Branch and bound for the cutwidth minimization problem

Martı́, Rafael; Pantrigo, Juan José; Duarte, Abraham

doi:10.1016/j.cor.2012.05.016

Cited by 20 publications

(7 citation statements)

References 30 publications

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“…Aside from using a MIP-based approach directly to solve the problem, a combinatorial branch-and-bound algorithm could be an interesting idea. Such approaches, which typically work with partial labelings and use problem-specific, graph-theoretic bounds for the problem at hand often are often quite effective for graph labeling problems (Caprara and Salazar-González 2005;Martí et al 2010Martí et al , 2013. In case of the ABP, even solving NP-hard problems within the branch-and-bound to provide bounds could be a viable option, since our computational study showed, that the stable set problem and graph coloring problem can be solved very quickly for the standard benchmark instances of the ABP.…”

Section: Discussionmentioning

confidence: 99%

“…In such problems, we are given a graph and we want to find a labeling (i.e., a numbering of its vertices), such that a given objective function is optimized. Problems in this class include the bandwidth problem (Cuthill and McKee 1969;Caprara and Salazar-González 2005) and variants of it like cyclic bandwidth (Rodriguez-Tello et al 2015), the linear arrangement problem (Caprara et al 2011;Rodriguez-Tello et al 2008) and the cutwidth problem (Martí et al 2013), see also the surveys (Díaz et al 2002;Gallian 2009). In this work, we consider the antibandwidth problem (ABP), also known as dual bandwidth problem (Yixun and JinJiang 2003), separation problem (Miller and Pritikin 1989) and maximum differential coloring problem (Bekos et al 2014).…”

Section: Introductionmentioning

confidence: 99%

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A note on computational approaches for the antibandwidth problem

Sinnl

2020

Cent Eur J Oper Res

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In this note, we consider the antibandwidth problem, also known as dual bandwidth problem, separation problem and maximum differential coloring problem. Given a labeled graph (i.e., a numbering of the vertices of a graph), the antibandwidth of a node is defined as the minimum absolute difference of its labeling to the labeling of all its adjacent vertices. The goal in the antibandwidth problem is to find a labeling maximizing the antibandwidth. The problem is NP-hard in general graphs and has applications in diverse areas like scheduling, radio frequency assignment, obnoxious facility location and map-coloring. There has been much work on deriving theoretical bounds for the problem and also in the design of metaheuristics in recent years. However, the optimality gaps between the best known solution values and reported upper bounds for the HarwellBoeing Matrix-instances, which are the commonly used benchmark instances for this problem, are often very large (e.g., up to 577%). Moreover, only for three of these 24 instances, the optimal solution is known, leading the authors of a state-of-the-art heuristic to conclude "HarwellBoeing instances are actually a challenge for modern heuristic methods". The upper bounds reported in literature are based on the theoretical bounds involving simple graph characteristics, i.e., size, order and degree, and a mixed-integer programming (MIP) model. We present new MIP models for the problem, together with valid inequalities, and design a branch-and-cut algorithm and an iterative solution algorithm based on them. These algorithms also include two starting heuristics and a primal heuristic. We also present a constraint programming approach, and calculate upper bounds based on the stability number and chromatic number. Our computational study shows that the developed approaches allow to find the proven optimal solution for eight instances from literature, where the optimal solution was unknown and also provide reduced gaps for eleven additional instances, including improved solution values for seven instances, the largest optimality gap is now 46%.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A note on computational approaches for the antibandwidth problem

Sinnl

2020

Cent Eur J Oper Res

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“…These optimization criteria ensure that the successor relations are mostly kept intact and at the same time successors are placed close to each other in the solution. Both problems the Minimum Feedback Arc Set Problem [ 37 ] and the Average Cut-Width Minimization Problem [ 59 – 61 ] are known to be NP-hard. Cutwidth minimization problems ask for a linear ordering of the vertices of a graph such that the average or maximum number of edges spanning across the gap between a pair of consecutive vertices is minimized.…”

