6,917 research outputs found

    On the practically interesting instances of MAXCUT

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    The complexity of a computational problem is traditionally quantified based on the hardness of its worst case. This approach has many advantages and has led to a deep and beautiful theory. However, from the practical perspective, this leaves much to be desired. In application areas, practically interesting instances very often occupy just a tiny part of an algorithm's space of instances, and the vast majority of instances are simply irrelevant. Addressing these issues is a major challenge for theoretical computer science which may make theory more relevant to the practice of computer science. Following Bilu and Linial, we apply this perspective to MAXCUT, viewed as a clustering problem. Using a variety of techniques, we investigate practically interesting instances of this problem. Specifically, we show how to solve in polynomial time distinguished, metric, expanding and dense instances of MAXCUT under mild stability assumptions. In particular, (1+ϵ)(1+\epsilon)-stability (which is optimal) suffices for metric and dense MAXCUT. We also show how to solve in polynomial time Ω(n)\Omega(\sqrt{n})-stable instances of MAXCUT, substantially improving the best previously known result

    Improved Cheeger's Inequality: Analysis of Spectral Partitioning Algorithms through Higher Order Spectral Gap

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    Let \phi(G) be the minimum conductance of an undirected graph G, and let 0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, \phi(G) = O(k) \lambda_2 / \sqrt{\lambda_k}, and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any k. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if \lambda_k$ is a constant for some constant k. This provides some theoretical justification to its empirical performance in image segmentation and clustering problems. We extend the analysis to other graph partitioning problems, including multi-way partition, balanced separator, and maximum cut

    Additive Approximation Algorithms for Modularity Maximization

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    The modularity is a quality function in community detection, which was introduced by Newman and Girvan (2004). Community detection in graphs is now often conducted through modularity maximization: given an undirected graph G=(V,E)G=(V,E), we are asked to find a partition C\mathcal{C} of VV that maximizes the modularity. Although numerous algorithms have been developed to date, most of them have no theoretical approximation guarantee. Recently, to overcome this issue, the design of modularity maximization algorithms with provable approximation guarantees has attracted significant attention in the computer science community. In this study, we further investigate the approximability of modularity maximization. More specifically, we propose a polynomial-time (cos(354π)1+58)\left(\cos\left(\frac{3-\sqrt{5}}{4}\pi\right) - \frac{1+\sqrt{5}}{8}\right)-additive approximation algorithm for the modularity maximization problem. Note here that cos(354π)1+58<0.42084\cos\left(\frac{3-\sqrt{5}}{4}\pi\right) - \frac{1+\sqrt{5}}{8} < 0.42084 holds. This improves the current best additive approximation error of 0.46720.4672, which was recently provided by Dinh, Li, and Thai (2015). Interestingly, our analysis also demonstrates that the proposed algorithm obtains a nearly-optimal solution for any instance with a very high modularity value. Moreover, we propose a polynomial-time 0.165980.16598-additive approximation algorithm for the maximum modularity cut problem. It should be noted that this is the first non-trivial approximability result for the problem. Finally, we demonstrate that our approximation algorithm can be extended to some related problems.Comment: 23 pages, 4 figure

    Finding Connected Dense kk-Subgraphs

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    Given a connected graph GG on nn vertices and a positive integer knk\le n, a subgraph of GG on kk vertices is called a kk-subgraph in GG. We design combinatorial approximation algorithms for finding a connected kk-subgraph in GG such that its density is at least a factor Ω(max{n2/5,k2/n2})\Omega(\max\{n^{-2/5},k^2/n^2\}) of the density of the densest kk-subgraph in GG (which is not necessarily connected). These particularly provide the first non-trivial approximations for the densest connected kk-subgraph problem on general graphs

    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 WVW\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(43)/13+ϵ)O(n0.697831+ϵ)O(n^{4(4-\sqrt{3})/13 + \epsilon}) \leq O(n^{0.697831+\epsilon})-approximation (for arbitrary constant ϵ>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

    Edge covering with budget constrains

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    We study two related problems: finding a set of k vertices and minimum number of edges (kmin) and finding a graph with at least m' edges and minimum number of vertices (mvms). Goldschmidt and Hochbaum \cite{GH97} show that the mvms problem is NP-hard and they give a 3-approximation algorithm for the problem. We improve \cite{GH97} by giving a ratio of 2. A 2(1+\epsilon)-approximation for the problem follows from the work of Carnes and Shmoys \cite{CS08}. We improve the approximation ratio to 2. algorithm for the problem. We show that the natural LP for \kmin has an integrality gap of 2-o(1). We improve the NP-completeness of \cite{GH97} by proving the pronlem are APX-hard unless a well-known instance of the dense k-subgraph admits a constant ratio. The best approximation guarantee known for this instance of dense k-subgraph is O(n^{2/9}) \cite{BCCFV}. We show that for any constant \rho>1, an approximation guarantee of \rho for the \kmin problem implies a \rho(1+o(1)) approximation for \mwms. Finally, we define we give an exact algorithm for the density version of kmin.Comment: 17 page
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