10,486 research outputs found

    The Complexity of Approximating Vertex Expansion

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    We study the complexity of approximating the vertex expansion of graphs G=(V,E)G = (V,E), defined as ΦV:=minSVnN(S)SV\S. \Phi^V := \min_{S \subset V} n \cdot \frac{|N(S)|}{|S| |V \backslash S|}. We give a simple polynomial-time algorithm for finding a subset with vertex expansion O(OPTlogd)O(\sqrt{OPT \log d}) where dd is the maximum degree of the graph. Our main result is an asymptotically matching lower bound: under the Small Set Expansion (SSE) hypothesis, it is hard to find a subset with expansion less than COPTlogdC\sqrt{OPT \log d} for an absolute constant CC. In particular, this implies for all constant ϵ>0\epsilon > 0, it is SSE-hard to distinguish whether the vertex expansion <ϵ< \epsilon or at least an absolute constant. The analogous threshold for edge expansion is OPT\sqrt{OPT} with no dependence on the degree; thus our results suggest that vertex expansion is harder to approximate than edge expansion. In particular, while Cheeger's algorithm can certify constant edge expansion, it is SSE-hard to certify constant vertex expansion in graphs. Our proof is via a reduction from the {\it Unique Games} instance obtained from the \SSE hypothesis to the vertex expansion problem. It involves the definition of a smoother intermediate problem we call {\sf Analytic Vertex Expansion} which is representative of both the vertex expansion and the conductance of the graph. Both reductions (from the UGC instance to this problem and from this problem to vertex expansion) use novel proof ideas

    Inapproximability of Maximum Biclique Problems, Minimum kk-Cut and Densest At-Least-kk-Subgraph from the Small Set Expansion Hypothesis

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    The Small Set Expansion Hypothesis (SSEH) is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small subset of vertices whose edge expansion is almost zero and one in which all small subsets of vertices have expansion almost one. In this work, we prove inapproximability results for the following graph problems based on this hypothesis: - Maximum Edge Biclique (MEB): given a bipartite graph GG, find a complete bipartite subgraph of GG with maximum number of edges. - Maximum Balanced Biclique (MBB): given a bipartite graph GG, find a balanced complete bipartite subgraph of GG with maximum number of vertices. - Minimum kk-Cut: given a weighted graph GG, find a set of edges with minimum total weight whose removal partitions GG into kk connected components. - Densest At-Least-kk-Subgraph (DALkkS): given a weighted graph GG, find a set SS of at least kk vertices such that the induced subgraph on SS has maximum density (the ratio between the total weight of edges and the number of vertices). We show that, assuming SSEH and NP \nsubseteq BPP, no polynomial time algorithm gives n1εn^{1 - \varepsilon}-approximation for MEB or MBB for every constant ε>0\varepsilon > 0. Moreover, assuming SSEH, we show that it is NP-hard to approximate Minimum kk-Cut and DALkkS to within (2ε)(2 - \varepsilon) factor of the optimum for every constant ε>0\varepsilon > 0. The ratios in our results are essentially tight since trivial algorithms give nn-approximation to both MEB and MBB and efficient 22-approximation algorithms are known for Minimum kk-Cut [SV95] and DALkkS [And07, KS09]. Our first result is proved by combining a technique developed by Raghavendra et al. [RST12] to avoid locality of gadget reductions with a generalization of Bansal and Khot's long code test [BK09] whereas our second result is shown via elementary reductions.Comment: A preliminary version of this work will appear at ICALP 2017 under a different title "Inapproximability of Maximum Edge Biclique, Maximum Balanced Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis

    The Complexity of Approximately Counting Tree Homomorphisms

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    We study two computational problems, parameterised by a fixed tree H. #HomsTo(H) is the problem of counting homomorphisms from an input graph G to H. #WHomsTo(H) is the problem of counting weighted homomorphisms to H, given an input graph G and a weight function for each vertex v of G. Even though H is a tree, these problems turn out to be sufficiently rich to capture all of the known approximation behaviour in #P. We give a complete trichotomy for #WHomsTo(H). If H is a star then #WHomsTo(H) is in FP. If H is not a star but it does not contain a certain induced subgraph J_3 then #WHomsTo(H) is equivalent under approximation-preserving (AP) reductions to #BIS, the problem of counting independent sets in a bipartite graph. This problem is complete for the class #RHPi_1 under AP-reductions. Finally, if H contains an induced J_3 then #WHomsTo(H) is equivalent under AP-reductions to #SAT, the problem of counting satisfying assignments to a CNF Boolean formula. Thus, #WHomsTo(H) is complete for #P under AP-reductions. The results are similar for #HomsTo(H) except that a rich structure emerges if H contains an induced J_3. We show that there are trees H for which #HomsTo(H) is #SAT-equivalent (disproving a plausible conjecture of Kelk). There is an interesting connection between these homomorphism-counting problems and the problem of approximating the partition function of the ferromagnetic Potts model. In particular, we show that for a family of graphs J_q, parameterised by a positive integer q, the problem #HomsTo(H) is AP-interreducible with the problem of approximating the partition function of the q-state Potts model. It was not previously known that the Potts model had a homomorphism-counting interpretation. We use this connection to obtain some additional upper bounds for the approximation complexity of #HomsTo(J_q)

    Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder

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    Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within ~~0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within ~~0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (~~0.878 for Max-Cut, and ~~0.940 for Max-2-Sat). The hardness for Max-2-Sat applies to monotone Max-2-Sat instances, meaning that we also obtain tight inapproximability for the Max-k-Vertex-Cover problem

    On the noncommutative geometry of tilings

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    This is a chapter in an incoming book on aperiodic order. We review results about the topology, the dynamics, and the combinatorics of aperiodically ordered tilings obtained with the tools of noncommutative geometry

    The positive semidefinite Grothendieck problem with rank constraint

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    Given a positive integer n and a positive semidefinite matrix A = (A_{ij}) of size m x m, the positive semidefinite Grothendieck problem with rank-n-constraint (SDP_n) is maximize \sum_{i=1}^m \sum_{j=1}^m A_{ij} x_i \cdot x_j, where x_1, ..., x_m \in S^{n-1}. In this paper we design a polynomial time approximation algorithm for SDP_n achieving an approximation ratio of \gamma(n) = \frac{2}{n}(\frac{\Gamma((n+1)/2)}{\Gamma(n/2)})^2 = 1 - \Theta(1/n). We show that under the assumption of the unique games conjecture the achieved approximation ratio is optimal: There is no polynomial time algorithm which approximates SDP_n with a ratio greater than \gamma(n). We improve the approximation ratio of the best known polynomial time algorithm for SDP_1 from 2/\pi to 2/(\pi\gamma(m)) = 2/\pi + \Theta(1/m), and we show a tighter approximation ratio for SDP_n when A is the Laplacian matrix of a graph with nonnegative edge weights.Comment: (v3) to appear in Proceedings of the 37th International Colloquium on Automata, Languages and Programming, 12 page
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