The quadratic cycle cover problem: special cases and efficient bounds

Meijer, Frank de; Sotirov, Renata

doi:10.1007/s10878-020-00547-7

Cited by 10 publications

(21 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Fischer et al (2009) show that the problem is N P-hard. This result is later on strengthened by De Meijer and Sotirov (2020), who prove that the QCCP is strongly N P-hard and not approximable within any constant factor.…”

Section: Introductionmentioning

confidence: 78%

See 1 more Smart Citation

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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 78%

“…The linearization problem of the QCCP is considered by De Meijer and Sotirov (2020). Several sufficient conditions for a QCCP instance to be linearizable are provided, which are used to construct strong linearization-based bounds for any instance of the problem.…”

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

Self Cite

View full text Add to dashboard Cite

show abstract

“…In particular, it searches for the best under-estimator of the quadratic objective function that is in the form of a weak sum matrix. Linearization based bounds are introduced by Hu and Sotirov [14] and further exploited by de Meijer and Sotirov [9]. Further, we consider the linearized QMSTP formulation from [1], which contains an exponential number of subtour elimination constraints.…”

Section: Main Results and Outlinementioning

confidence: 99%

The quadratic minimum spanning tree problem: lower bounds via extended formulations

Sotirov¹,

Verchére²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

The quadratic minimum spanning tree problem (QMSTP) is the problem of finding a spanning tree of a graph such that the total interaction cost between pairs of edges in the tree is minimized. We first show that most of the bounding approaches for the QMSTP are closely related. Then, we exploit an extended formulation for the minimum spanning tree problem to derive a sequence of relaxations for the QMSTP with increasing complexity. The resulting relaxations differ from the relaxations in the literature. Namely, our relaxations have a polynomial number of constraints and can be efficiently solved by a cutting plane algorithm. Moreover our bounds outperform most of the bounds from the literature.

show abstract

“…This cost structure is similar to the angle-distance costs considered in Fischer et al [33] and De Meijer and Sotirov [49]. In total, we consider 9 TSPLIB instances with n ranging from 15 to 70.…”

Section: Design Of Numerical Experimentsmentioning

confidence: 99%

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

Meijer¹,

Sotirov²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

The quadratic cycle cover problem: special cases and efficient bounds

Cited by 10 publications

References 36 publications

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

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

The quadratic minimum spanning tree problem: lower bounds via extended formulations

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

Contact Info

Product

Resources

About