Covering Tours and Cycle Covers with Turn Costs: Hardness and Approximation

Fekete, Sándor P.; Krupke, Dominik

doi:10.1007/978-3-030-17402-6_19

Cited by 7 publications

(7 citation statements)

References 21 publications

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“…Let us consider (ISDP 1 ). The set F 1 , see (28), consists of all tuples (y, X) where X represents a node-disjoint cycle cover in G. Our B&C algorithm starts with optimizing over the set F 1 , where we are allowed to relax the integrality of y at no cost, see Remark 1. If an integer point (ŷ, X) is found in the branching tree, it is verified whether λ min βI n + αJ n − 1 2 X + X ≥ 0.…”

Section: Chvátal-gomory Cuts For the Isdps Of The Qtspmentioning

confidence: 99%

“…• CG1 : In this setting we solve (ISDP 1 ) where we initially optimize over F 1 , see (28). In the separation routine we add the CG cut of the form (32) for each subtour present in the current candidate solution.…”

Section: Design Of Numerical Experimentsmentioning

confidence: 99%

“…Jäger and Molitor [43] introduce the QTSP as the problem of finding a Hamiltonian cycle in a graph that minimizes the total interaction costs among consecutive arcs. The problem is motivated by an important application in bioinformatics [43,33], but has also applications in telecommunication, precision farming and robotics, see e.g., [64,28,1]. The QTSP is N P-hard in the strong sense and is currently considered as one of the hardest combinatorial optimization problems to solve in practice.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Chvátal-Gomory Procedure for Integer SDPs with Applications in Combinatorial Optimization

Meijer¹,

Sotirov²

2022

Preprint

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In this paper we study the well-known Chvátal-Gomory (CG) procedure for the class of integer semidefinite programs (ISDPs). We prove several results regarding the hierarchy of relaxations obtained by iterating this procedure. We also study different formulations of the elementary closure of spectrahedra. A polyhedral description of the elementary closure for a specific type of spectrahedra is derived by exploiting total dual integrality for SDPs. Moreover, we show how to exploit (strengthened) CG cuts in a branch-and-cut framework for ISDPs. Different from existing algorithms in the literature, the separation routine in our approach exploits both the semidefinite and the integrality constraints. We provide separation routines for several common classes of binary SDPs resulting from combinatorial optimization problems. In the second part of the paper we present a comprehensive application of our approach to the quadratic traveling salesman problem (QTSP). Based on the algebraic connectivity of the directed Hamiltonian cycle, two ISDPs that model the QTSP are introduced. We show that the CG cuts resulting from these formulations contain several well-known families of cutting planes. Numerical results illustrate the practical strength of the CG cuts in our branch-and-cut algorithm, which outperforms alternative ISDP solvers and is able to solve large QTSP instances to optimality.

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Section: Chvátal-gomory Cuts For the Isdps Of The Qtspmentioning

confidence: 99%

Section: Design Of Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Chvátal-Gomory Procedure for Integer SDPs with Applications in Combinatorial Optimization

Meijer¹,

Sotirov²

2022

Preprint

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“…Covering tour problems play an important role in manufacturing, automatic inspection, spray-painting operations, etc. For a detailed overview of cycle cover problems with turn costs and their applications, we refer the reader to Fekete and Krupke (2019).…”

Section: Introductionmentioning

confidence: 99%

“…Several local search algorithms for the MINRC3 problem are given by Galbiati et al (2014). Approximation algorithms for the QCCP and its variants are studied by Aggarwal et al (1999), Arkin et al (2005), and Fekete and Krupke (2019).…”

Section: Introductionmentioning

confidence: 99%

SDP-Based Bounds for the Quadratic Cycle Cover Problem via Cutting-Plane Augmented Lagrangian Methods and Reinforcement Learning

