A new view on Rural Postman based on Eulerian Extension and Matching

Sorge, Manuel; Bevern, René van; Niedermeier, Rolf; Weller, Mathias

doi:10.1016/j.jda.2012.04.007

Cited by 27 publications

(32 citation statements)

References 27 publications

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“…To this end, observe that a feasible solution T for a DRPP instance (G, c, R) enters each vertex v of G as often as it leaves. Thus, if we consider the multigraph G[T ] that contains each arc of G with same multiplicity as T, then G[T ] is a supermultigraph of G [R] in which every vertex is balanced [16,39]:…”

Section: Directed Rural Postmanmentioning

confidence: 99%

A parameterized approximation algorithm for the mixed and windy capacitated arc routing problem: Theory and experiments

2017

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We prove that any polynomial‐time α ( n ) ‐approximation algorithm for the n‐vertex metric asymmetric Traveling Salesperson Problem yields a polynomial‐time O ( α ( C ) ) ‐approximation algorithm for the mixed and windy Capacitated Arc Routing Problem, where C is the number of weakly connected components in the subgraph induced by the positive‐demand arcs—a small number in many applications. In conjunction with known results, we obtain constant‐factor approximations for C ∈ O ( log n ) and O ( log C / log log C ) ‐approximations in general. Experiments show that our algorithm, together with several heuristic enhancements, outperforms many previous polynomial‐time heuristics. Finally, since the solution quality achievable in polynomial time appears to mainly depend on C and since C = 1 in almost all benchmark instances, we propose the Ob benchmark set, simulating cities that are divided into several components by a river. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(3), 262–278 2017

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Section: Directed Rural Postmanmentioning

confidence: 99%

A parameterized approximation algorithm for the mixed and windy capacitated arc routing problem: Theory and experiments

2017

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“…Let G = ( V , E ) be a general bidirected graph. A subgraph g of G is a weakly connected component if g is a connected component when all directions are removed [ 33 ]. In addition, g is maximal if g is not a subgraph of a larger weakly connected component.…”

Section: Resultsmentioning

confidence: 99%

Inferring the global structure of chromosomes from structural variations

Yasuda

Miyano

2015

BMC Genomics

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BackgroundNext generation sequencing (NGS) technologies have made it possible to exhaustively detect structural variations (SVs) in genomes. Although various methods for detecting SVs have been developed, the global structure of chromosomes, i.e., how segments in a reference genome are extracted and ordered in an unknown target genome, cannot be inferred by detecting only individual SVs.ResultsHere, we formulate the problem of inferring the global structure of chromosomes from SVs as an optimization problem on a bidirected graph. This problem takes into account the aberrant adjacencies of genomic regions, the copy numbers, and the number and length of chromosomes. Although the problem is NP-complete, we propose its polynomial-time solvable variation by restricting instances of the problem using a biologically meaningful condition, which we call the weakly connected constraint. We also explain how to obtain experimental data that satisfies the weakly connected constraint.ConclusionOur results establish a theoretical foundation for the development of practical computational tools that could be used to infer the global structure of chromosomes based on SVs. The computational complexity of the inference can be reduced by detecting the segments of the reference genome at the ends of the chromosomes of the target genome and also the segments that are known to exist in the target genome.

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“…The Chinese Postman Problem is equivalent to finding a minimum-weight set of edges whose addition makes a connected graph Eulerian [14,20,53]. For a disconnected graph, this is exactly RPP [9,19,56].…”

Section: Related Work 121 Classical Complexitymentioning

confidence: 99%

On approximate data reduction for the Rural Postman Problem: Theory and experiments

2020

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Given an undirected graph with edge weights and a subset R of its edges, the Rural Postman Problem (RPP) is to find a closed walk of minimum total weight containing all edges of R. We prove that RPP is WK[1]-complete parameterized by the number and weight d of edges traversed additionally to the required ones. Thus RPP instances cannot be polynomial-time compressed to instances of size polynomial in d unless the polynomial-time hierarchy collapses. In contrast, denoting by b ≤ 2d the number of vertices incident to an odd number of edges of R and by c ≤ d the number of connected components formed by the edges in R, we show how to reduce any RPP instance I to an RPP instance I ′ with 2b + O(c/ε) vertices in O(n 3) time so that any α-approximate solution for I ′ gives an α(1 + ε)-approximate solution for I, for any α ≥ 1 and ε > 0. That is, we provide a polynomial-size approximate kernelization scheme (PSAKS). We experimentally evaluate it on widespread benchmark data sets as well as on two real snow plowing instances from Berlin. We also make first steps toward a PSAKS for the parameter c.

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A new view on Rural Postman based on Eulerian Extension and Matching

Cited by 27 publications

References 27 publications

A parameterized approximation algorithm for the mixed and windy capacitated arc routing problem: Theory and experiments

A parameterized approximation algorithm for the mixed and windy capacitated arc routing problem: Theory and experiments

Inferring the global structure of chromosomes from structural variations

On approximate data reduction for the Rural Postman Problem: Theory and experiments

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