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We study a class of hypothesis testing problems in which, upon observing the realization of an n-dimensional Gaussian vector, one has to decide whether the vector was drawn from a standard normal distribution or, alternatively, whether there is a subset of the components belonging to a certain given class of sets whose elements have been " contaminated, "… (More)

A hole in a graph is an induced subgraph which is a cycle of length at least four. A hole is called even if it has an even number of vertices. An even-hole-free graph is a graph with no even holes. A vertex of a graph is bisimplicial if the set of its neighbours is the union of two cliques. In this paper we prove that every even-hole-free graph has a… (More)

We consider the minimum-weight path between any pair of nodes of the n-vertex complete graph in which the weights of the edges are i.i.d. exponentially distributed random variables. We show that the longest of these minimum-weight paths has about α log n edges where α ≈ 3.5911 is the unique solution of the equation α log α − α = 1. This answers a question… (More)

In this paper, we present new structural results about the existence of a subgraph where the degrees of the vertices are pre-specified. Further, we use these results to prove a 16-edge-weighting version of a conjecture by Karo´nski, Luczak and Thoma-son, an asymptotic 2-edge-weighting version of the same conjecture, and a 7/8 version of Louigi's Conjecture.

A weighting w of the edges of a graph G induces a colouring of the vertices of G where the colour of vertex v, denoted c v , is ev w(e). We show that the edges of every graph that does not contain a component isomorphic to K 2 can be weighted from the set {1,. .. , 30} such that in the resulting vertex-colouring of G, for every edge (u, v) of G, c u = c v .

Maximum likelihood (ML) (Neyman, 1971) is an increasingly popular optimality criterion for selecting evolutionary trees. Finding optimal ML trees appears to be a very hard computational task--in particular, algorithms and heuristics for ML take longer to run than algorithms and heuristics for maximum parsimony (MP). However, while MP has been known to be… (More)

Two-player win-lose games have a simple directed graph representation. Exploiting this, we develop graph theoretic techniques for finding Nash equilibria in such games. In particular, we give a polynomial time algorithm for finding a Nash equilibrium in a two-player win-lose game whose graph representation is planar.

" Small worlds " are large systems in which any given node has only a few connections to other points, but possessing the property that all pairs of points are connected by a short path, typically logarithmic in the number of nodes. The use of random walks for sampling a uniform element from a large state space is by now a classical technique; to prove that… (More)