easily approximated within factor kk. Our construction makes use of the biased Long-Code and is analyzed using combinatorial properties of ss-wise tt-intersecting families of subsets

    Generic Expression Hardness Results for Primitive Positive Formula Comparison

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    We study the expression complexity of two basic problems involving the comparison of primitive positive formulas: equivalence and containment. In particular, we study the complexity of these problems relative to finite relational structures. We present two generic hardness results for the studied problems, and discuss evidence that they are optimal and yield, for each of the problems, a complexity trichotomy

    Comparative Performance of Tabu Search and Simulated Annealing Heuristics for the Quadratic Assignment Problem

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    For almost two decades the question of whether tabu search (TS) or simulated annealing (SA) performs better for the quadratic assignment problem has been unresolved. To answer this question satisfactorily, we compare performance at various values of targeted solution quality, running each heuristic at its optimal number of iterations for each target. We find that for a number of varied problem instances, SA performs better for higher quality targets while TS performs better for lower quality targets

    Near-Optimal UGC-hardness of Approximating Max k-CSP_R

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    In this paper, we prove an almost-optimal hardness for Max kk-CSPR_R based on Khot's Unique Games Conjecture (UGC). In Max kk-CSPR_R, we are given a set of predicates each of which depends on exactly kk variables. Each variable can take any value from 1,2,,R1, 2, \dots, R. The goal is to find an assignment to variables that maximizes the number of satisfied predicates. Assuming the Unique Games Conjecture, we show that it is NP-hard to approximate Max kk-CSPR_R to within factor 2O(klogk)(logR)k/2/Rk12^{O(k \log k)}(\log R)^{k/2}/R^{k - 1} for any k,Rk, R. To the best of our knowledge, this result improves on all the known hardness of approximation results when 3k=o(logR/loglogR)3 \leq k = o(\log R/\log \log R). In this case, the previous best hardness result was NP-hardness of approximating within a factor O(k/Rk2)O(k/R^{k-2}) by Chan. When k=2k = 2, our result matches the best known UGC-hardness result of Khot, Kindler, Mossel and O'Donnell. In addition, by extending an algorithm for Max 2-CSPR_R by Kindler, Kolla and Trevisan, we provide an Ω(logR/Rk1)\Omega(\log R/R^{k - 1})-approximation algorithm for Max kk-CSPR_R. This algorithm implies that our inapproximability result is tight up to a factor of 2O(klogk)(logR)k/212^{O(k \log k)}(\log R)^{k/2 - 1}. In comparison, when 3k3 \leq k is a constant, the previously known gap was O(R)O(R), which is significantly larger than our gap of O(polylog R)O(\text{polylog } R). Finally, we show that we can replace the Unique Games Conjecture assumption with Khot's dd-to-1 Conjecture and still get asymptotically the same hardness of approximation
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