Distance Transformation for Network Design Problems

International audienceWe propose a new generic way to construct extended formulations for a large class of network design problems with given connectivity requirements. The approach is based on a graph transformation that maps any graph into a layered graph according to a given distance function. The original connectivity requirements are in turn transformed into equivalent connectivity requirements in the layered graph. The mapping is extended to the graphs induced by fractional vectors through an extended linear integer programming formulation. While graphs induced by binary vectors are mapped to isomorphic layered graphs, those induced by fractional vectors are mapped to a set of graphs having worse connectivity properties. Hence, the connectivity requirements in the layered graph may cut off fractional vectors that were feasible for the problem formulated in the original graph. Experiments over instances of the Steiner Forest and Hop-constrained Survivable Network Design problems show that significant gap reductions over the state-of-the art formulations can be obtained

Integer programming approaches for minimum stabbing problems

The problem of finding structures with minimum stabbing number has received considerable attention from researchers. Particularly, [10] study the minimum stabbing number of perfect matchings (mspm), spanning trees (msst) and triangulations (mstr) associated to set of points in the plane. The complexity of the mstr remains open whilst the other two are known to be . This paper presents integer programming (ip) formulations for these three problems, that allowed us to solve them to optimality through ip branch-and-bound (b&b) or branch-and-cut (b&c) algorithms. Moreover, these models are the basis for the development of Lagrangian heuristics. Computational tests were conducted with instances taken from the literature where the performance of the Lagrangian heuristics were compared with that of the exact b&b and b&c algorithms. The results reveal that the Lagrangian heuristics yield solutions with minute, and often null, duality gaps for instances with several hundreds of points in small computation times. To our knowledge, this is the first computational study ever reported in which these three stabbing problems are considered and where provably optimal solutions are given. © 2014 EDP Sciences, ROADEF, SMAI.The problem of finding structures with minimum stabbing number has received considerable attention from researchers. Particularly, [10] study the minimum stabbing number of perfect matchings (mspm), spanning trees (msst) and triangulations (mstr) associat482211233CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO301732/2007-8; 473867/2010-9; 147619/2010-607/52015-

Hybrid heuristics for a maritime short sea inventory routing problem

We consider a short sea fuel oil distribution problem where an oil company is responsible for the routing and scheduling of ships between ports such that the demand for various fuel oil products is satisfied during the planning horizon. The inventory management has to be considered at the demand side only, and the consumption rates are given and assumed to be constant within the planning horizon. The objective is to determine distribution policies that minimize the routing and operating costs, while the inventory levels are maintained within their limits. We propose an arc-load flow formulation for the problem which is tightened with valid inequalities. In order to obtain good feasible solutions for planning horizons of several months, we compare different hybridization strategies. Computational results are reported for real small-size instances

Reformulations and solution algorithms for the maximum leaf spanning tree problem

