Mathieu Lacroix

Robust location transportation problems under uncertain demands

In robust optimization, the multi-stage context (or dynamic decision-making) assumes that the information is revealed in stages. So, part of the decisions must be taken before knowing the real values of the uncertain parameters, and another part, called recourse decisions, is taken when the information is known. In this paper, we are interested in a robust version of the location transportation problem with an uncertain demand using a 2-stage formulation. The obtained robust formulation is a convex (not linear) program, and we apply a cutting plane algorithm to exactly solve the problem. At each iteration, we have to solve an NP-hard recourse problem in an exact way, which is time-consuming. Here, we go further in the analysis of the recourse problem of the location transportation problem. In particular, we propose a mixed integer program formulation to solve the quadratic recourse problem and we define a tight bound for this reformulation. We present an extensive computation analysis of the 2-stage robust location transportation problem. The proposed tight bound allows us to solve large size instances.

The splittable pickup and delivery problem with reloads

In this paper, we consider a variant of the Pickup and Delivery Problem (PDP), where any demand may be dropped off elsewhere other than its destination, picked up later by the same or another vehicle, and so on until it has reached its destination. We discuss the complexity of this problem and present two mixed-integer linear programming formulations based on a space-time graph. We describe some valid inequalities for the problem along with separation routines. Based on these results, we develop a branch-and-cut algorithm for the problem, and present some computational results. [Received 31 January 2007; Revised 08 August 2007; Accepted 02 October 2007]

On the NP-completeness of the perfect matching free subgraph problem

Given a bipartite graph G=(U@?V,E) such that |U|=|V| and every edge is labelled true or false or both, the perfect matching free subgraph problem is to determine whether or not there exists a subgraph of G containing, for each node u of U, either all the edges labelled true or all the edges labelled false incident to u, and which does not contain a perfect matching. This problem arises in the structural analysis of differential-algebraic systems. The purpose of this paper is to show that this problem is NP-complete. We show that the problem is equivalent to the stable set problem in a restricted case of tripartite graphs. Then we show that the latter remains NP-complete in that case. We also prove the NP-completeness of the related minimum blocker problem in bipartite graphs with perfect matching.

Models for the single-vehicle preemptive pickup and delivery problem

In this paper, we study a variant of the well-known single-vehicle pickup and delivery problem where the demands can be unloaded/reloaded at any node. By proving new complexity results, we give the minimum information which is necessary to represent feasible solutions. Using this, we present integer linear programs for both the unitary and the general versions. We then show that the associated linear relaxations are polynomial-time solvable and present some computational results.

Polyhedral results and a branch-and-cut algorithm for the double traveling Salesman problem with multiple stacks

Abstract In the double TSP with multiple stacks, one performs a Hamiltonian circuit to pick up n items, storing them in a vehicle with s stacks of finite capacity q satisfying last-in-first-out constraints, and then delivers every item by performing a Hamiltonian circuit. We introduce an integer linear programming formulation with arc and precedence variables. We show that the underlying polytope shares some polyhedral properties with the ATSP polytope, which let us characterize large number of facets of our polytope. We convert these theoretical results into a branch-and-cut algorithm for the double TSP with two stacks. Our algorithm outperforms the existing exact methods and solves instances that were previously unsolved.

The uncapacitated asymmetric traveling salesman problem with multiple stacks

In the uncapacitated asymmetric traveling salesman problem with multiple stacks, one first performs a hamiltonian circuit to pick up n items, storing them in a vehicle with k stacks satisfying last-in-first-out constraints, and then delivers every item by performing a second hamiltonian circuit. Here, we are interested in the convex hull of the arc-incidence vectors of such couples of hamiltonian circuits. For the general case, we determine the dimension of this polytope, and show that every facet of the asymmetric traveling salesman polytope defines one of its facets. For the special case with two stacks, we provide an integer linear programming formulation whose linear relaxation is polynomial-time solvable, and we propose new families of valid inequalities to reinforce the latter.

Dependency Parsing with Bounded Block Degree and Well-nestedness via Lagrangian Relaxation and Branch-and-Bound

We present a novel dependency parsing method which enforces two structural properties on dependency trees: bounded block degree and well-nestedness. These properties are useful to better represent the set of admissible dependency structures in treebanks and connect dependency parsing to context-sensitive grammatical formalisms. We cast this problem as an Integer Linear Program that we solve with Lagrangian Relaxation from which we derive a heuristic and an exact method based on a Branch-and-Bound search. Experimentally, we see that these methods are efficient and competitive compared to a baseline unconstrained parser, while enforcing structural properties in all cases.

Mathematical formulations for the Balanced Vertex k-Separator Problem

Given an indirected graph G = (V;E), a Vertex k-Separator is a subset of the vertex set V such that, when the separator is removed from the graph, the remaining vertices can be partitioned into k subsets that are pairwise edge-disconnected. In this paper we focus on the Balanced Vertex k-Separator Problem, i.e., the problem of finding a minimum cardinality separator such that the sizes of the resulting disconnected subsets are balanced. We present a compact Integer Linear Programming formulation for the problem, and present a polyhedral study of the associated polytope. We also present an Exponential-Size formulation, for which we derive a column generation and a branching scheme. Preliminary computational results are reported comparing the performance of the two formulations on a set of benchmark instances.

Combinatorial optimization model and MIP formulation for the structural analysis of conditional differential-algebraic systems

In this paper we consider the structural analysis problem for differential-algebraic systems with conditional equations. This problem consists, given a conditional differential-algebraic system, in verifying if the system is structurally nonsingular for every state, and if not in finding a state in which the system is structurally singular. We give a formulation for this problem as an integer linear program. This is based on a transformation of the problem into a matching problem in an auxiliary graph. We also show that the linear relaxation of that formulation can be solved in polynomial time. Using this, we develop a Branch-and-Cut algorithm for solving the problem and present some experimental results.

Recourse problem of the 2-stage robust location transportation problem

In this paper, we are interested in the recourse problem of the 2-stage robust location transportation problem. We propose a solution process using a mixed-integer formulation with an appropriate tight bound.