Approximating vertex cover in dense hypergraphs

Cardinal, Jean; Karpiński, Marek; Schmied, Richard; Viehmann, Claus

doi:10.1016/j.jda.2012.01.003

Cited by 10 publications

(8 citation statements)

References 24 publications

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“…We sort each variable

in decreasing order and we add the corresponding vertex to the solution if the new vertex covers at least one hyperedge that was not covered previously. This greedy algorithm is a 4-approximation algorithm (see below), which is the best known approximation achievable in polynomial time for hypergraph with hyperedges of constant size ( Cardinal et al , 2012 ).…”

Section: Methodsmentioning

confidence: 99%

Novo&Stitch: accurate reconciliation of genome assemblies via optical maps

et al. 2018

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Motivation De novo genome assembly is a challenging computational problem due to the high repetitive content of eukaryotic genomes and the imperfections of sequencing technologies (i.e. sequencing errors, uneven sequencing coverage and chimeric reads). Several assembly tools are currently available, each of which has strengths and weaknesses in dealing with the trade-off between maximizing contiguity and minimizing assembly errors (e.g. mis-joins). To obtain the best possible assembly, it is common practice to generate multiple assemblies from several assemblers and/or parameter settings and try to identify the highest quality assembly. Unfortunately, often there is no assembly that both maximizes contiguity and minimizes assembly errors, so one has to compromise one for the other.ResultsThe concept of assembly reconciliation has been proposed as a way to obtain a higher quality assembly by merging or reconciling all the available assemblies. While several reconciliation methods have been introduced in the literature, we have shown in one of our recent papers that none of them can consistently produce assemblies that are better than the assemblies provided in input. Here we introduce Novo&Stitch, a novel method that takes advantage of optical maps to accurately carry out assembly reconciliation (assuming that the assembled contigs are sufficiently long to be reliably aligned to the optical maps, e.g. 50 Kbp or longer). Experimental results demonstrate that Novo&Stitch can double the contiguity (N50) of the input assemblies without introducing mis-joins or reducing genome completeness.Availability and implementationNovo&Stitch can be obtained from https://github.com/ucrbioinfo/Novo_Stitch.

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“…We sort each variable

Section: Methodsmentioning

confidence: 99%

Novo&Stitch: accurate reconciliation of genome assemblies via optical maps

et al. 2018

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“…As m is odd by Theorem 3.4, k is necessarily even, which implies that n m by Corollary 3.7(3). If G is d-regular, d is even by Theorem 3.4 (4). Surely nd = mk, so d k as n m.…”

Section: Minimal Non-odd-bipartite Regular Hypergraphsmentioning

confidence: 95%

“…A subset U of V (G) is called a transversal (also called vertex cover [4] or hitting set [17]) of G if each edge of G has a nonempty intersection with U . The transversal number of G is the minimum size of transversals in G, which was well studied by Alon [2], Chvátal and McDiarmid [7], Henning and Yeo [16].…”

Section: Introductionmentioning

confidence: 99%

Minimal Non-Odd-Transversal Hypergraphs and Minimal Non-Odd-Bipartite Hypergraphs

Fan¹,

Wang²,

Wan³

2020

Electron. J. Combin.

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Among all uniform hypergraphs with even uniformity, the odd-transversal or odd-bipartite hypergraphs are closer to bipartite simple graphs than bipartite hypergraphs from the viewpoint of both structure and spectrum. A hypergraph is called odd-transversal if it contains a subset of the vertex set such that each edge intersects the subset in an odd number of vertices, and it is called minimal non-odd-transversal if it is not odd-transversal but deleting any edge results in an odd-transversal hypergraph. In this paper we give an equivalent characterization of the minimal non-odd-transversal hypergraphs by means of the degrees and the rank of its incidence matrix over $\mathbb{Z}_2$. If a minimal non-odd-transversal hypergraph is uniform, then it has even uniformity, and hence is minimal non-odd-bipartite. We characterize $2$-regular uniform minimal non-odd-bipartite hypergraphs, and give some examples of $d$-regular uniform hypergraphs which are minimal non-odd-bipartite. Finally we give upper bounds for the least H-eigenvalue of the adjacency tensor of minimal non-odd-bipartite hypergraphs.

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“…More precisely, it was shown in [4] that Max-k-CSP admits a polynomial-time approximation scheme (PTAS) on dense instances, that is, an algorithm which for any constant ε > 0 can in time polynomial in n produce an assignment that satisfies (1 − ε)OPT constraints. Subsequent work produced a stream of positive [17,5,2,10,9,21,3,20,23] (and some negative [16,1]) results on approximating CSPs which are in general APX-hard, showing that dense instances form an island of tractability where many optimization problems which are normally APX-hard admit a PTAS.…”

Section: Introductionmentioning

confidence: 99%

Sub-exponential Approximation Schemes for CSPs: from Dense to Almost Sparse

Fotakis,

Lampis,

Paschos

2015

Preprint

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It has long been known, since the classical work of (Arora, Karger, Karpinski, JCSS 99), that Max-CUT admits a PTAS on dense graphs, and more generally, Max-k-CSP admits a PTAS on "dense" instances with Ω(n k ) constraints. In this paper we extend and generalize their exhaustive sampling approach, presenting a framework for (1 − ε)-approximating any Max-k-CSP problem in sub-exponential time while significantly relaxing the denseness requirement on the input instance. Specifically, we prove that for any constants δ ∈ (0, 1] and ε > 0, we can approximate Max-k-CSP problems with Ω(n k−1+δ ) constraints within a factor of (1 − ε) in time 2 O(n 1−δ ln n/ε 3 ) . The framework is quite general and includes classical optimization problems, such as Max-CUT, Max-DICUT, Max-k-SAT, and (with a slight extension) k-Densest Subgraph, as special cases. For Max-CUT in particular (where k = 2), it gives an approximation scheme that runs in time sub-exponential in n even for "almostsparse" instances (graphs with n 1+δ edges). We prove that our results are essentially best possible, assuming the ETH. First, the density requirement cannot be relaxed further: there exists a constant r < 1 such that for all δ > 0, Max-k-SAT instances with O(n k−1 ) clauses cannot be approximated within a ratio better than r in time 2 O(n 1−δ ) . Second, the running time of our algorithm is almost tight for all densities. Even for Max-CUT there exists r < 1 such that for all δ ′ > δ > 0, Max-CUT instances with n 1+δ edges cannot be approximated within a ratio better than r in time 2 n 1−δ ′ .

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Approximating vertex cover in dense hypergraphs

Cited by 10 publications

References 24 publications

Novo&Stitch: accurate reconciliation of genome assemblies via optical maps

Novo&Stitch: accurate reconciliation of genome assemblies via optical maps

Minimal Non-Odd-Transversal Hypergraphs and Minimal Non-Odd-Bipartite Hypergraphs

Sub-exponential Approximation Schemes for CSPs: from Dense to Almost Sparse

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