Gautier Stauffer

The stable set polytope of quasi-line graphs

It is a long standing open problem to find an explicit description of the stable set polytope of claw-free graphs. Yet more than 20 years after the discovery of a polynomial algorithm for the maximum stable set problem for claw-free graphs, there is even no conjecture at hand today.Such a conjecture exists for the class of quasi-line graphs. This class of graphs is a proper superclass of line graphs and a proper subclass of claw-free graphs for which it is known that not all facets have 0/1 normal vectors. The Ben Rebea conjecture states that the stable set polytope of a quasi-line graph is completely described by clique-family inequalities. Chudnovsky and Seymour recently provided a decomposition result for claw-free graphs and proved that the Ben Rebea conjecture holds, if the quasi-line graph is not a fuzzy circular interval graph.In this paper, we give a proof of the Ben Rebea conjecture by showing that it also holds for fuzzy circular interval graphs. Our result builds upon an algorithm of Bartholdi, Orlin and Ratliff which is concerned with integer programs defined by circular ones matrices.

On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs

We deal with non-rank facets of the stable set polytope of claw-free graphs. We extend results of Giles and Trotter [7] by (i) showing that for any nonnegative integer a there exists a circulant graph whose stable set polytope has a facet-inducing inequality with (a,a+1)-valued coefficients (rank facets have only coefficients 0, 1), and (ii) providing new facets of the stable set polytope with up to five different non-zero coefficients for claw-free graphs. We prove that coefficients have to be consecutive in any facet with exactly two different non-zero coefficients (assuming they are relatively prime). Last but not least, we present a complete description of the stable set polytope for graphs with stability number 2, already observed by Cook [3] and Shepherd [18].

Circular ones matrices and the stable set polytope of quasi-line graphs

It is a long standing open problem to find an explicit description of the stable set polytope of claw-free graphs. Yet more than 20 years after the discovery of a polynomial algorithm for the maximum stable set problem for claw-free graphs, there is even no conjecture at hand today. Such a conjecture exists for the class of quasi-line graphs. This class of graphs is a proper superclass of line graphs and a proper subclass of claw-free graphs for which it is known that not all facets have 0/1 normal vectors. Ben Rebea’s conjecture states that the stable set polytope of a quasi-line graph is completely described by clique-family inequalities. Chudnovsky and Seymour recently provided a decomposition result for claw-free graphs and proved that Ben Rebea’s conjecture holds, if the quasi-line graph is not a fuzzy circular interval graph. In this paper, we give a proof of Ben Rebea’s conjecture by showing that it also holds for fuzzy circular interval graphs. Our result builds upon an algorithm of Bartholdi, Orlin and Ratliff which is concerned with integer programs defined by circular ones matrices.

Solving the Weighted Stable Set Problem in Claw-Free Graphs via Decomposition

We propose an algorithm for solving the maximum weighted stable set problem on claw-free graphs that runs in O(|V|(|E| + |V| log|V|))-time, drastically improving the previous best known complexity bound. This algorithm is based on a novel decomposition theorem for claw-free graphs, which is also introduced in the present article. Despite being weaker than the structural results for claw-free graphs given by Chudnovsky and Seymour [2005, 2008a, 2008b] our decomposition theorem is, on the other hand, algorithmic, that is, it is coupled with an O(|V||E|)-time algorithm that actually produces the decomposition.

The p-median polytope of Y-free graphs: An application of the matching theory

Recently, Baiou and Barahona [M. Baiou, F. Barahona, On the p-median polytope of Y-free graphs. Discrete Optimization (in press, available online October 2007)] gave a characterization of the p-median polytope for Y-free graphs. In this paper, we give an alternative proof of this result by reducing the p-median problem in those graphs to a matching problem and then by building upon powerful results from the matching theory.

