arXiv subjects

Frédéric Meunier

Publications and source records attributed to Frédéric Meunier.

At least 19 recordsLinked to original sources

arXiv2026

Quadratic Degree Sequence Optimization and the Critical Roots of a Graph

The degree sequence optimization problem is to find a subgraph of a given graph which maximizes the sum over all vertices of a given function evaluated at the subgraph degree of that vertex. Here we study this problem and its complexity for quadratic functions. In particular, we introduce the critical roots of a graph, and show they define intervals over which the optimal value of the problem, as the quadratic root varies, is convex piecewise affine.

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arXiv2026

Fair and efficient allocation of indivisible items under category constraints

We study the problem of fairly allocating indivisible goods and chores under category constraints. Specifically, there are $n$ agents and $m$ indivisible items which are partitioned into categories with associated capacities. An allocation is considered feasible if each bundle satisfies the capacity constraints of its respective categories. For the case of two agents, Shoshan et al. (2023) recently developed a polynomial-time algorithm to find a Pareto-optimal allocation satisfying a relaxed version of envy-freeness, called EF$[1,1]$. Extending such guarantees beyond two agents has remained open. We make progress toward this question by proving that for any number $n$ of agents, there always exists a Pareto-optimal allocation in which each agent can be made envy-free by reallocating at most $\min \{k+1,n\}(n-1)$ items. Moreover, when the number of agents is constant, we give a polynomial-time algorithm to compute such an allocation. Our results rely on a new application of the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma to a simplex of agent weights, which may be of independent interest for fair division problems involving indivisible items.

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arXiv2026

Bucket Brigades: Uniqueness of the Fixed Point and Three-Worker Asymptotics

A standard organization of production lines exhibiting self-balancing behavior is given by bucket brigades. Their study in operations research was initiated by the foundational work of Bartholdi and Eisenstein ({\em Operations Research}, 1996), where a simplified version of the model is considered. Their main result shows that when workers are ordered from the slowest to the fastest, the system is stable and converges to a ``fixed point,'' where each worker oscillates between two limiting positions. They also observe that the dynamics can become highly complex when this ordering condition is not satisfied. The {\em no-station} setting, in which work is distributed continuously and uniformly along the production line, is given special attention in their work. In a subsequent paper with Bunimovich ({\em Operations Research}, 1999), they characterize all stable behaviors of this setting for up to three workers. In this work, we extend their analysis for three workers beyond the stable regime, providing a complete description when workers are ordered from the fastest to the slowest. We also show that, due to their restrictive notion of stability, some of their conclusions must be revisited. Finally, for an arbitrary number of workers, we prove that the fixed point is always unique in the no-station setting.

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arXiv2026

An Exact Method for a Problem of Time Slot Pricing

A company provides a service at different time slots, each slot being endowed with a capacity. A non-atomic population of users is willing to purchase this service. The population is modeled as a continuous measure over the preferred times. Every user looks at the time slot that minimizes the sum of the price assigned by the company to this time slot and the distance to their preferred time. If this sum is non-negative, then the user chooses this time slot for getting the service. If this sum is positive, then the user rejects the service. We address the problem of finding prices that ensure that the volume of users choosing each time slot is below capacity, while maximizing the revenue of the company. For the case where the distance function is convex, we propose an exact algorithm for solving this problem in time $O(n^3|P|^3)$, where $P$ is the set of possible prices and $n$ is the number of time slots. For the case where the prices can be any real numbers, this algorithm can also be used to find asymptotically optimal solutions in polynomial time under mild extra assumptions on the distance function and the measure modeling the population.

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arXiv2025

Aircraft routing: periodicity and complexity

The aircraft routing problem is one of the most studied problems of operations research applied to aircraft management. It involves assigning flights to aircraft while ensuring regular visits to maintenance bases. This paper examines two aspects of the problem. First, we explore the relationship between periodic instances, where flights are the same every day, and periodic solutions. The literature has implicitly assumed-without discussion-that periodic instances necessitate periodic solutions, and even periodic solutions in a stronger form, where every two airplanes perform either the exact same cyclic sequence of flights, or completely disjoint cyclic sequences. However, enforcing such periodicity may eliminate feasible solutions. We prove that, when regular maintenance is required at most every four days, there always exist periodic solutions of this form. Second, we consider the computational hardness of the problem. Even if many papers in this area refer to the NP-hardness of the aircraft routing problem, such a result is only available in the literature for periodic instances. We establish its NP-hardness for a non-periodic version. Polynomiality of a special but natural case is also proven.

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arXiv2025

Geoffrion's theorem beyond finiteness and rationality

Geoffrion's theorem is a fundamental result from mathematical programming assessing the quality of Lagrangian relaxation, a standard technique to get bounds for integer programs. An often implicit condition is that the set of feasible solutions is finite or described by rational linear constraints. However, we show through concrete examples that the conclusion of Geoffrion's theorem does not necessarily hold when this condition is dropped. We then provide sufficient conditions ensuring the validity of the result even when the feasible set is not finite and cannot be described using finitely-many linear constraints.

