7 research outputs found

    The Densest k-Subhypergraph Problem

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    The Densest kk-Subgraph (DkkS) problem, and its corresponding minimization problem Smallest pp-Edge Subgraph (SppES), have come to play a central role in approximation algorithms. This is due both to their practical importance, and their usefulness as a tool for solving and establishing approximation bounds for other problems. These two problems are not well understood, and it is widely believed that they do not an admit a subpolynomial approximation ratio (although the best known hardness results do not rule this out). In this paper we generalize both DkkS and SppES from graphs to hypergraphs. We consider the Densest kk-Subhypergraph problem (given a hypergraph (V,E)(V, E), find a subset W⊆VW\subseteq V of kk vertices so as to maximize the number of hyperedges contained in WW) and define the Minimum pp-Union problem (given a hypergraph, choose pp of the hyperedges so as to minimize the number of vertices in their union). We focus in particular on the case where all hyperedges have size 3, as this is the simplest non-graph setting. For this case we provide an O(n4(4−3)/13+Ï”)≀O(n0.697831+Ï”)O(n^{4(4-\sqrt{3})/13 + \epsilon}) \leq O(n^{0.697831+\epsilon})-approximation (for arbitrary constant Ï”>0\epsilon > 0) for Densest kk-Subhypergraph and an O~(n2/5)\tilde O(n^{2/5})-approximation for Minimum pp-Union. We also give an O(m)O(\sqrt{m})-approximation for Minimum pp-Union in general hypergraphs. Finally, we examine the interesting special case of interval hypergraphs (instances where the vertices are a subset of the natural numbers and the hyperedges are intervals of the line) and prove that both problems admit an exact polynomial time solution on these instances.Comment: 21 page

    Vertex Sparsifiers: New Results from Old Techniques

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    Given a capacitated graph G=(V,E)G = (V,E) and a set of terminals K⊆VK \subseteq V, how should we produce a graph HH only on the terminals KK so that every (multicommodity) flow between the terminals in GG could be supported in HH with low congestion, and vice versa? (Such a graph HH is called a flow-sparsifier for GG.) What if we want HH to be a "simple" graph? What if we allow HH to be a convex combination of simple graphs? Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC 2010], we give efficient algorithms for constructing: (a) a flow-sparsifier HH that maintains congestion up to a factor of O(log⁡k/log⁡log⁡k)O(\log k/\log \log k), where k=∣K∣k = |K|, (b) a convex combination of trees over the terminals KK that maintains congestion up to a factor of O(log⁡k)O(\log k), and (c) for a planar graph GG, a convex combination of planar graphs that maintains congestion up to a constant factor. This requires us to give a new algorithm for the 0-extension problem, the first one in which the preimages of each terminal are connected in GG. Moreover, this result extends to minor-closed families of graphs. Our improved bounds immediately imply improved approximation guarantees for several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), 2010. Final version to appear in SIAM J. Computin

    A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs

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    A (k×l)(k \times l)-birthday repetition Gk×l\mathcal{G}^{k \times l} of a two-prover game G\mathcal{G} is a game in which the two provers are sent random sets of questions from G\mathcal{G} of sizes kk and ll respectively. These two sets are sampled independently uniformly among all sets of questions of those particular sizes. We prove the following birthday repetition theorem: when G\mathcal{G} satisfies some mild conditions, val(Gk×l)val(\mathcal{G}^{k \times l}) decreases exponentially in Ω(kl/n)\Omega(kl/n) where nn is the total number of questions. Our result positively resolves an open question posted by Aaronson, Impagliazzo and Moshkovitz (CCC 2014). As an application of our birthday repetition theorem, we obtain new fine-grained hardness of approximation results for dense CSPs. Specifically, we establish a tight trade-off between running time and approximation ratio for dense CSPs by showing conditional lower bounds, integrality gaps and approximation algorithms. In particular, for any sufficiently large ii and for every k≄2k \geq 2, we show the following results: - We exhibit an O(q1/i)O(q^{1/i})-approximation algorithm for dense Max kk-CSPs with alphabet size qq via Ok(i)O_k(i)-level of Sherali-Adams relaxation. - Through our birthday repetition theorem, we obtain an integrality gap of q1/iq^{1/i} for Ω~k(i)\tilde\Omega_k(i)-level Lasserre relaxation for fully-dense Max kk-CSP. - Assuming that there is a constant Ï”>0\epsilon > 0 such that Max 3SAT cannot be approximated to within (1−ϔ)(1-\epsilon) of the optimal in sub-exponential time, our birthday repetition theorem implies that any algorithm that approximates fully-dense Max kk-CSP to within a q1/iq^{1/i} factor takes (nq)Ω~k(i)(nq)^{\tilde \Omega_k(i)} time, almost tightly matching the algorithmic result based on Sherali-Adams relaxation.Comment: 45 page

    The Densest kk-Subhypergraph Problem

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