International audience

Abstract. Linear complementarity programming is a generalization of linear programming which encompasses the computation of Nash equilibria for bimatrix games. While the latter problem is PPAD-complete, we show that the tropical analogue of the complementarity problem associated with Nash equilibria can be solved in polynomial time. Moreover, we prove that the Lemke–Howson algorithm carries over the tropical setting and performs a linear number of pivots in the worst case. A consequence of this result is a new class of (classical) bimatrix games for which Nash equilibria computation can be done in polynomial time.

Haviv [ European J. Combin., 81 (2019), pp. 84--97] has recently proved that some topological lower bounds on the chromatic number of graphs are also lower bounds on their orthogonality dimension over $\mathbb{R}$. We show that this actually holds for all known topological lower bounds and all fields. We also improve the topological bound he obtained for the minrank parameter over $\mathbb{R}$---an important graph invariant from coding theory---and show that this bound is actually valid for all fields as well. The notion of independent representation over a matroid is introduced and used in a general theorem having these results as corollaries. Related complexity results are also discussed.

Within a partnership with a consulting company, we address a production problem modeled as a stochastic multi-item lot-sizing problem with bounded numbers of setups per period and without setup cost. While this formulation seems to be rather non-standard in the lot-sizing landscape, it is motivated by concrete missions of the company. Since the deterministic version of the problem is NP-hard and its full stochastic version clearly intractable, we turn to approximate methods and propose a repeated two-stage stochastic programming approach to solve it. Using simulations on real-world instances, we show that our method gives better results than current heuristics used in industry. Moreover, our method provides lower bounds proving the quality of the approach. Since the computational times are small and the method easy to use, our contribution constitutes a promising response to the original industrial problem.

This paper deals with the multiple vehicle balancing problem (MVBP). Given a fleet of vehicles of limited capacity, a set of vertices with initial and target inventory levels and a distribution network, the MVBP requires to design a set of routes along with pickup and delivery operations such that inventory is redistributed among the vertices without exceeding capacities, and routing costs are minimized. The MVBP is NP‐hard, generalizing several problems in transportation, and arising in bike‐sharing systems. Using theoretical properties of the problem, we propose an integer linear programming formulation and introduce strengthening valid inequalities. Lower bounds are computed by column generation embedding an ad‐hoc pricing algorithm, while upper bounds are obtained by a memetic algorithm that separate routing from pickup and delivery operations. We combine these bounding routines in both exact and matheuristic algorithms, obtaining proven optimal solutions for MVBP instances with up to 25 stations.

Let C1,…, Cd+i be d + 1 point sets in ℝd, each containing the origin in its convex hull. A subset C of is called a colorful choice (or rainbow) for C1,…, Cd+1, if it contains exactly one point from each set Ci. The colorful Carathéodory theorem states that there always exists a colorful choice for C1,…, Cd+1 that has the origin in its convex hull. This theorem is very general and can be used to prove several other existence theorems in high-dimensional discrete geometry, such as the centerpoint theorem or Tverberg's theorem. The colorful Carathéodory problem (ColorfulCarathéodory) is the computational problem of finding such a colorful choice. Despite several efforts in the past, the computational complexity of ColorfulCarathéodory in arbitrary dimension is still open. We show that ColorfulCarathéodory lies in the intersection of the complexity classes PPAD and PLS. This makes it one of the few geometric problems in PPAD and PLS that are not known to be solvable in polynomial time. Moreover, it implies that the problem of computing centerpoints, computing Tverberg partitions, and computing points with large simplicial depth is contained in PPAD Π PLS. This is the first nontrivial upper bound on the complexity of these problems. Finally, we show that our PPAD formulation leads to a polynomial-time algorithm for a special case of ColorfulCarathéodory in which we have only two color classes C1 and C2 in d dimensions, each with the origin in its convex hull, and we would like to find a set with half the points from each color class that contains the origin in its convex hull.

At the occasion of ATMOS 2012, Tim Nonner and Alexander Souza defined a new train shunting problem that can roughly be described as follows. We are given a train visiting stations in a given order and cars located at some source stations. Each car has a target station. During the trip of the train, the cars are added to the train at their source stations and removed from it at their target stations. An addition or a removal of a car in the strict interior of the train incurs a cost higher than when the operation is performed at the end of the train. The problem consists in minimizing the total cost, and thus, at each source station of a car, the position the car takes in the train must be carefully decided. Among other results, Nonner and Souza showed that this problem is polynomially solvable by reducing the problem to the computation of a minimum independent set in a bipartite graph. They worked in the offline setting, i.e. the sources and the targets of all cars are known before the trip of the train starts. We study the online version of the problem, in which cars become known at their source stations. We derive a 2-competitive algorithm and prove than no better ratios are achievable. Other related questions are also addressed.

The colourful simplicial depth conjecture states that any point in the convex hull of each of $d+1$ sets, or colours, of $d+1$ points in general position in $\mathbb{R}^d$ is contained in at least $d^2+1$ simplices with one vertex from each set. We verify the conjecture in dimension 4 and strengthen the known lower bounds in higher dimensions. These results are obtained using a combinatorial generalization of colourful point configurations called octahedral systems. We present properties of octahedral systems generalizing earlier results on colourful point configurations and exhibit an octahedral system which cannot arise from a colourful point configuration. The number of octahedral systems is also given.