Given a graph G = (V, E), the maximum leaf spanning tree problem (MLSTP) is to find a spanning tree of G with as many leaves as possible. The problem is easy to solve when G is complete. However, for the general case, when the graph is sparse, it is proven to be NP-hard. In this paper, two reformulations are proposed for the problem. The first one is a reinforced directed graph version of a formulation found in the literature. The second recasts the problem as a Steiner arborescence problem over an associated directed graph. Branch-and-Cut algorithms are implemented for these two reformulations. Additionally, we also implemented an improved version of a MLSTP Branch-and-Bound algorithm, suggested in the literature. All of these algorithms benefit from pre-processing tests and a heuristic suggested in this paper. Computational comparisons between the three algorithms indicate that the one associated with the first reformulation is the overall best. It was shown to be faster than the other two algorithms and is capable of solving much larger MLSTP instances than previously attempted in the literature. © 2010 Springer-Verlag.73289311Aneja, Y.P., An integer linear programming approach to the Steiner problem in graphs (1980) Networks, 10, pp. 167-178Chopra, S., Gorres, E., Rao, M.R., Solving Steiner tree problem on a graph using branch and cut (1992) ORSA J Comput, 4 (3), pp. 320-335Edmonds, J., Matroids and the greedy algorithm (1971) Math Prog, 1, pp. 127-136Fernandes, M.L., Gouveia, L., Minimal spanning trees with a constraint on the number of leaves (1998) Eur J Oper Res, 104, pp. 250-261Fujie, T., An exact algorithm for the maximum-leaf spanning tree problem (2003) Comput Oper Res, 30, pp. 1931-1944Fujie, T., The maximum-leaf spanning tree problem: Formulations and facets (2004) Networks, 43 (4), pp. 212-223Galbiati, G., Maffioli, F., Morzenti, A., A short note on the approximability of the maximum leaves spanning tree problem (1994) Info Proc Lett, 52, pp. 45-49Garey, M.R., Johnson, D.S., (1979) Computers and Intractability: A Guide to the Theory of NP-Completeness, , New York: W. H. FreemanGuha, S., Khuller, S., Approximation algorithms for connected dominating sets (1998) Algorithmica, 20 (4), pp. 374-387Koch, T., Martin, A., Solving Steiner tree problems in graphs to optimality (1998) Networks, 33, pp. 207-232Lu, H., Ravi, R., Approximating maximum leaf spanning trees in almost linear time (1998) J Algo, 29, pp. 132-141Poggi de Aragão, M., Uchoa, E., Werneck, R., Dual heuristics on the exact solution of large Steiner problems (2001) Electron Notes Discret Math, 7, pp. 150-153Polzin, T., Daneshmand, S.V., Improved algorithms for the Steiner problem in networks (2001) Discret Appl Math, 112 (1-3), pp. 263-300Resende, M.G.C., Pardalos, P.M., (2006) Handbook of Optimization in Telecommunications, , New York: SpringerSolis-Oba, S., 2-approximation algorithm for finding a spanning tree with maximum number of leaves (1998) Lect Notes Comput Sci, 1461, pp. 441-452Wong, R., A dual ascent approach for Steiner tree problems on a directed graph (1984) Math Prog, 28, pp. 271-28

A Branch-and-Cut and MIP-based heuristics for the Prize-Collecting Travelling Salesman Problem

The Prize Collecting Traveling Salesman Problem (PCTSP) represents a generalization of the well-known Traveling Salesman Problem. The PCTSP can be associated with a salesman that collects a prize in each visited city and pays a penalty for each unvisited city, with travel costs among the cities. The objective is to minimize the sum of the costs of the tour and penalties, while collecting a minimum amount of prize. This paper suggests MIP-based heuristics and a branch-and-cut algorithm to solve the PCTSP. Experiments were conducted with instances of the literature, and the results of our methods turned out to be quite satisfactory

Hop-level flow formulation for the survivable network design with hop constraints problem

International audienceThe hop-constrained survivable network design problem consists of finding a minimum cost subgraph containing K edge-disjoint paths with length at most H joining each pair of vertices in a given demand set. When all demands have a common vertex, the instance is said to be rooted. We propose a new extended formulation for the rooted case, called hop-level multicommodity flow (MCF), that can be significantly stronger than the previously known formulations, at the expense of having a larger number of variables and constraints, growing linearly with the number of edges and demands and quadratically with H. However, for the particular case where H = 2, it can be specialized into a very compact and efficient formulation. Even when H = 3, hop-level-MCF can still be quite efficient and it has solved several instances from the literature for the first time

Hybrid heuristics for a short sea inventory routing problem

We consider a short sea fuel oil distribution problem where an oil company is responsible for the routing and scheduling of ships between ports such that the demand for various fuel oil products is satisfied during the planning horizon. The inventory management has to be considered at the demand side only, and the consumption rates are given and assumed to be constant within the planning horizon. The objective is to determine distribution policies that minimize the routing and operating costs, while the inventory levels are maintained within their limits. We propose an arc-load flow formulation for the problem which is tightened with valid inequalities. In order to obtain good feasible solutions for planning horizons of several months, we compare different hybridization strategies. Computational results are reported for real small-size instances