Constant approximation algorithms for the one warehouse multiple retailers problem with backlog or lost-sales

We consider the One Warehouse Multi-Retailer (OWMR) problem with deterministic time-varying demand in the case where shortages are allowed. Demand may be either backlogged or lost. We present a simple combinatorial algorithm to build an approximate solution from a decomposition of the system into single-echelon subproblems. We establish that the algorithm has a performance guarantee of 3 for the OWMR with backlog under mild assumptions on the cost structure. In addition, we improve this guarantee to 2 in the special case of the Joint-Replenishment Problem (JRP) with backlog. As a by-product of our approach, we show that our decomposition provides a new lower bound of the optimal cost. A similar technique also leads to a 2-approximation for the OWMR problem with lost-sales. In all cases, the complexity of the algorithm is linear in the number of retailers and quadratic in the number of time periods, which makes it a valuable tool for practical applications. To the best of our knowledge, these are the first constant approximations for the OWMR with shortages.

France Telecom Workforce Scheduling problem : A Challenge

In this paper, we describe the methodology used to tackle France Telecom workforce scheduling problem (the subject of the Roadef Challenge 2007) and we report the results obtained on the different data sets provided for the competition. Since the problem at hand appears to be NP-hard and due to the high dimensions of the instance sets, we use a two-step heuristical approach. We first devise a problem-tailored heuristic that provides good feasible solutions and then we use a meta-heuristic scheme to improve the current results. The tailored heuristic makes use of sophisticated integer programming models and the corresponding sub-problems are solved using CPLEX while the meta-heuristic framework is a randomized local search algorithm. The approach herein described allowed us to rank 5th in this challenge.

Fast Approximation Algorithms for the One-Warehouse Multi-Retailer Problem Under General Cost Structures and Capacity Constraints

We consider a well-studied multi-echelon (deterministic) inventory control problem, known in the literature as the one-warehouse multi-retailer (OWMR) problem. We propose a simple and fast 2-approximation algorithm for this NP-hard problem, by recombining the solutions of single-echelon relaxations at the warehouse and at the retailers. We then show that our approach remains valid under quite general assumptions on the cost structures and under capacity constraints at some retailers. In particular, we present the first approximation algorithms for the OWMR problem with nonlinear holding costs, truckload discount on procurement costs, or with capacity constraints at some retailers. In all cases, the procedure is purely combinatorial and can be implemented to run in low polynomial time.

Minimum clique cover in claw-free perfect graphs and the weak Edmonds-Johnson property

We give new algorithms for the minimum (weighted) clique cover in a claw-free perfect graph G, improving the complexity from O(|V(G)|5) to O(|V(G)|3). The new algorithms build upon neat reformulations of the problem: it basically reduces either to solving a 2-SAT instance (in the unweighted case) or to testing if a polyhedra associated with the edge-vertex incidence matrix of a bidirected graph has an integer solution (in the weighted case). The latter question was elegantly answered using neat polyhedral arguments by Schrijver in 1994. We give an alternative approach to this question combining pure combinatorial arguments (using techniques from 2-SAT and shortest paths) with polyhedral ones. Our approach is inspired by an algorithm from the Constraint Logic Programming community and we give as a side benefit a formal proof that the corresponding algorithm is correct (apparently answering an open question in this community). Interestingly, the systems we study have properties closely connected with the so-called Edmonds-Johnson property and we study some interesting related questions.

A constant approximation for the one-warehouse multi-retailer problem with backorder

Abstract We consider two deterministic inventory models : the One-Warehouse Multi-Retailer problem and its special case the Joint Replenishment Problem. Both models have been studied extensively in their classical version, where demand has to be satisfied on time, but there are very few results in the case where backlog is allowed. This work aims to develop a simple and fast method to provide an approximate solution for both models. More precisely, our algorithm has a worst case guarantee of three for the One-Warehouse Multi-Retailer problem with backlog. To the best of our knowledge, it is the first constant approximation for such a general and complex model. We also introduce a new lower bound based on a decomposition into single-echelon systems of these problems and improve the approximation guarantee to two in the special case of the Joint Replenishment model.