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arXiv2025

Colorful circuits and colorful topes in oriented matroids

We provide a short proof of a conic version of the colorful Carathéodory theorem for oriented matroids. Holmsen's extension of the colorful Carathéodory theorem to oriented matroids (Advances in Mathematics, 2016) already encompasses several generalizations of the original result, but not its conic version. Our approach relies on a common generalization of Sperner's lemma and Meshulam's lemma-two closely related results from combinatorial topology that have found a number of applications in discrete geometry and combinatorics. This generalization may be of independent interest. Using a similar approach, we also establish the following colorful theorem for topes, whose special geometric case had not been considered before: Given $n$ topes from a uniform oriented matroid with $n$ elements, if they agree on some element, then there is a way to select a distinct element from each tope, together with its sign, so as to form another tope of the oriented matroid. Motivated by this theorem, we further explore other conditions leading to the same conclusion.

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arXiv2025

Matroids, intersecting bases, and Borsuk property

A subset $S$ of $\mathbb R^d$ has the Borsuk property if it can be decomposed into at most $d+1$ parts of diameter smaller than $S$. This is an important geometric property, inspired by a conjecture of Borsuk from the 1930s, which has attracted considerable attention over the years. In this paper, we define and investigate the Borsuk property for matroids, providing a purely combinatorial approach to the Borsuk property for matroid polytopes, a well-studied family of $(0,1)$-polytopes associated with matroids. We show that a sufficient condition for a matroid -- and thus its matroid polytope -- to have the Borsuk property is that the matroid or its dual has two disjoint bases. However, we show that this condition is not necessary by exhibiting infinite families of matroids having the Borsuk property and yet being such that every two bases intersect and every two cobases intersect. Kneser graphs, which form an important object from topological combinatorics, play a crucial role in most proofs.

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arXiv2025

Balanced assignments of periodic tasks

This work addresses the problem of assigning periodic tasks to workers in a balanced way, i.e., so that each worker performs every task with the same frequency over the long term. The input consists of a list of tasks to be repeated weekly at fixed times and a number of indistinguishable workers. In the basic version, the sole constraint is that no worker performs two tasks simultaneously. In the extended version, additional constraints can be introduced, such as limits on the total number of working hours per week. Regarding the basic version, a necessary and sufficient condition for the existence of a balanced assignment is established. This condition can be verified in polynomial time. For the extended version, it is demonstrated that whenever a balanced assignment exists, a periodic balanced assignment exists as well, with a tighter bound on the period for the basic version.

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arXiv2025

Finite adaptability in two-stage robust optimization: asymptotic optimality and tractability

Two-stage robust optimization is a fundamental paradigm for modeling and solving optimization problems with uncertain parameters. A now classical method within this paradigm is finite adaptability, introduced by Bertsimas and Caramanis (IEEE Transactions on Automatic Control, 2010). It consists in restricting the recourse to a finite number $k$ of possible values. In this work, we point out that the continuity assumption they stated to ensure the convergence of the method when $k$ goes to infinity is not correct, and we propose an alternative assumption for which we prove the desired convergence. Bertsimas and Caramanis also established that finite adaptability is NP-hard, even in the special case when $k=2$, the variables are continuous, and only specific parameters are subject to uncertainty. We provide a theorem showing that this special case becomes polynomial when the uncertainty set is a polytope with a bounded number of vertices, and we extend this theorem for $k=3$ as well. On our way, we establish new geometric results on coverings of polytopes with convex sets, which might be interesting for their own sake.

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arXiv2025

Revisiting classical results on kernels in digraphs

In a digraph, a kernel is a subset of vertices that is both independent and absorbing. Kernels have important applications in combinatorics and outside. Kernels do not always exist and finding sufficient conditions ensuring their existence is a key theoretical challenge. In this work, we revisit and generalize a few classical results of this sort, especially the Sands--Sauer--Woodrow theorem and the Galeana-Sánchez--Neumann-Lara theorem.

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arXiv2024

Quasi-kernels in split graphs

In a digraph, a quasi-kernel is a subset of vertices that is independent and such that the shortest path from every vertex to this subset is of length at most two. The ``small quasi-kernel conjecture,'' proposed by Erdős and Székely in 1976, postulates that every sink-free digraph has a quasi-kernel whose size is within a fraction of the total number of vertices. The conjecture is even more precise with a $1/2$ ratio, but even with larger ratio, this property is known to hold only for few classes of graphs. The focus here is on small quasi-kernels in split graphs. This family of graphs has played a special role in the study of the conjecture since it was used to disprove a strengthening that postulated the existence of two disjoint quasi-kernels. The paper proves that every sink-free split digraph $D$ has a quasi-kernel of size at most $\frac{2}{3}|V(D)|$, and even of size at most two when the graph is an orientation of a complete split graph. It is also shown that computing a quasi-kernel of minimal size in a split digraph is W[2]-hard.

