New Models for and Numerical Tests of the Hamiltonian p-Median Problem

Gollowitzer, Stefan; Pereira, Dilson Lucas; Wojciechowski, Adam

doi:10.1007/978-3-642-21527-8_43

Cited by 5 publications

(20 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The third model consists of assignment constraints (deg) and the two sets of constraints introduced in this section. This latter model has already been proposed in Gollowitzer et al [10]. When p = 1, constraint sets (nat≤) and (z≤) are equivalent and, as noted before, constraints (nat≥) and (z≥) are redundant.…”

Section: Assigning Nodes To Cyclesmentioning

confidence: 82%

“…iii. As pointed out in [10], the partition inequalities for more than p + 1 subsets can also be strengthened by adding one unit to the right-hand side of the inequalities implied by (4). Figure 3a, v i = 1/6 for all i ∈ I 12 and v ij = 1/6 for i, j ∈ S k and v ij = 1/24 for i ∈ S k , j ∈ S l and k = l. Then, (x, v, y) ∈ PM ≤ but constraint (nat≤) for P = {S 1 , S 2 , S 3 } is violated, thus x / ∈ NAT ≤ .…”

Section: Corollary 2 Lrmentioning

confidence: 95%

“…As proposed in [10], we apply symmetry breaking constraints to variables z. We require that the cycles are numbered according to the smallest node index in each respective cycle.…”

Section: Symmetry Breaking and Variable Reductionmentioning

confidence: 99%

“…We use a simple primal heuristic originally proposed in [10] to search for incumbent solutions. It is called at every node of the B&C search tree and consists of the following steps:…”

Section: Primal Heuristicmentioning

confidence: 99%

“…We tried using the heuristic algorithm of [10] in preliminary experiments, but very few violated (nat≥) inequalities were found with this procedure and it was not included in the final code.…”

mentioning

confidence: 99%

See 4 more Smart Citations

A comparison of several models for the hamiltonian p‐median problem

et al. 2014

Self Cite

View full text Add to dashboard Cite

The Hamiltonian p-median problem consists of determining p disjoint cycles of minimum total cost covering all vertices of a graph. We present several new and existing models for this problem, provide a hierarchy with respect to the quality of the lower bounds yielded by their linear programming relaxations, and compare their computational performance on a set of benchmark instances. We conclude that three of the models are superior from a computational point of view, two of which are introduced in this article.

show abstract

Section: Assigning Nodes To Cyclesmentioning

confidence: 82%

Section: Corollary 2 Lrmentioning

confidence: 95%

“…As proposed in [10], we apply symmetry breaking constraints to variables z. We require that the cycles are numbered according to the smallest node index in each respective cycle.…”

Section: Symmetry Breaking and Variable Reductionmentioning

confidence: 99%

“…We use a simple primal heuristic originally proposed in [10] to search for incumbent solutions. It is called at every node of the B&C search tree and consists of the following steps:…”

Section: Primal Heuristicmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

A comparison of several models for the hamiltonian p‐median problem

et al. 2014

Self Cite

View full text Add to dashboard Cite

show abstract

Node based compact formulations for the Hamiltonian p‐median problem

et al. 2023

View full text Add to dashboard Cite

In this paper, we introduce, study and analyze several classes of compact formulations for the symmetric Hamiltonian ‐median problem (HMP). Given a positive integer and a weighted complete undirected graph with weights on the edges, the HMP on is to find a minimum weight set of elementary cycles partitioning the vertices of . The advantage of developing compact formulations is that they can be readily used in combination with off‐the‐shelf optimization software, unlike other types of formulations possibly involving the use of exponentially sized sets of variables or constraints. The main part of the paper focuses on compact formulations for eliminating solutions with less than cycles. Such formulations are less well known and studied than formulations which prevent solutions with more than cycles. The proposed formulations are based on a common motivation, that is, the formulations contain variables that assign labels to nodes, and prevent less than cycles by stating that different depots must have different labels and that nodes in the same cycle must have the same label. We introduce and study aggregated formulations (which consider integer variables that represent the label of the node) and disaggregated formulations (which consider binary variables that assign each node to a given label). The aggregated models are new. The disaggregated formulations are not, although in all of them new enhancements have been included to make them more competitive with the aggregated models. The two main conclusions of this study are: (i) in the context of compact formulations, it is worth looking at the models with integer node variables, which have a smaller size. Despite their weaker LP relaxation bounds, the fewer variables and constraints lead to faster integer resolution, especially when solving instances with more than 50 nodes; (ii) the best of our compact models exhibit a performance that, overall, is comparable to that of the best methods known for the HMP (including branch‐and‐cut algorithms), solving to optimality instances with up to 226 nodes within 1 h. This corroborates our message that the knowledge of the inequalities for preventing less than cycles is much less well understood.

show abstract

A hybrid genetic algorithm for the Hamiltonian p‐median problem

He,

Hao,

2023

Networks

View full text Add to dashboard Cite

The Hamiltonian p‐median problem consists of finding p( is given) non‐intersecting Hamiltonian cycles in a complete edge‐weighted graph such that each cycle visits at least three vertices and each vertex belongs to exactly one cycle, while minimizing the total cost of pcycles. In this work, we present an effective and scalable hybrid genetic algorithm to solve this computationally challenging problem. The algorithm combines an edge‐assembly crossover to generate promising offspring solutions from high‐quality parents, and a multiple neighborhood local search to improve each offspring solution. To promote population diversity, the algorithm applies a mutation operator to the offspring solutions and a quality‐and‐distance update strategy to manage the population. We compare the method to the best reference algorithms in the literature based on three sets of 145 popular benchmark instances (with up to 318 vertices), and report improved best upper bounds for eight instances. To evaluate the scalability of the method, we perform experiments on a new set of 70 large instances (with up to 1060 vertices). We examine the contributions of key components of the algorithm.

show abstract

New Models for and Numerical Tests of the Hamiltonian p-Median Problem

Cited by 5 publications

References 5 publications

A comparison of several models for the hamiltonian p‐median problem

A comparison of several models for the hamiltonian p‐median problem

Node based compact formulations for the Hamiltonian p‐median problem

A hybrid genetic algorithm for the Hamiltonian p‐median problem

Contact Info

Product

Resources

About