Integer programming formulations for the elementary shortest path problem
•We compare integer programming formulations for the elementary shortest path problem.•We describe IP formulations with exponentially-many SECs and MIP extended formulations of polynomial size.•We study the polyhedral structure of the two strongest formulations and prove their LP bounds are equivale...
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| Published in: | European journal of operational research Vol. 252; no. 1; pp. 122 - 130 |
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| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Amsterdam
Elsevier B.V
01.07.2016
Elsevier Sequoia S.A |
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| ISSN: | 0377-2217, 1872-6860 |
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| Abstract | •We compare integer programming formulations for the elementary shortest path problem.•We describe IP formulations with exponentially-many SECs and MIP extended formulations of polynomial size.•We study the polyhedral structure of the two strongest formulations and prove their LP bounds are equivalent.•We report computational results for the LP relaxations and for the full branch-and-cut.•Formulation with dynamically generated cutset inequalities is the most effective.
Given a directed graph G=(V,A) with arbitrary arc costs, the Elementary Shortest Path Problem (ESPP) consists of finding a minimum-cost path between two nodes s and t such that each node of G is visited at most once. If negative costs are allowed, the problem is NP-hard. In this paper, several integer programming formulations for the ESPP are compared. We present analytical results based on a polyhedral study of the formulations, and computational experiments where we compare their linear programming relaxation bounds and their behavior within a branch-and-cut framework. The computational results show that a formulation with dynamically generated cutset inequalities is the most effective. |
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| AbstractList | Given a directed graph G=(V,A)G=(V,A) with arbitrary arc costs, the Elementary Shortest Path Problem (ESPP) consists of finding a minimum-cost path between two nodes s and t such that each node of G is visited at most once. If negative costs are allowed, the problem is NPNP-hard. In this paper, several integer programming formulations for the ESPP are compared. We present analytical results based on a polyhedral study of the formulations, and computational experiments where we compare their linear programming relaxation bounds and their behavior within a branch-and-cut framework. The computational results show that a formulation with dynamically generated cutset inequalities is the most effective. •We compare integer programming formulations for the elementary shortest path problem.•We describe IP formulations with exponentially-many SECs and MIP extended formulations of polynomial size.•We study the polyhedral structure of the two strongest formulations and prove their LP bounds are equivalent.•We report computational results for the LP relaxations and for the full branch-and-cut.•Formulation with dynamically generated cutset inequalities is the most effective. Given a directed graph G=(V,A) with arbitrary arc costs, the Elementary Shortest Path Problem (ESPP) consists of finding a minimum-cost path between two nodes s and t such that each node of G is visited at most once. If negative costs are allowed, the problem is NP-hard. In this paper, several integer programming formulations for the ESPP are compared. We present analytical results based on a polyhedral study of the formulations, and computational experiments where we compare their linear programming relaxation bounds and their behavior within a branch-and-cut framework. The computational results show that a formulation with dynamically generated cutset inequalities is the most effective. Given a directed graph G=(V,A) with arbitrary arc costs, the Elementary Shortest Path Problem (ESPP) consists of finding a minimum-cost path between two nodes s and t such that each node of G is visited at most once. If negative costs are allowed, the problem is NP-hard. In this paper, several integer programming formulations for the ESPP are compared. We present analytical results based on a polyhedral study of the formulations, and computational experiments where we compare their linear programming relaxation bounds and their behavior within a branch-and-cut framework. The computational results show that a formulation with dynamically generated cutset inequalities is the most effective. |
| Author | Taccari, Leonardo |
| Author_xml | – sequence: 1 givenname: Leonardo surname: Taccari fullname: Taccari, Leonardo email: leonardo.taccari@polimi.it, leonardo.taccari@gmail.com organization: Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Italy |
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| Keywords | Integer programming Subtour elimination constraints Extended formulations Branch-and-cut Elementary shortest path |
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| Snippet | •We compare integer programming formulations for the elementary shortest path problem.•We describe IP formulations with exponentially-many SECs and MIP... Given a directed graph G=(V,A) with arbitrary arc costs, the Elementary Shortest Path Problem (ESPP) consists of finding a minimum-cost path between two nodes... Given a directed graph G=(V,A)G=(V,A) with arbitrary arc costs, the Elementary Shortest Path Problem (ESPP) consists of finding a minimum-cost path between two... |
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| SubjectTerms | Branch-and-cut Comparative analysis Computation Costs Effectiveness studies Elementary shortest path Extended formulations Formulations Graph theory Graphs Inequalities Integer programming Linear programming Mathematical analysis Mathematical problems Polyhedra Shortest path algorithms Shortest-path problems Subtour elimination constraints |
| Title | Integer programming formulations for the elementary shortest path problem |
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