A Polyhedral Study of the Quadratic Traveling Salesman Problem

Fischer, Anja

doi:10.1007/978-3-319-28697-6_21

Cited by 11 publications

(26 citation statements)

References 10 publications

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“…Finally, let y = [µ T , γ T ] T , where µ and γ are the dual variables belonging to constraints (2). By substitution of these variables and combining constraints (19) and (22), we obtain the problem (DCGL ILP ), i.e., the dual of (CGL ILP ). Thus, we have v GL LBB = v GL .…”

Section: The Classical Gl Type Boundmentioning

confidence: 99%

The quadratic cycle cover problem: special cases and efficient bounds

Meijer

Sotirov

2020

J Comb Optim

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The quadratic cycle cover problem is the problem of finding a set of node-disjoint cycles visiting all the nodes such that the total sum of interaction costs between consecutive arcs is minimized. In this paper we study the linearization problem for the quadratic cycle cover problem and related lower bounds.In particular, we derive various sufficient conditions for the quadratic cost matrix to be linearizable, and use these conditions to compute bounds. We also show how to use a sufficient condition for linearizability within an iterative bounding procedure. In each step, our algorithm computes the best equivalent representation of the quadratic cost matrix and its optimal linearizable matrix with respect to the given sufficient condition for linearizability. Further, we show that the classical Gilmore-Lawler type bound belongs to the family of linearization based bounds, and therefore apply the above mentioned iterative reformulation technique. We also prove that the linearization vectors resulting from this iterative approach satisfy the constant value property.The best among here introduced bounds outperform existing lower bounds when taking both quality and efficiency into account.

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Section: The Classical Gl Type Boundmentioning

confidence: 99%

The quadratic cycle cover problem: special cases and efficient bounds

Meijer

Sotirov

2020

J Comb Optim

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“…Our benchmark instances are based on the instance specification developed in [9] and [12] and also used in [2] from where we take over the integer point sets. As in [2] we do not round the costs to integers.…”

Section: Benchmark Instancesmentioning

confidence: 99%

“…, 500}. Then we compute the turning angles α, multiply by 1000, in the same way as [9] and [2], and then we round them to 12 decimal places.…”

Section: Benchmark Instancesmentioning

confidence: 99%

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Geometric and LP-based heuristics for angular travelling salesman problems in the plane

Staněk

Greistorfer

Ladner

et al. 2019

Computers & Operations Research

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A generalization of the classical TSP is the so-called quadratic travelling salesman problem (QTSP), in which a cost coefficient is associated with the transition in every vertex, i.e. with every pair of edges traversed in succession. In this paper we consider two geometrically motivated special cases of the QTSP known from the literature, namely the angular-metric TSP, where transition costs correspond to turning angles in every vertex, and the angular-distance-metric TSP, where a linear combination of turning angles and Euclidean distances is considered.At first we introduce a wide range of heuristic approaches, motivated by the typical geometric structure of optimal solutions. In particular, we exploit lens-shaped neighborhoods of edges and a decomposition of the graph into layers of convex hulls, which are then merged into a tour by a greedy-type procedure or by utilizing an ILP model. Secondly, we consider an ILP model for a standard linearization of QTSP and compute fractional solutions of a relaxation. By rounding we obtain a collection of subtours, paths and isolated points, which are combined into a tour by various strategies, all of them involving auxiliary ILP models. Finally, different improvement heuristics are proposed, most notably a matheuristic which locally reoptimizes the solution for rectangular sectors of the given point set by an ILP approach.Extensive computational experiments for benchmark instances from the literature and extensions thereof illustrate the Pareto-efficient frontier of algorithms in a (running time, objective value)-space. It turns out that our new methods clearly dominate the previously published heuristics. *

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“…Klein [17] introduced two more inequality classes, and proved them to be facet-defining (see Theorems 6.2.2 and 6.2.3 in [17]). They are indexed by subsets S Ó and S Ò of nodes (see Figure 1), defined via and read xpErSsq`y ď 1 2 p|S|´1q for all S P S Ó and (6) xpErSsq`x e1`xe2´y ď 1 2 |S| for all S P S Ò . Klein [17] even conjectured, that Constraints (1) and (3)- (7) completely describe the mentioned face of P 1Q match .…”

Section: Introductionmentioning

confidence: 99%

Complete Description of Matching Polytopes with One Linearized Quadratic Term for Bipartite Graphs

Walter¹

2019

SIAM J. Discrete Math.

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We consider, for complete bipartite graphs, the convex hulls of characteristic vectors of all matchings, extended by a binary entry indicating whether the matching contains two specific edges. These polytopes are associated to the quadratic matching problems with a single linearized quadratic term. We provide a complete irredundant inequality description, which settles a conjecture by Klein (Ph.D. thesis, TU Dortmund, 2015). In addition, we also derive facetness and separation results for the polytopes. The completeness proof is based on a geometric relationship to a matching polytope of a nonbipartite graph. Using standard techniques, we finally extend the result to capacitated b-matchings.S Ó :" tS Ď U 9 YW : |S| odd and either S X V˚" tu 1 , u 2 u and |S X U | " |S X W |`1 or S X V˚" tw 1 , w 2 u and |S X W | " |S X U |`1u, and S Ò :" tS Ď U 9 YW : |S X U | " |S X W | and either S X V˚" tu 1 , w 2 u or S X V˚" tu 2 , w 1 uu,

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A Polyhedral Study of the Quadratic Traveling Salesman Problem

Cited by 11 publications

References 10 publications

The quadratic cycle cover problem: special cases and efficient bounds

The quadratic cycle cover problem: special cases and efficient bounds

Geometric and LP-based heuristics for angular travelling salesman problems in the plane

Complete Description of Matching Polytopes with One Linearized Quadratic Term for Bipartite Graphs

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