529 research outputs found

    Metro-Line Crossing Minimization: Hardness, Approximations, and Tractable Cases

    Full text link
    Crossing minimization is one of the central problems in graph drawing. Recently, there has been an increased interest in the problem of minimizing crossings between paths in drawings of graphs. This is the metro-line crossing minimization problem (MLCM): Given an embedded graph and a set L of simple paths, called lines, order the lines on each edge so that the total number of crossings is minimized. So far, the complexity of MLCM has been an open problem. In contrast, the problem variant in which line ends must be placed in outermost position on their edges (MLCM-P) is known to be NP-hard. Our main results answer two open questions: (i) We show that MLCM is NP-hard. (ii) We give an O(logL)O(\sqrt{\log |L|})-approximation algorithm for MLCM-P

    On the Complexity of the Single Individual SNP Haplotyping Problem

    Full text link
    We present several new results pertaining to haplotyping. These results concern the combinatorial problem of reconstructing haplotypes from incomplete and/or imperfectly sequenced haplotype fragments. We consider the complexity of the problems Minimum Error Correction (MEC) and Longest Haplotype Reconstruction (LHR) for different restrictions on the input data. Specifically, we look at the gapless case, where every row of the input corresponds to a gapless haplotype-fragment, and the 1-gap case, where at most one gap per fragment is allowed. We prove that MEC is APX-hard in the 1-gap case and still NP-hard in the gapless case. In addition, we question earlier claims that MEC is NP-hard even when the input matrix is restricted to being completely binary. Concerning LHR, we show that this problem is NP-hard and APX-hard in the 1-gap case (and thus also in the general case), but is polynomial time solvable in the gapless case.Comment: 26 pages. Related to the WABI2005 paper, "On the Complexity of Several Haplotyping Problems", but with more/different results. This papers has just been submitted to the IEEE/ACM Transactions on Computational Biology and Bioinformatics and we are awaiting a decision on acceptance. It differs from the mid-August version of this paper because here we prove that 1-gap LHR is APX-hard. (In the earlier version of the paper we could prove only that it was NP-hard.

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

    Full text link
    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

    Approximate Graph Coloring by Semidefinite Programming

    Full text link
    We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on nn vertices with min O(Delta^{1/3} log^{1/2} Delta log n), O(n^{1/4} log^{1/2} n) colors where Delta is the maximum degree of any vertex. Besides giving the best known approximation ratio in terms of n, this marks the first non-trivial approximation result as a function of the maximum degree Delta. This result can be generalized to k-colorable graphs to obtain a coloring using min O(Delta^{1-2/k} log^{1/2} Delta log n), O(n^{1-3/(k+1)} log^{1/2} n) colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovasz theta-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the theta-function

    Robustly Solvable Constraint Satisfaction Problems

    Full text link
    An algorithm for a constraint satisfaction problem is called robust if it outputs an assignment satisfying at least (1g(ε))(1-g(\varepsilon))-fraction of the constraints given a (1ε)(1-\varepsilon)-satisfiable instance, where g(ε)0g(\varepsilon) \rightarrow 0 as ε0\varepsilon \rightarrow 0. Guruswami and Zhou conjectured a characterization of constraint languages for which the corresponding constraint satisfaction problem admits an efficient robust algorithm. This paper confirms their conjecture

    Bisect and Conquer: Hierarchical Clustering via Max-Uncut Bisection

    Full text link
    Hierarchical Clustering is an unsupervised data analysis method which has been widely used for decades. Despite its popularity, it had an underdeveloped analytical foundation and to address this, Dasgupta recently introduced an optimization viewpoint of hierarchical clustering with pairwise similarity information that spurred a line of work shedding light on old algorithms (e.g., Average-Linkage), but also designing new algorithms. Here, for the maximization dual of Dasgupta's objective (introduced by Moseley-Wang), we present polynomial-time .4246 approximation algorithms that use Max-Uncut Bisection as a subroutine. The previous best worst-case approximation factor in polynomial time was .336, improving only slightly over Average-Linkage which achieves 1/3. Finally, we complement our positive results by providing APX-hardness (even for 0-1 similarities), under the Small Set Expansion hypothesis
    corecore