Section: Theorymentioning

confidence: 99%

Coordinate systems for supergenomes

Gärtner

Siederdissen

Müller

et al. 2018

Algorithms Mol Biol

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BackgroundGenome sequences and genome annotation data have become available at ever increasing rates in response to the rapid progress in sequencing technologies. As a consequence the demand for methods supporting comparative, evolutionary analysis is also growing. In particular, efficient tools to visualize-omics data simultaneously for multiple species are sorely lacking. A first and crucial step in this direction is the construction of a common coordinate system. Since genomes not only differ by rearrangements but also by large insertions, deletions, and duplications, the use of a single reference genome is insufficient, in particular when the number of species becomes large.ResultsThe computational problem then becomes to determine an order and orientations of optimal local alignments that are as co-linear as possible with all the genome sequences. We first review the most prominent approaches to model the problem formally and then proceed to showing that it can be phrased as a particular variant of the Betweenness Problem. It is NP hard in general. As exact solutions are beyond reach for the problem sizes of practical interest, we introduce a collection of heuristic simplifiers to resolve ordering conflicts.ConclusionBenchmarks on real-life data ranging from bacterial to fly genomes demonstrate the feasibility of computing good common coordinate systems.Electronic supplementary materialThe online version of this article (10.1186/s13015-018-0133-4) contains supplementary material, which is available to authorized users.

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“…The problem of minimizing the average cut width [6] is also known to be NP-hard. A good heuristic [7] is necessary for good results. In our case, starting our procedure with the "primary path" taken by the reference genome is a natural choice.…”

Section: Problem Statementmentioning

confidence: 99%

“…The appendix provides a dependence of the number of feedback arcs and cut widths of number of variations of the same type. For this study, the number of variations of all types, except the examined, were fixed at level 7, and the number of investigated variations were changing according to the following list: 7,9,11,13,15,17…”

Section: Test Data Set Modellingmentioning

confidence: 99%

A Flow Procedure for the Linearization of Genome Sequence Graphs

Haussler¹,

Smuga-Otto²,

Paten³

et al. 2017

Preprint

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Efforts to incorporate human genetic variation into the reference human genome have converged on the idea of a graph representation of genetic variation within a species, a genome sequence graph. A sequence graph represents a set of individual haploid reference genomes as paths in a single graph. When that set of reference genomes is sufficiently diverse, the sequence graph implicitly contains all frequent human genetic variations, including translocations, inversions, deletions, and insertions.In representing a set of genomes as a sequence graph one encounters certain challenges. One of the most important is the problem of graph linearization, essential both for efficiency of storage and access, as well as for natural graph visualization and compatibility with other tools. The goal of graph linearization is to order nodes of the graph in such a way that operations such as access, traversal and visualization are as efficient and effective as possible.A new algorithm for the linearization of sequence graphs, called the flow procedure, is proposed in this paper. Comparative experimental evaluation of the flow procedure against other algorithms shows that it outperforms its rivals in the metrics most relevant to sequence graphs. Keywordssequence graph · linearization · flow procedure · feedback arcs · cut width · backbone · grooming . CC-BY 4.0 International license peer-reviewed) is the author/funder. It is made available under a The copyright holder for this preprint (which was not . http://dx.doi.org/10.1101/101501 doi: bioRxiv preprint first posted online Jan. 18, 2017; 2 MotivationThe current human reference genome consists essentially of a single representative of each of the human chromosomes. In essence, an arbitrary person's genome is chosen to represent all of humanity. This leads to loss of information and bias. Efforts to incorporate human genetic variation into the reference human genome have converged on the idea of a graph representation of genetic variation within a species, a genome sequence graph [1]. In its mathematically most simple form, each node of a sequence graph contains a single DNA base that occurs at an orthologous locus in one or more of the haploid genomes represented in the graph. Each arc represents an adjacency (chemically, a covalent bond) that occurs between consecutive instances of bases in those genomes. Excluding reversing joins (see below) each arc is directed according to the default strand direction of the DNA sequence used to build the graph, connecting the 3' side of the previous base (the tail of the edge) with the 5' side of the next base (the head of the edge). At points where the individual genomes differ to the right (i.e., 3', downstream) of an orthologous base, the node representing that base will have two or more outgoing arcs. For example, the graph in Figure 2 was built from the DNA strands CATCGCT, CATCGT, CAGCGAT and CATCGAGAGAGCT aligned at orthologous bases as shown in Figure 1. Figure 2. An example of a sequence graph reflecting SNP (T/G), tandem...

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Branch and bound for the cutwidth minimization problem

Cited by 20 publications

References 30 publications

A note on computational approaches for the antibandwidth problem

A note on computational approaches for the antibandwidth problem

Coordinate systems for supergenomes

A Flow Procedure for the Linearization of Genome Sequence Graphs

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