ABSTRACT In the minimum clique routing problem on cycles mcrpc, we are given a cycle together with a set of demands (weighted terminals pairs) and the goal is to route all the pairs minimizing the maximum weight clique of the intersection graph induced by the routing. The nodes of this graph are the demands with their corresponding weights and two demands are adjacent when their routes share at least one arc. In this work, we are not only interested in the mcrpc but also in two natural subproblems. First, we consider the situation where the demands are disjoint, in the sense that every two demands do not share any of their corresponding terminals. Second, we analyze the subproblem where the weights of the routes are all equal. We first show that the problem is NP‐hard even in the subproblem of disjoint demands. For the case of arbitrary weights, we exhibit a simple combinatorial 2‐approximation algorithm and a ‐approximation algorithm based on rounding a solution of a relaxation of an integer linear programming formulation of our problem. Finally, we give a fixed parameter tractable algorithm for the case of uniform weights, whose parameter is the maximum number of demands for which a demand exists whose terminals alternate in the cycle with the terminals of each of them.

It is a classic result that a set of n non-collinear points in the Euclidean plane defines at least n different lines. Chen and Chvátal conjectured in 2008 that the same results is true in metric spaces for an adequate definition of line. More recently, this conjecture was studied in the context of quasi-metric spaces. One way to study lines in an space is though its betweenness. Given a quasi-metric space (V,ρ), its induced quasi-metric be-tweenness is the set of triples (x, y, z) ϵ V3 such that ρ(x, z) = ρ(x, y) +ρ(y, z). In this work, we prove the existence of a quasi-metric space on four points a, b, c and d whose quasi-metric betweenness is ẞ = {(c, a, b), (a, b, c), (d, b, a), (b, a, d)}. This space has only three lines, none of which has four points. Moreover, we show that the betweenness of any quasi-metric space on four points with this property is isomorphic to B. Since B is not metric, we conclude that Chen and Chvatal's conjecture is valid for any metric space on four points.

A digraph D = (V, A) is semi-complete if for each pair of distinct vertices x and y in V, either xy or yx belong to A. A subset ℓ of vertices is a line of D if there are two distinct vertices x and y such that for any vertex z ε V, z ε ℓ if and only if a directed shortest path exists containing x, y and z. A classic result proved by Erdös says that any set of n points in the Euclidean plane endowed with the Euclidean distance defines a metric space with at least n different lines unless there is a line containing the n points. Chen and Chvátal in 2008 conjectured that the same results is true for any metric spaces where lines are defined in a manner similar to above. In this paper we prove that in any semi-complete digraphs with n vertices the number of lines defined by vertices connected by an arc is at least n. Then, the quasi-metric spaces defined by semi-complete digraphs fulfill Chen and Chvátal conjecture in a stronger manner as, on the one hand, they always have at least n lines, and on the other hand, these n lines are defined by vertices at distance one.

We consider the problem of coloring a grid using k colors with the restriction that each row and each column has a specific number of cells of each color. This problem has been known as the $(k-1)$-atom problem in the discrete tomography community. In an already classical result, Ryser obtained a necessary and sufficient condition for the existence of such a coloring when two colors are considered. This characterization yields a linear time algorithm for constructing such a coloring when it exists. Gardner et al. showed that for $k\geqslant 7$ the problem is NP-hard. Afterward Chrobak and Dürr improved this result by proving that it remains NP-hard for $k\geqslant 4$. We close the gap by showing that for $k=3$ colors the problem is already NP-hard. In addition, we give some results on tiling tomography problems.

In this paper we ask which properties of a distributed network can be computed from a few amount of local information provided by its nodes. The distributed model we consider is a restriction of the classical CONGEST (distributed) model and it is close to the simultaneous messages (communication complexity) model defined by Babai, Kimmel and Lokam. More precisely, each of these n nodes-which only knows its own ID and the IDs of its neighbors- is allowed to send a message of O(log n) bits to some central entity, called the referee. Is it possible for the referee to decide some basic structural properties of the network topology G? We show that simple questions like, "does G contain a square?", "does G contain a triangle?" or "Is the diameter of G at most 3?" cannot be solved in general. On the other hand, the referee can decode the messages in order to have full knowledge of G when G belongs to many graph classes such as planar graphs, bounded tree width graphs and, more generally, bounded degeneracy graphs. We leave open questions related to the connectivity of arbitrary graphs.

Let $G=(V,E)$ be a graph and T be a spanning tree of G. We consider the following strategy in advancing in G from a vertex x towards a target vertex y: from a current vertex z (initially, $z=x$), unless $z=y$, go to a neighbor of z in G that is closest to y in T (breaking ties arbitrarily). In this strategy, each vertex has full knowledge of its neighborhood in G and can use the distances in T to navigate in G. Thus, additionally to standard local information (the neighborhood $N_G(v)$), the only global information that is available to each vertex v is the topology of the spanning tree T (in fact, v can know only a very small piece of information about T and still be able to infer from it the necessary tree-distances). For each source vertex x and target vertex y, this way, a path, called a greedy routing path, is produced. Denote by $g_{G,T}(x,y)$ the length of a longest greedy routing path that can be produced for x and y using this strategy and T. We say that a spanning tree T of a graph G is an additive r-carcass for G if $g_{G,T}(x,y)\leq d_G(x,y)+r$ for each ordered pair $x,y\in V$. In this paper, we investigate the problem, given a graph family $\mathcal{F}$, of whether a small integer r exists such that any graph $G\in\mathcal{F}$ admits an additive r-carcass. We show that rectilinear $p\times q$ grids, hypercubes, distance-hereditary graphs, dually chordal graphs (and, therefore, strongly chordal graphs and interval graphs) all admit additive 0-carcasses. Furthermore, every chordal graph G admits an additive $(\omega+1)$-carcass (where $\omega$ is the size of a maximum clique of G), each 3-sun-free chordal graph admits an additive 2-carcass, and each chordal bipartite graph admits an additive 4-carcass. In particular, any k-tree admits an additive $(k+2)$-carcass. All those carcasses are easy to construct in sequential as well as in distributed settings.

In this paper we study the dynamical behavior of a class of neural networks where the local transition rules are max or min functions. We prove that sequential updates define dynamics which reach the equilibrium in O(n2) steps, where n is the size of the network. For synchronous updates the equilibrium is reached in O(n) steps. It is shown that the number of fixed points of the sequential update is at most n. Moreover, given a set of p < or = n vectors, we show how to build a network of size n such that all these vectors are fixed points.

We present dynamical results concerning neural networks with high order arguments. More precisely, we study the family of block-sequential iteration of neural networks with polynomial arguments. In this context, we prove that, under a symmetric hypothesis, the sequential iteration is the only one of this family to converge to fixed points. The other iteration modes present a highly complex dynamical behavior: non-bounded cycles and simulation of arbitrary non-symmetric linear neural network. We also study a high order memory iteration scheme which accepts an energy functional and bounded cycles in the size of the memory steps.