Vertex adjacencies in the set covering polyhedron

Aguilera, Nstor E.; Katz, Ricardo D.; Tolomei, Paola B.

doi:10.1016/j.dam.2016.10.024

Cited by 15 publications

(8 citation statements)

References 20 publications

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“…For some polytopes, there are easily verifiable necessary and sufficient conditions for the vertex adjacency in 1-skeletons. See, for example, vertex covering, partitioning, linear and partial ordering [29], set covering [1], constrained assignment problem [2], fractional stable set polytope [38], etc. Unfortunately, Papadimitriou [44] proved that verifying vertex adjacency in 1-skeleton of the travelling salesperson polytope is a hard problem.…”

Section: Travelling Salesperson Polytopementioning

confidence: 99%

Finding a second Hamiltonian decomposition of a 4-regular multigraph by integer linear programming

Nikolaev¹,

Klimov²

2022

Preprint

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A Hamiltonian decomposition of a regular graph is a partition of its edge set into Hamiltonian cycles. We consider the second Hamiltonian decomposition problem: for a 4-regular multigraph find 2 edge-disjoint Hamiltonian cycles different from the given ones. This problem arises in polyhedral combinatorics as a sufficient condition for non-adjacency in the 1-skeleton of the travelling salesperson polytope.We introduce two integer linear programming models for the problem based on the classical Dantzig-Fulkerson-Johnson and Miller-Tucker-Zemlin formulations for the travelling salesperson problem. To enhance the performance on feasible problems, we supplement the algorithm with a variable neighbourhood descent heuristic w.r.t. two neighbourhood structures, and a chain edge fixing procedure. Based on the computational experiments, the Dantzig-Fulkerson-Johnson formulation showed the best results on directed multigraphs, while on undirected multigraphs, the variable neighbourhood descent heuristic was especially effective.

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Section: Travelling Salesperson Polytopementioning

confidence: 99%

Finding a second Hamiltonian decomposition of a 4-regular multigraph by integer linear programming

Nikolaev¹,

Klimov²

2022

Preprint

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“…одномерные грани. Исследование полиэдральных графов представляет интерес, так как, с одной стороны, некоторые комбинаторные алгоритмы для таких задач как совершенное паросочетание, покрытие множества, независимое множество, ранжирование объектов, задачи с нечёткими мерами и ряд других основаны на отношении смежности вершин в полиэдральном графе и технике локального поиска (когда от текущего решения переход осуществляется к «лучшему» решению среди смежных) [9][10][11][12][13]. С другой стороны, различные характеристики полиэдрального графа задачи, такие как диаметр и кликовое число (число вершин в наибольшей клике), служат оценками сложности для различных моделей вычислений и классов алгоритмов [14][15][16].…”

Section: многогранник коммивояжёраunclassified

Backtracking Algorithms for Constructing the Hamiltonian Decomposition of a 4-regular Multigraph

Nikolaev

2021

Model. anal. inf. sist.

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We consider a Hamiltonian decomposition problem of partitioning a regular graph into edge-disjoint Hamiltonian cycles. It is known that verifying vertex non-adjacency in the 1-skeleton of the symmetric and asymmetric traveling salesperson polytopes is an NP-complete problem. On the other hand, a suffcient condition for two vertices to be non-adjacent can be formulated as a combinatorial problem of finding a Hamiltonian decomposition of a 4-regular multigraph. We present two backtracking algorithms for verifying vertex non-adjacency in the 1-skeleton of the traveling salesperson polytope and constructing a Hamiltonian decomposition: an algorithm based on a simple path extension and an algorithm based on the chain edge fixing procedure. Based on the results of the computational experiments for undirected multigraphs, both backtracking algorithms lost to the known heuristic general variable neighborhood search algorithm. However, for directed multigraphs, the algorithm based on chain fixing of edges showed comparable results with heuristics on instances with existing solutions, and better results on instances of the problem where the Hamiltonian decomposition does not exist.

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“…The 1-skeleton of a polytope P is the graph whose vertex set is the vertex set of P and edge set is the set of geometric edges or one-dimensional faces of P . The study of 1-skeleton is of interest, since, on the one hand, there are algorithms for perfect matching, set covering, independent set, a ranking of objects, problems with fuzzy measures, and many others that are based on the vertex adjacency relation in 1-skeleton and the local search technique (see, for example, [1,6,11,12,15]). On the other hand, some characteristics of 1-skeleton, such as the diameter and clique number, estimate the time complexity for different computation models and classes of algorithms [7,9,18].…”

Section: Traveling Salesperson Polytopementioning

confidence: 99%

An iterative ILP approach for constructing a Hamiltonian decomposition of a regular multigraph

Kostenko,

Nikolaev

2021

Preprint

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A Hamiltonian decomposition of a regular graph is a partition of its edge set into Hamiltonian cycles. The problem of finding edge-disjoint Hamiltonian cycles in a given regular graph has many applications in combinatorial optimization and operations research. Our motivation for this problem comes from the field of polyhedral combinatorics, as a sufficient condition for vertex nonadjacency in the 1-skeleton of the traveling salesperson polytope can be formulated as the Hamiltonian decomposition problem in a 4-regular multigraph. In our approach, the algorithm starts by solving the relaxed 2-matching problem, then iteratively generates subtour elimination constraints for all subtours in the solution and solves the corresponding ILP-model to optimality. The procedure is enhanced by the local search heuristic based on chain edge fixing and cycle merging operations. In the computational experiments, the iterative ILP algorithm showed comparable results with the previously known heuristics on undirected multigraphs and significantly better performance on directed multigraphs.

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Vertex adjacencies in the set covering polyhedron

Cited by 15 publications

References 20 publications

Finding a second Hamiltonian decomposition of a 4-regular multigraph by integer linear programming

Finding a second Hamiltonian decomposition of a 4-regular multigraph by integer linear programming

Backtracking Algorithms for Constructing the Hamiltonian Decomposition of a 4-regular Multigraph

An iterative ILP approach for constructing a Hamiltonian decomposition of a regular multigraph

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