arXiv subjects

Frédéric Meunier

Publications and source records attributed to Frédéric Meunier.

48 records · Page 3Linked to original sources

arXiv2014

Hedetniemi's conjecture for Kneser hypergraphs

One of the most famous conjecture in graph theory is Hedetniemi's conjecture stating that the chromatic number of the categorical product of graphs is the minimum of their chromatic numbers. Using a suitable extension of the definition of the categorical product, Zhu proposed in 1992 a similar conjecture for hypergraphs. We prove that Zhu's conjecture is true for the usual Kneser hypergraphs of same rank. It provides to the best of our knowledge the first non-trivial and explicit family of hypergraphs with rank larger than two satisfying this conjecture (the rank two case being Hedetniemi's conjecture). We actually prove a more general result providing a lower bound on the chromatic number of the categorical product of any Kneser hypergraphs as soon as they all have same rank. We derive from it new families of graphs satisfying Hedetniemi's conjecture. The proof of the lower bound relies on the $Z_p$-Tucker lemma.

math.CO↗

arXiv2014

Stochastic Shortest Paths and Risk Measures

We consider three shortest path problems in directed graphs with random arc lengths. For the first and the second problems, a risk measure is involved. While the first problem consists in finding a path minimizing this risk measure, the second one consists in finding a path minimizing a deterministic cost, while satisfying a constraint on the risk measure. We propose algorithms solving these problems for a wide range of risk measures, which includes among several others the $CVaR$ and the probability of being late. Their performances are evaluated through experiments. One of the key elements in these algorithms is the use of stochastic lower bounds that allow to discard partial solutions. Good stochastic lower bounds are provided by the so-called Stochastic Ontime Arrival Problem. This latter problem is the third one studied in this paper and we propose a new and very efficient algorithm solving it. Complementary discussions on the complexity of the problems are also provided.

cs.DS↗

arXiv2014

A further generalization of the colourful Carathéodory theorem

Given $d+1$ sets, or colours, $S_1, S_2,...,S_{d+1}$ of points in $\mathbb{R}^d$, a {\em colourful} set is a set $S\subseteq\bigcup_i S_i$ such that $|S\cap S_i|\leq 1$ for $i=1,...,d+1$. The convex hull of a colourful set $S$ is called a {\em colourful simplex}. Bárány's colourful Carathéodory theorem asserts that if the origin 0 is contained in the convex hull of $S_i$ for $i=1,...,d+1$, then there exists a colourful simplex containing 0. The sufficient condition for the existence of a colourful simplex containing 0 was generalized to 0 being contained in the convex hull of $S_i\cup S_j$ for $1\leq i< j \leq d+1$ by Arocha et al. and by Holmsen et al. We further generalize the sufficient condition and obtain new colourful Carathéodory theorems. We also give an algorithm to find a colourful simplex containing 0 under the generalized condition. In the plane an alternative, and more general, proof using graphs is given. In addition, we observe that any condition implying the existence of a colourful simplex containing 0 actually implies the existence of $\min_i|S_i|$ such simplices.

cs.CG↗

arXiv2013

The uniqueness property for networks with several origin-destination pairs

We consider congestion games on networks with nonatomic users and user-specific costs. We are interested in the uniqueness property defined by Milchtaich [Milchtaich, I. 2005. Topological conditions for uniqueness of equilibrium in networks. Math. Oper. Res. 30 225-244] as the uniqueness of equilibrium flows for all assignments of strictly increasing cost functions. He settled the case with two-terminal networks. As a corollary of his result, it is possible to prove that some other networks have the uniqueness property as well by adding common fictitious origin and destination. In the present work, we find a necessary condition for networks with several origin-destination pairs to have the uniqueness property in terms of excluded minors or subgraphs. As a key result, we characterize completely bidirectional rings for which the uniqueness property holds: it holds precisely for nine networks and those obtained from them by elementary operations. For other bidirectional rings, we exhibit affine cost functions yielding to two distinct equilibrium flows. Related results are also proven. For instance, we characterize networks having the uniqueness property for any choice of origin-destination pairs.

cs.GT↗

arXiv2013

Colorful hypergraphs in Kneser hypergraphs

Using a $Z_q$-generalization of a theorem of Ky Fan, we extend to Kneser hypergraphs a theorem of Simonyi and Tardos that ensures the existence of multicolored complete bipartite graphs in any proper coloring of a Kneser graph. It allows to derive a lower bound for the local chromatic number of Kneser hypergraphs (using a natural definition of what can be the local chromatic number of a hypergraph).

math.CO↗

arXiv2013

A combinatorial approach to colourful simplicial depth

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 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 can not arise from a colourful point configuration. The number of octahedral systems is also given.

