Pierre Pesneau

Two Edge-Disjoint Hop-Constrained Paths and Polyhedra

Given a graph G with distinguished nodes s and t, a cost on each edge of G, and a fixed integer L \geq 2, the two edge-disjoint hop-constrained paths problem is to find a minimum cost subgraph such that between s and t there exist at least two edge-disjoint paths of length at most L. In this paper, we consider that problem from a polyhedral point of view. We give an integer programming formulation for the problem when L = 2,3. An extension of this result to the more general case where the number of required paths is arbitrary and L = 2,3 is also given. We discuss the associated polytope, P(G,L), for L = 2,3. In particular, we show in this case that the linear relaxation of P(G,L), Q(G,L), given by the trivial, the st-cut, and the so-called L-path-cut inequalities, is integral. As a consequence, we obtain a polynomial time cutting plane algorithm for the problem when L = 2,3. We also give necessary and sufficient conditions for these inequalities to define facets of P(G,L) for L \geq 2 when G is complete. We finally investigate the dominant of P(G,L) and give a complete description of this polyhedron for L \geq 2 when P(G,L) = Q(G,L).

Two-edge connected subgraphs with bounded rings: Polyhedral results and Branch-and-Cut

We consider the network design problem which consists in determining at minimum cost a 2-edge connected network such that the shortest cycle (a “ring”) to which each edge belongs, does not exceed a given length K. We identify a class of inequalities, called cycle inequalities, valid for the problem and show that these inequalities together with the so-called cut inequalities yield an integer programming formulation of the problem in the space of the natural design variables. We then study the polytope associated with that problem and describe further classes of valid inequalities. We give necessary and sufficient conditions for these inequalities to be facet defining. We study the separation problem associated with these inequalities. In particular, we show that the cycle inequalities can be separated in polynomial time when K≤4. We develop a Branch-and-Cut algorithm based on these results and present extensive computational results.

Natural and extended formulations for the Time-Dependent Traveling Salesman Problem

In this paper, we present a new formulation for the Time-Dependent Traveling Salesman Problem (TDTSP). We start by reviewing well known natural formulations with some emphasis on the formulation by Picard and Queyranne (1978) [22]. The main feature of this formulation is that it uses, as a subproblem, an exact description of the n-circuit problem. Then, we present a new formulation that uses more variables and is based on using, for each node, a stronger subproblem, namely an n-circuit subproblem with the additional constraint that the corresponding node is not repeated in the circuit. Although the new model has more variables and constraints than the original PQ model, the results given from our computational experiments show that the linear programming relaxation of the new model gives, for many of the instances tested, gaps that are close to zero. Thus, the new model is worth investigating for solving TDTSP instances. We have also provided a complete characterization of the feasible set of the corresponding linear programming relaxation in the space of the variables of the PQ model. This characterization permits us to suggest alternative methods of using the proposed formulations.

On the k edge-disjoint 2-hop-constrained paths polytope

The k edge-disjoint 2-hop-constrained paths problem consists in finding a minimum cost subgraph such that between two given nodes s and t there exist at least k edge-disjoint paths of at most 2 edges. We give an integer programming formulation for this problem and characterize the associated polytope.

ON THE STEINER 2-EDGE CONNECTED SUBGRAPH POLYTOPE

In this paper, we study the Steiner 2-edge connected subgraph polytope. We introduce a large class of valid inequalities for this polytope called the generalized Steiner F-partition inequalities, that generalizes the so-called Steiner F-partition inequalities. We show that these inequalities together with the trivial and the Steiner cut inequalities completely describe the polytope on a class of graphs that generalizes the wheels. We also describe necessary conditions for these inequalities to be facet defining, and as a consequence, we obtain that the separation problem over the Steiner 2-edge connected subgraph polytope for that class of graphs can be solved in polynomial time. Moreover, we discuss that polytope in the graphs that decompose by 3-edge cutsets. And we show that the generalized Steiner F-partition inequalities together with the trivial and the Steiner cut inequalities suffice to describe the polytope in a class of graphs that generalizes the class of Halin graphs when the terminals have a particular disposition. This generalizes a result of Barahona and Mahjoub [4] for Halin graphs. This also yields a polynomial time cutting plane algorithm for the Steiner 2-edge connected subgraph problem in that class of graphs.

