Ricardo Fukasawa

Robust Branch-and-Cut-and-Price for the Capacitated Vehicle Routing Problem

The best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) have been based on either branch-and-cut or Lagrangean relaxation/column generation. This paper presents an algorithm that combines both approaches: it works over the intersection of two polytopes, one associated with a traditional Lagrangean relaxation over q-routes, the other defined by bound, degree and capacity constraints. This is equivalent to a linear program with exponentially many variables and constraints that can lead to lower bounds that are superior to those given by previous methods. The resulting branch-and-cut-and-price algorithm can solve to optimality all instances from the literature with up to 135 vertices. This more than doubles the size of the instances that can be consistently solved.

Robust branch-cut-and-price for the Capacitated Minimum Spanning Tree problem over a large extended formulation

This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to q-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arborescence problem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms: powerful new cuts expressed over a very large set of variables are added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the OR-Library show very significant improvements over previous algorithms. Several open instances could be solved to optimality.

The time dependent traveling salesman problem: polyhedra and algorithm

The time dependent traveling salesman problem (TDTSP) is a generalization of the classical traveling salesman problem (TSP), where arc costs depend on their position in the tour with respect to the source node. While TSP instances with thousands of vertices can be solved routinely, there are very challenging TDTSP instances with less than 100 vertices. In this work, we study the polytope associated to the TDTSP formulation by Picard and Queyranne, which can be viewed as an extended formulation of the TSP. We determine the dimension of the TDTSP polytope and identify several families of facet-defining cuts. We obtain good computational results with a branch-cut-and-price algorithm using the new cuts, solving almost all instances from the TSPLIB with up to 107 vertices.

Numerically Safe Gomory Mixed-Integer Cuts

We describe a simple process for generating numerically safe cutting planes using floating-point arithmetic and the mixed-integer rounding procedure. Applying this method to the rows of the simplex tableau permits the generation of Gomory mixed-integer cuts that are guaranteed to be satisfied by all feasible solutions to a mixed-integer programming problem (MIP). We report on tests with the MIPLIB 3.0 and MIPLIB 2003 test collections as well as with MIP instances derived from the TSPLIB traveling salesman library.

A Branch-Cut-and-Price Algorithm for the Energy Minimization Vehicle Routing Problem

We study a variant of the capacitated vehicle routing problem where the cost over each arc is defined as the product of the arc length and the weight of the vehicle when it traverses that arc. We propose two new mixed-integer linear programming formulations for the problem: an arc-load formulation and a set partitioning formulation based on q-routes with additional constraints. A family of cycle elimination constraints are derived for the arc-load formulation. We then compare the linear programming LP relaxations of these formulations with the two-index one-commodity flow formulation proposed in the literature. In particular, we show that the arc-load formulation with the new cycle elimination constraints gives the same LP bound as the set partitioning formulation based on 2-cycle-free q-routes, which is stronger than the LP bound given by the two-index one-commodity flow formulation. We propose a branch-and-cut algorithm for the arc-load formulation, and a branch-cut-and-price algorithm for the set partitioning formulation strengthened by additional constraints. Computational results on instances from the literature demonstrate that a significant improvement can be achieved by the branch-cut-and-price algorithm over other methods.

Strengthening lattice-free cuts using non-negativity

In recent years there has been growing interest in generating valid inequalities for mixed-integer programs using sets with two or more constraints. In particular, Andersen et al. (2007) [2] and Borozan and Cornuejols (2009) [3] have studied sets defined by equations that contain exactly one integer variable per row. The integer variables are not restricted in sign. Cutting planes based on this approach have already been computationally studied by Espinoza (2008) [8] for general mixed-integer problems, and there is ongoing computational research in this area. In this paper, we extend the model studied in the earlier papers and require the integer variables to be non-negative. We extend the results in [2] and [3] to our case, and show that cuts generated by their approach can be strengthened by using the non-negativity of the integer variables. In particular, it is possible to obtain cuts which have negative coefficients for some variables.

On the mixing set with a knapsack constraint

We study a substructure appearing in mixed-integer programming reformulations of chance-constrained programs with stochastic right-hand-sides over a finite discrete distribution, which we call the mixing set with a knapsack constraint. Recently, Luedtke et al. (Math. Program. 122(2):247–272, 2010) and Küçükyavuz (Math Program 132(1):31–56, 2012) studied valid inequalities for such sets. However, most of their results were focused on the equal probabilities case (when the knapsack constraint reduces to a cardinality constraint). In this paper, we focus on the general probabilities case (general knapsack constraint). We characterize the valid inequalities that do not come from the knapsack polytope and use this characterization to generalize the results previously derived for the equal probabilities case. Our results allow for a deep understanding of the relationship that the set under consideration has with the knapsack polytope. Moreover, they allow us to establish benchmarks that can be used to identify when a relaxation will be useful for the considered types of reformulations of chance-constrained programs.

On a generalization of the master cyclic group polyhedron

AbstractWe study the master equality polyhedron (MEP) which generalizes the master cyclic group polyhedron (MCGP) and the master knapsack polyhedron (MKP). We present an explicit characterization of the polar of the nontrivial facet-defining inequalities for MEP. This result generalizes similar results for the MCGP by Gomory (1969) and for the MKP by Araóz (1974). Furthermore, this characterization gives a polynomial time algorithm for separating an arbitrary point from MEP. We describe how facet-defining inequalities for the MCGP can be lifted to obtain facet-defining inequalities for MEP, and also present facet-defining inequalities for MEP that cannot be obtained in such a way. Finally, we study the mixed-integer extension of MEP and present an interpolation theorem that produces valid inequalities for general mixed integer programming problems using facets of MEP.

The time dependent traveling salesman problem: polyhedra and branch-cut-and-price algorithm

The Time Dependent Traveling Salesman Problem (TDTSP) is a generalization of the classical Traveling Salesman Problem (TSP), where arc costs depend on their position in the tour with respect to the source node. While TSP instances with thousands of vertices can be solved routinely, there are very challenging TDTSP instances with less than 60 vertices. In this work, we study the polytope associated to the TDTSP formulation by Picard and Queyranne, which can be viewed as an extended formulation of the TSP. We determine the dimension of the TDTSP polytope and identify several families of facet defining cuts. In particular, we also show that some facet defining cuts for the usual Asymmetric TSP formulation define low dimensional faces of the TDTSP formulation and give a way to lift them. We obtain good computational results with a branch-cut-and-price algorithm using the new cuts, solving several instances of reasonable size at the root node.

Lifting, tilting and fractional programming revisited

Lifting, tilting and fractional programming, though seemingly different, reduce to a common optimization problem. This connection allows us to revisit key properties of these three problems on mixed integer linear sets. We introduce a simple common framework for these problems, and extend known results from each to the other two.