Another Note on Dilworth's Decomposition Theorem

Pijls, Wim; Potharst, Rob

doi:10.1155/2013/692645

Cited by 9 publications

(23 citation statements)

References 14 publications

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“…As a side result of this paper, we obtain a simple algorithm for the classical minimum weight MPC Problem running in time O ( n 2 log n + nm ), based on a recent reduction to a network flow problem [36]. This improves the current best bound O ( n 2 log n + nt ( G )), where

t ()(, G) MathClass-rel\in (}{, m MathClass-punc, m MathClass-bin+ 1 MathClass-punc, MathClass-op\dots MathClass-punc, ()(, \begin{matrix} arraynonefalsefalse \\ arraycenter n \\ arraycenter 2 \end{matrix}))

is the number of edges in the transitive closure of G , arising from the reduction in [29].…”

Section: Resultsmentioning

confidence: 99%

“…A recent solution for the MPC Problem reduces it instead to a min-flow problem [ 36 ], as follows. Each node of G is replaced by an arc with lower bound 1 (all other edges of G have lower bound 0), and a new global source s and sink t are added to G and connected to all sources and sinks of G , respectively (see Figures 1 (a) and 1(c)).…”

Section: Methodsmentioning

confidence: 99%

“…A min-flow on this digraph is a flow of minimum value satisfying all lower bounds. The value of the min-flow on this network equals the maximum size of an anti-chain of G , and any decomposition of it into paths gives a MPC [ 36 ]. A decomposition of a flow on a DAG into paths can be computed in time linear in the number of edges, by traversing the edges used by the flow [ 45 ].…”

Section: Methodsmentioning

confidence: 99%

“…If in the unweighted case, the complexity of the method of [ 36 ] is incomparable with the complexity of solving the MPC Problem by a maximum matching problem, in the weighted case, the method of [ 36 ] leads to one of improved complexity. This is obtained by an algorithm for the following restricted variant of the min-cost circulation problem [ 45 , 47 ]: given a directed graph, and a flow lower bound for each edge and a cost per flow unit for each edge, the task is to find a circulation of minimum total cost satisfying all lower bounds.…”

Section: Methodsmentioning

confidence: 99%

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On the complexity of Minimum Path Cover with Subpath Constraints for multi-assembly

2014

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BackgroundMulti-assembly problems have gathered much attention in the last years, as Next-Generation Sequencing technologies have started being applied to mixed settings, such as reads from the transcriptome (RNA-Seq), or from viral quasi-species. One classical model that has resurfaced in many multi-assembly methods (e.g. in Cufflinks, ShoRAH, BRANCH, CLASS) is the Minimum Path Cover (MPC) Problem, which asks for the minimum number of directed paths that cover all the nodes of a directed acyclic graph. The MPC Problem is highly popular because the acyclicity of the graph ensures its polynomial-time solvability.ResultsIn this paper, we consider two generalizations of it dealing with integrating constraints arising from long reads or paired-end reads; these extensions have also been considered by two recent methods, but not fully solved. More specifically, we study the two problems where also a set of subpaths, or pairs of subpaths, of the graph have to be entirely covered by some path in the MPC. We show that in the case of long reads (subpaths), the generalized problem can be solved in polynomial-time by a reduction to the classical MPC Problem. We also consider the weighted case, and show that it can be solved in polynomial-time by a reduction to a min-cost circulation problem. As a side result, we also improve the time complexity of the classical minimum weight MPC Problem. In the case of paired-end reads (pairs of subpaths), the generalized problem becomes NP-hard, but we show that it is fixed-parameter tractable (FPT) in the total number of constraints. This computational dichotomy between long reads and paired-end reads is also a general insight into multi-assembly problems.

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t ()(, G) MathClass-rel\in (}{, m MathClass-punc, m MathClass-bin+ 1 MathClass-punc, MathClass-op\dots MathClass-punc, ()(, \begin{matrix} arraynonefalsefalse \\ arraycenter n \\ arraycenter 2 \end{matrix}))

is the number of edges in the transitive closure of G , arising from the reduction in [29].…”

Section: Resultsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

See 2 more Smart Citations

On the complexity of Minimum Path Cover with Subpath Constraints for multi-assembly

2014

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“…An anti-chain is a subset of a partially ordered set such that any two elements in the subset are unrelated, and a chain is a totally ordered subset of a partial ordered set. Although Dilworth's Theorem is non-constructive, there exists constructive versions that solve the minimum chain cover problem either via the maximum matching problem in a bipartite graph [16] or via a max-flow problem [28]. Both problems are optimally solvable in polynomial time.…”

Section: A Polynomial-time Misp Algorithmmentioning

confidence: 99%

Automatic index selection for large-scale datalog computation

et al. 2018

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Datalog has been applied to several use cases that require very high performance on large rulesets and factsets. It is common to create indexes for relations to improve search performance. However, the existing indexing schemes either require manual index selection or result in insufficient performance on very large tasks. In this paper, we propose an automatic scheme to select indexes. We automatically create the minimum number of indexes to speed up all the searches in a given Datalog program. We have integrated our indexing scheme into an open-source Datalog engine SOUFFLÉ. We obtain performance on a par with what users have accepted from hand-optimized Datalog programs running on state-of-the-art Datalog engines, while we do not require the effort of manual index selection. Extensive experiments on large real Datalog programs demonstrate that our indexing scheme results in considerable speedups (up to 2x) and significantly less memory usage (up to 6x) compared with other automated index selections.

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Safety in Multi-Assembly via Paths Appearing in All Path Covers of a DAG

Cáceres¹,

Mumey²,

Husić³

et al. 2022

IEEE/ACM Trans. Comput. Biol. and Bioinf.

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A multi-assembly problem asks to reconstruct multiple genomic sequences from mixed reads sequenced from all of them. Standard formulations of such problems model a solution as a path cover in a directed acyclic graph, namely a set of paths that together cover all vertices of the graph. Since multi-assembly problems admit multiple solutions in practice, we consider an approach commonly used in standard genome assembly: output only partial solutions (contigs, or safe paths), that appear in all path cover solutions. We study constrained path covers, a restriction on the path cover solution that incorporate practical constraints arising in multi-assembly problems. We give efficient algorithms finding all maximal safe paths for constrained path covers. We compute the safe paths of splicing graphs constructed from transcript annotations of different species. Our algorithms run in less than 15 seconds per species and report RNA contigs that are over 99% precise and are up to 8 times longer than unitigs. Moreover, RNA contigs cover over 70% of the transcripts and their coding sequences in most cases. With their increased length to unitigs, high precision, and fast construction time, maximal safe paths can provide a better base set of sequences for transcript assembly programs.

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Another Note on Dilworth's Decomposition Theorem

Abstract: This paper proposes a new proof of Dilworth's theorem. The proof is based upon the minflow/maxcut property in flow networks. In relation to this proof, a new method to find both a Dilworth decomposition and a maximal antichain is presented.

Cited by 9 publications

References 14 publications

On the complexity of Minimum Path Cover with Subpath Constraints for multi-assembly

On the complexity of Minimum Path Cover with Subpath Constraints for multi-assembly

Automatic index selection for large-scale datalog computation

Safety in Multi-Assembly via Paths Appearing in All Path Covers of a DAG

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