Integer Programming Formulations for the k-Edge-Connected 3-Hop-Constrained Network Design Problem

In this article, we study the k-edge-connected L-hop-constrained network design problem. Given a weighted graph G = (V,E), a set D of pairs of nodes, two integers L ≥ 2 and k ≥ 2, the problem consists in finding a minimum weight subgraph of G containing at least k edge-disjoint paths of length at most L between every pair {s, t } ∈ D. We consider the problem in the case where L = 2, 3 and |D| ≥ 2. We first discuss integer programming formulations introduced in the literature. Then, we introduce new integer programming formulations for the problem that are based on the transformation of the initial undirected graph into directed layered graphs. We present a theoretical comparison of these formulations in terms of LP-bound. Finally, these formulations are tested using CPLEX and compared in a computational study for k = 3, 4, 5.

A multi scalable model based on a connexity graph representation

Train operations will be greatly enhanced with the development of new decision support systems. However, when considering problems such as online rescheduling of trains, experience shows a pitfall in the communication between the different elements that compose them, namely simulation software (in charge of projection, conflict detection, validation) and optimization tools (in charge of scheduling and decision making). The main problem is the inadequacy of the infrastructures monolithic description and the inability to manage together different description levels. Simulation uses a very precise description, while the optimization of a mathematical problem usually does not. Indeed, an exhaustive description of the whole network is usually counter-productive in optimization problems; the description must be accurate, but should rely on a less precise representation. Unfortunately, the usual model representing the railway system does not guarantee compatibility between two different description levels; a representation usually corresponds to a given (unique) description level, designed in most cases with a specific application in mind, such as platforming. Moreover, further modifications that could improve performances or precision are usually impossible. We propose, therefore, a model with a new description of the infrastructure that permits one to scroll between different description levels. These operations can be automated via dynamic aggregation and disaggregation methods. They allow one to manage heterogeneous descriptions and cooperation between various tools using different description levels. This model is based on the connexity graph representation of the infrastructure resources. We will present how to generate corresponding mathematical models based on resource occupancy and will show how the aggregation of resources leads to the aggregation of properties (e.g. capacity) that can be translated into mathematical constraints in the optimization problem.

Reliable Service Allocation in Clouds with Memory and Capacity Constraints

We consider allocation problems that arise in the context of service allocation in Clouds. More specifically, on the one part we assume that each Physical Machine (denoted as PM) is offering resources (memory, CPU, disk, network). On the other part, we assume that each application in the IaaS Cloud comes as a set of services running as Virtual Machines (VMs) on top of the set of PMs. In turn, each service requires a given quantity of each resource on each machine where it runs (memory footprint, CPU, disk, network). Moreover, there exists a Service Level Agreement (SLA) between the Cloud provider and the client that can be expressed as follows: the client requires a minimal number of service instances which must be alive at the end of a time period, with a given reliability (that can be converted into penalties paid by the provider). In this context, the goal for the Cloud provider is to find an allocation of VMs onto PMs so as to satisfy, at minimal cost, both capacity and reliability constraints for each service. In this paper, we propose a simple model for reliability constraints and we prove that it is possible to derive efficient heuristics.

Circuit and bond polytopes on series-parallel graphs

In this paper, we describe the circuit polytope on series-parallel graphs. We first show the existence of a compact extended formulation. Though not being explicit, its construction process helps us to inductively provide the description in the original space. As a consequence, using the link between bonds and circuits in planar graphs, we also describe the bond polytope on series-parallel graphs.

Hop-indexed Circuit-based formulations for the Traveling Salesman Problem

Abstract We discuss a new Hop-indexed Circuit-based formulation for the Traveling Salesman Problem (TSP). We show that the new formulation enhanced with some valid inequalities dominates the previous best (compact) formulations from the literature and that it produces very tight linear bounds (with emphasis on the so-called cumulative TSP).