k-SAT is one of the best known among a wide class of random constraint satisfaction problems believed to exhibit a threshold phenomenon where the control parameter is the ratio, number of constraints to number of variables. There has been a large amount of work towards estimating the 3-SAT threshold. We present a new structural (or syntactic) approach aimedâ€¦ (More)

An old problem on random graphs remaining open to this day is that of the existence and determination of a k-colourability threshold, already posed in the paper [10] which launched the whole subject. Using the uniformly distributed model G (n,m) of graphs with m edges on n vertices, it reads: does there exist a constant ck such that if m âˆ¼ (ck âˆ’ Îµ)n forâ€¦ (More)

A constructive scheme for determining pure states at very low temperature in the 3-spins glass model on a random lattice is provided, in full agreement with Parisi's one step replica symmetry breaking (RSB) scheme. Proof is based on the analysis of a partial decimation procedure and of the statistical properties of its output, i.e., a reduced Hamiltonianâ€¦ (More)

A long-standing conjecture in combinatorial optimization says that the integrality gap of the famous Held-Karp relaxation of the symmetric TSP is precisely 4/3. In this paper, we show that a slight strengthening of this conjecture implies a tight 4/3 integrality gap for a linear programming relaxation of the asymmetric TSP. This is surprising since noâ€¦ (More)

An additive decomposition of a set I of nonnegative integers is an expression of I as the arithmetic sum of two other such sets. If the smaller of these has p elements, we have a p-decomposition. If I is obtained by randomly removing nÎ± integers from {0, ..., n âˆ’ 1}, decomposability translates into a balls-and-urns problem which we start to investigate (forâ€¦ (More)