Minimal vertex covers on finite-connectivity random graphs: A hard-sphere lattice-gas picture

Abstract

The minimal vertex-cover (or maximal independent-set) problem is studied on random graphs of finite connectivity. Analytical results are obtained by a mapping to a lattice gas of hard spheres of (chemical) radius one, and they are found to be in excellent agreement with numerical simulations. We give a detailed description of the replica-symmetric phase, including the size and the entropy of the minimal vertex covers, and the structure of the unfrozen component which is found to percolate at connectivity c ≃ 1.43. The replica-symmetric solution breaks down at c = e ≃ 2.72. We give a simple one-step replica symmetry broken solution, and discuss the problems in interpretation and generalization of this solution.

Solving satisfiability problems by fluctuations: The dynamics of stochastic local search algorithms

“…The most striking results so far were obtained in the description of the solution-space structure of the random satisfiability problem [3,4,5,6,7], of the number partitioning problem [8,9], of vertex covers [10,11,12] or colorings [13] of random graphs. In these cases, equilibrium methods from statistical mechanics can be applied directly, including e.g.…”

Message passing for vertex covers

“…It can be justified using the more detailed belief propagation calculating the full single site probability π i [8], or via the replica method [16]. The WP equations can be used to construct a vertex cover, i.e.…”

“…In these models, hard particles are to be positioned on a lattice under geometrical packing constraints representing hard-core interactions. These models are considered as simple lattice models for the glass transition due to geometric frustration, and their closest packing correspond to minimal VCs [16]. This paper is organized as follows: the vertex cover problem is defined in Sec.…”

“…The typical properties of VCs on such graphs have already been analyzed both with rigorous mathematical tools [14,15] and with statistical-physics methods [16,17,18,19,20]. We therefore have a pretty complete knowledge about the phase diagram of this problem, and can systematically compare the algorithmic performance of the message-passing techniques on single, finite, randomly generated graphs to the average behavior in the thermodynamic limit.…”

Core organization of directed complex networks

“…It is this giant component that is usually called the core of a network [12]. The cores are related to a wide range of topical problems for networks such as controllability of networks [13], localization on random graphs [14], and some combinatorial optimization problems like maximum matching [15,16] and minimum vertex cover [17][18][19].…”