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arXiv2024

Box complexes: at the crossroad of graph theory and topology

Various simplicial complexes can be associated with a graph. Box complexes form an important families of such simplicial complexes and are especially useful for providing lower bounds on the chromatic number of the graph via some of their topological properties. They provide thus a fascinating topic mixing topology and discrete mathematics. This paper is intended to provide an up-do-date survey on box complexes. It is based on classical results and recent findings from the literature, but also establishes new results improving our current understanding of the topic, and identifies several challenging open questions.

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arXiv2023

Maximal workload, minimal workload, maximal workload difference: optimizing all criteria at once

In a simple model of assigning workers to tasks, every solution that minimizes the load difference between the most loaded worker and the least loaded one actually minimizes the maximal load and maximizes the minimal load. This can be seen as a consequence of standard results of optimization over polymatroids. We show that similar phenomena still occur in close models, simple to state, and that do not enjoy any polymatroid structure.

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arXiv2022

Envy-free division of multi-layered cakes

We study the problem of dividing a multi-layered cake under non-overlapping constraints. This problem, recently proposed by Hosseini et al. (IJCAI, 2020), captures several natural scenarios such as the allocation of multiple facilities over time where each agent can utilize at most one facility simultaneously, and the allocation of tasks over time where each agent can perform at most one task simultaneously. We establish the existence of an envy-free multi-division that is both non-overlapping and contiguous within each layer when the number of agents is a prime power and the number of layers is at most the number of agents, providing a positive partial answer to an open question by Hosseini et al. To achieve this, we employ a new approach based on a general fixed point theorem, originally proven by Volovikov (1996), and recently applied by Jojić, Panina, and Živaljević (2020) to the envy-free division problem of a cake. We further design a fully polynomial-time approximation scheme (FPTAS) to find an approximate envy-free solution that is both non-overlapping and contiguous for a two-layered cake division among three agents with monotone preferences. More generally, we establish all the results for divisions among groups of almost the same size. When there are three groups and one layer, our FPTAS is actually able to deal with groups of any predetermined sizes, still under the monotone preference assumptions. For three groups, this provides thus an algorithmic version of a recent theorem by Segal-Halevi and Suksompong (2021). A classical lemma from topology ensures that every self-map of a simplex mapping each facet to itself is surjective. This lemma has often been used in economics theory and our algorithm can be interpreted as an FPTAS counterpart of its two-dimensional version.

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arXiv2022

Tropical complementarity problems and Nash equilibria

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.

math.OC↗

arXiv2022

Linear lexicographic optimization and preferential bidding system

Some airlines use the preferential bidding system to construct the schedules of their pilots. In this system, the pilots bid on the different activities and the schedules that lexicographically maximize the scores of the pilots according to their seniority are selected. A sequential approach to solve this maximization problem is natural: the problem is first solved with the bids of the most senior pilot; then it is solved with those of the second most senior without decreasing the score of the most senior, and so on. The literature admits that the structure of the problem somehow imposes such an approach. The problem can be modeled as an integer linear lexicographic program. We propose a new exact method, which relies on column generation for solving its continuous relaxation. To design this column generation, we prove that bounded linear lexicographic programs admit "primal-dual" feasible bases and we show how to compute such bases efficiently. Another contribution on which our exact method relies consists in the extension of standard tools for resource-constrained longest path problems to their lexicographic versions. This is useful in our context since the generation of new columns is modeled as a lexicographic resource-constrained longest path problem. Numerical experiments show that this new method is already able to solve industrial instances provided by Air France, with up to 150 pilots. By adding a last ingredient in the resolution of the longest path problems, which exploits the specificity of the preferential bidding system, the method achieves for these instances computational times that are compatible with operational constraints.

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arXiv2021

Algorithmic aspects of quasi-kernels

In a digraph, a quasi-kernel is a subset of vertices that is independent and such that every vertex can reach some vertex in that set via a directed path of length at most two. Whereas Chvátal and Lovász proved in 1974 that every digraph has a quasi-kernel, very little is known so far about the complexity of finding small quasi-kernels. In 1976 Erdős and Székely conjectured that every sink-free digraph $D = (V, A)$ has a quasi-kernel of size at most $|V|/2$. Obviously, if $D$ has two disjoint quasi-kernels then it has a quasi-kernel of size at most $|V|/2$, and in 2001 Gutin, Koh, Tay and Yeo conjectured that every sink-free digraph has two disjoint quasi-kernels. Yet, they constructed in 2004 a counterexample, thereby disproving this stronger conjecture. We shall show that, not only sink-free digraphs occasionally fail to contain two disjoint quasi-kernels, but it is computationally hard to distinguish those that do from those that do not. We also prove that the problem of computing a small quasi-kernel is polynomial time solvable for orientations of trees but is computationally hard in most other cases (and in particular for restricted acyclic digraphs).

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