Meijer

Sotirov

2021

INFORMS Journal on Computing

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We study the quadratic cycle cover problem (QCCP), which aims to find a node-disjoint cycle cover in a directed graph with minimum interaction cost between successive arcs. We derive several semidefinite programming (SDP) relaxations and use facial reduction to make these strictly feasible. We investigate a nontrivial relationship between the transformation matrix used in the reduction and the structure of the graph, which is exploited in an efficient algorithm that constructs this matrix for any instance of the problem. To solve our relaxations, we propose an algorithm that incorporates an augmented Lagrangian method into a cutting-plane framework by utilizing Dykstra’s projection algorithm. Our algorithm is suitable for solving SDP relaxations with a large number of cutting-planes. Computational results show that our SDP bounds and efficient cutting-plane algorithm outperform other QCCP bounding approaches from the literature. Finally, we provide several SDP-based upper bounding techniques, among which is a sequential Q-learning method that exploits a solution of our SDP relaxation within a reinforcement learning environment. Summary of Contribution: The quadratic cycle cover problem (QCCP) is the problem of finding a set of node-disjoint cycles covering all the nodes in a graph such that the total interaction cost between successive arcs is minimized. The QCCP has applications in many fields, among which are robotics, transportation, energy distribution networks, and automatic inspection. Besides this, the problem has a high theoretical relevance because of its close connection to the quadratic traveling salesman problem (QTSP). The QTSP has several applications, for example, in bioinformatics, and is considered to be among the most difficult combinatorial optimization problems nowadays. After removing the subtour elimination constraints, the QTSP boils down to the QCCP. Hence, an in-depth study of the QCCP also contributes to the construction of strong bounds for the QTSP. In this paper, we study the application of semidefinite programming (SDP) to obtain strong bounds for the QCCP. Our strongest SDP relaxation is very hard to solve by any SDP solver because of the large number of involved cutting-planes. Because of that, we propose a new approach in which an augmented Lagrangian method is incorporated into a cutting-plane framework by utilizing Dykstra’s projection algorithm. We emphasize an efficient implementation of the method and perform an extensive computational study. This study shows that our method is able to handle a large number of cuts and that the resulting bounds are currently the best QCCP bounds in the literature. We also introduce several upper bounding techniques, among which is a distributed reinforcement learning algorithm that exploits our SDP relaxations.

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Covering Tours and Cycle Covers with Turn Costs: Hardness and Approximation

Fekete

Krupke

2019

Lecture Notes in Computer Science

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We investigate a variety of problems of finding tours and cycle covers with minimum turn cost. Questions of this type have been studied in the past, with complexity and approximation results, and open problems dating back to work by Arkin et al. in 2001. A wide spectrum of practical applications have renewed the interest in these questions, and spawned variants: for full coverage, every point has to be covered, for subset coverage, specific points have to be covered, and for penalty coverage, points may be left uncovered by incurring an individual penalty.We make a number of contributions. We first show that finding a minimum-turn (full) cycle cover is NP-hard even in 2-dimensional grid graphs, solving the long-standing open Problem 53 in The Open Problems Project edited by Demaine, Mitchell and O'Rourke. We also prove NPhardness of finding a subset cycle cover of minimum turn cost in thin grid graphs, for which Arkin et al. gave a polynomial-time algorithm for full coverage; this shows that their boundary techniques cannot be applied to compute exact solutions for subset and penalty variants.On the positive side, we establish the first constant-factor approximation algorithms for all considered subset and penalty problem variants for very general classes of instances, making use of LP/IP techniques. For these generalized graph problems with many possible edge directions (and thus, turn angles, such as in hexagonal grids or higher-dimensional variants), our approximation factors also improve the combinatorial ones of Arkin et al. Our approach can also be extended to other geometric variants, such as scenarios with obstacles and linear combinations of turn and distance costs.

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Covering Tours and Cycle Covers with Turn Costs: Hardness and Approximation

Cited by 7 publications

References 21 publications

The Chvátal-Gomory Procedure for Integer SDPs with Applications in Combinatorial Optimization

The Chvátal-Gomory Procedure for Integer SDPs with Applications in Combinatorial Optimization

SDP-Based Bounds for the Quadratic Cycle Cover Problem via Cutting-Plane Augmented Lagrangian Methods and Reinforcement Learning

Covering Tours and Cycle Covers with Turn Costs: Hardness and Approximation

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