On the difficulty of finding walks of length k

Basagni, Stefano; Bruschi, Danilo; Ravasio, F.

doi:10.1051/ita/1997310504291

Cited by 3 publications

(1 citation statement)

References 7 publications

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“…Currently, the best known algorithms for matrix multiplication operate in approximately O(n 2.3 ) [4], so for large values of k the resultant complexity can be quite high at O(kn 2.3 ). Basagni et al [1] have also noted that the problem of calculating a u-v-walk of length k can be solved in polynomial time when using unweighted graphs, providing that k = n O (1) ; however, the problem is NP-hard with edge-weighted graphs.…”

Section: Introductionmentioning

confidence: 99%

A Heuristic Algorithm for Finding Attractive Fixed-Length Circuits in Street Maps

Lewis

2020

Lecture Notes in Computer Science

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In this paper we consider the problem of determining fixedlength routes on a street map that start and end at the same location. We propose a heuristic for this problem based on finding pairs of edgedisjoint shortest paths, which can then be combined into a circuit. Various heuristics and filtering techniques are also proposed for improving the algorithm's performance. Definition 2. A u-v-walk/trail/path is considered closed whenever u = v. Closed trails are usually known as circuits; closed paths are usually known as cycles.

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Section: Introductionmentioning

confidence: 99%

A Heuristic Algorithm for Finding Attractive Fixed-Length Circuits in Street Maps

Lewis

2020

Lecture Notes in Computer Science

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Finding fixed-length circuits and cycles in undirected edge-weighted graphs: an application with street networks

Lewis

Corcoran

2022

J Heuristics

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This paper proposes two heuristic algorithms for finding fixed-length circuits and cycles in undirected edge-weighted graphs. It focusses particularly on a largely unresearched practical application where we are seeking attractive round trips for pedestrians and joggers in urban street networks. Our first method is based on identifying suitable pairs of paths that are combined to form a solution; our second is based on local search techniques. Both algorithms display high levels of accuracy, producing solutions within just a few meters of the target. Run times for the local search algorithm are also short, with solutions in large cities often being found in less than one second.

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Methods for determining cycles of a specific length in undirected graphs with edge weights

Lewis,

Corcoran,

Gagarin

2023

J Comb Optim

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In this paper, we consider the $${{\mathcal{N}\mathcal{P}}}$$ N P -hard problem of determining fixed-length cycles in undirected edge-weighted graphs. Two solution methods are proposed, one based on integer programming (IP) and one that uses bespoke local search operators. These methods are executed under a common algorithmic framework that seeks to partition problem instances into a series of smaller sub-problems. Large-scale empirical tests indicate that the local search algorithm is generally preferable to IP, even with short run times. However, it can still produce suboptimal solutions, even with relatively small graphs.

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On the difficulty of finding walks of length k

Cited by 3 publications

References 7 publications

A Heuristic Algorithm for Finding Attractive Fixed-Length Circuits in Street Maps

A Heuristic Algorithm for Finding Attractive Fixed-Length Circuits in Street Maps

Finding fixed-length circuits and cycles in undirected edge-weighted graphs: an application with street networks

Methods for determining cycles of a specific length in undirected graphs with edge weights

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