math.CO↗

arXiv2009

The chromatic number of almost stable Kneser hypergraphs

Let $V(n,k,s)$ be the set of $k$-subsets $S$ of $[n]$ such that for all $i,j\in S$, we have $|i-j|\geq s$ We define almost $s$-stable Kneser hypergraph $KG^r{{[n]}\choose k}_{s{\tiny{\textup{-stab}}}}^{\displaystyle\sim}$ to be the $r$-uniform hypergraph whose vertex set is $V(n,k,s)$ and whose edges are the $r$-uples of disjoint elements of $V(n,k,s)$. With the help of a $Z_p$-Tucker lemma, we prove that, for $p$ prime and for any $n\geq kp$, the chromatic number of almost 2-stable Kneser hypergraphs $KG^p {{[n]}\choose k}_{2{\tiny{\textup{-stab}}}}^{\displaystyle\sim}$ is equal to the chromatic number of the usual Kneser hypergraphs $KG^p{{[n]}\choose k}$, namely that it is equal to $\lceil\frac{n-(k-1)p}{p-1}\rceil.$ Defining $μ(r)$ to be the number of prime divisors of $r$, counted with multiplicities, this result implies that the chromatic number of almost $2^{μ(r)}$-stable Kneser hypergraphs $KG^r{{[n]}\choose k}_{2^{μ(r)}{\tiny{\textup{-stab}}}}^{\displaystyle\sim}$ is equal to the chromatic number of the usual Kneser hypergraphs $KG^r{{[n]}\choose k}$ for any $n\geq kr$, namely that it is equal to $\lceil\frac{n-(k-1)r}{r-1}\rceil.$

math.CO↗

arXiv2009

Completely symmetric configurations for sigma-games on grid graphs

The paper deals with sigma-games on grid graphs (in dimension 2 and more) and conditions under which any completely symmetric configuration of lit vertices can be reached -- in particular the completely lit configuration -- when starting with the all-unlit configuration. The answer is complete in dimension 2. In dimension greater than or equal to 3, the answer is complete for the sigma^+ -game, and for the sigma^- -game if at least one of the sizes is even. The case sigma^-, dimension greater than or equal to 3 and all sizes odd remains open.

math.CO↗

arXiv2009

Directed paths on a tree: coloring, multicut and kernel

In the present paper, we study algorithmic questions for the arc-intersection graph of directed paths on a tree. Such graphs are known to be perfect (proved by Monma and Wei in 1986). We present faster algorithms than all previously known algorithms for solving the minimum coloring and the minimum clique cover problems. They both run in $O(np)$ time, where $n$ is the number of vertices of the tree and $p$ the number of paths. Another result is a polynomial algorithm computing a kernel in the intersection graph, when its edges are oriented in a clique-acyclic way. Indeed, such a kernel exists for any perfect graph by a theorem of Boros and Gurvich. Such algorithms computing kernels are known only for few classes of perfect graphs.

cs.DM↗

arXiv2008

Dynamic assignment: there is an equilibrium !

Given a network with a continuum of users at some origins, suppose that the users wish to reach specific destinations, but that they are not indifferent to the time needed to reach their destination. They may have several possibilities (of routes or deparure time), but their choices modify the travel times on the network. Hence, each user faces the following problem: given a pattern of travel times for the different possible routes that reach the destination, find a shortest path. The situation in a context of perfect information is a so-called Nash equilibrium, and the question whether there is such an equilibrium and of finding it if it exists is the so-called equilibrium assignment problem. It arises for various kind of networks, such as computers, communication or transportation network. When each user occupies permanently the whole route from the origin to its destination, we call it the static assignment problem, which has been extensively studied with pioneers works by Wardrop or Beckmann. A less studied, but more realistic, and maybe more difficult, problem is when the time needed to reach an arc is taken into account. We speak then of a dynamic assignment problem. Several models have been proposed. For some of them, the existence of an equilibrium has been proved, but always under some technical assumptions or in a very special case (a network with one arc for the case when the users may chose their departure time). The present paper proposes a compact model, with minimal and natural assumptions. For this model, we prove that there is always an equilibrium. To our knowledge, this imply all previous results about existence of an equilibrium for the dynamic assignment problem.

cs.GT↗

arXiv2008

Polytopal complexes: maps, chain complexes and... necklaces

The notion of polytopal map between two polytopal complexes is defined. Surprisingly, this definition is quite simple and extends naturally those of simplicial and cubical maps. It is then possible to define an induced chain map between the associated chain complexes. Finally, we use this new tool to give the first combinatorial proof of the splitting necklace theorem of Alon. The paper ends with open questions, such as the existence of Sperner's lemma for a polytopal complex or the existence of a cubical approximation theorem.

math.CO↗

arXiv2008

Carathéodory, Helly and the others in the max-plus world

Carathéodory's, Helly's and Radon's theorems are three basic results in discrete geometry. Their max-plus counterparts have been proved by various authors. In this paper, more advanced results in discrete geometry are shown to have also their max-plus counterparts: namely, the colorful Carathéodory theorem and the Tverberg theorem. A conjecture connected to the Tverberg theorem -- Sierksma's conjecture --, although still open for the usual convexity, is shown to be true in the max-plus settings.

math.CO↗