Finite-connectivity Random Graphs Research Articles

Spin glass approach to the feedback vertex set problem

A feedback vertex set (FVS) of an undirected graph is a set of vertices that contains at least one vertex of each cycle of the graph. The feedback vertex set problem consists of constructing a FVS of size less than a certain given value. This combinatorial optimization problem has many practical applications, but it is in the nondeterministic polynomial-complete class of worst-case computational complexity. In this paper we define a spin glass model for the FVS problem and then study this model on the ensemble of finite-connectivity random graphs. In our model the global cycle constraints are represented through the local constraints on all the edges of the graph, and they are then treated by distributed message-passing procedures such as belief propagation. Our belief propagation-guided decimation algorithm can construct nearly optimal feedback vertex sets for single random graph instances and regular lattices. We also design a spin glass model for the FVS problem on a directed graph. Our work will be very useful for identifying the set of vertices that contribute most significantly to the dynamical complexity of a large networked system.

Replica theory for Levy spin glasses**Dedicated to Professor Thomas Nattermann on the occasion of his 60th anniversary.

Infinite-range spin-glass models with Levy-distributed interactions show a spin-glasstransition with similarities to both the Sherrington–Kirkpatrick model and to disorderedspin systems on finite connectivity random graphs. Despite the diverging moments of thecoupling distribution the transition can be analyzed within the replica approach by workingat imaginary temperature. Within the replica symmetric approximation a self-consistentequation for the distribution of local fields is derived and from the instability of theparamagnetic solution to this equation the glass transition temperature is determined. Therole of the percolation of rare strong bonds for the transition is elucidated. The resultspartly agree and partly disagree with those obtained within the cavity approach. Numericalsimulations using parallel tempering are in agreement with the transition temperaturesfound.

Vertex cover problem studied by cavity method: Analytics and population dynamics

We study the vertex cover problem on finite connectivity random graphs by zero-temperature cavity method. The minimum vertex cover corresponds to the ground state(s) of a proposed Ising spin model. When the connectivity c>e=2.718282, there is no state for this system as the reweighting parameter y, which takes a similar role as the inverse temperature \beta in conventional statistical physics, approaches infinity; consequently the ground state energy is obtained at a finite value of y when the free energy function attains its maximum value. The minimum vertex cover size at given c is estimated using population dynamics and compared with known rigorous bounds and numerical results. The backbone size is also calculated.

Dynamics of heuristic optimization algorithms on random graphs

In this paper, the dynamics of heuristic algorithms for constructing small vertex covers (or independent sets) of finite-connectivity random graphs is analysed. In every algorithmic step, a vertex is chosen with respect to its vertex degree. This vertex, and some environment of it, is covered and removed from the graph. This graph reduction process can be described as a Markovian dynamics in the space of random graphs of arbitrary degree distribution. We discuss some solvable cases, including algorithms already analysed using different techniques, and develop approximation schemes for more complicated cases. The approximations are corroborated by numerical simulations.

Statistical mechanics perspective on the phase transition in vertex covering of finite-connectivity random graphs

The vertex-cover problem is studied for random graphs G N,cN having N vertices and cN edges. Exact numerical results are obtained by a branch-and-bound algorithm. It is found that a transition in the coverability at a c-dependent threshold x=x c(c) appears, where xN is the cardinality of the vertex cover. This transition coincides with a sharp peak of the typical numerical effort, which is needed to decide whether there exists a cover with xN vertices or not. For small edge concentrations c⪡0.5, a cluster expansion is performed, giving very accurate results in this regime. These results are extended using methods developed in statistical physics. The so-called annealed approximation reproduces a rigorous bound on x c(c) which was known previously. The main part of the paper contains an application of the replica method. Within the replica symmetric ansatz the threshold x c(c) and various statistical properties of minimal vertex covers can be calculated. For c<e/2 the results show an excellent agreement with the numerical findings. At average vertex degree 2c=e, an instability of the simple replica symmetric solution occurs.

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

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 1, 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 entropy of the minimal vertex covers, and the structure of the unfrozen component which is found to percolate at a connectivity c approximately 1.43. The replica-symmetric solution breaks down at c=e approximately 2.72. We give a simple one-step replica-symmetry-broken solution, and discuss the problems in the interpretation and generalization of this solution.

Typical solution time for a vertex-covering algorithm on finite-connectivity random graphs.

We analytically describe the typical solution time needed by a backtracking algorithm to solve the vertex-cover problem on finite-connectivity random graphs. We find two different transitions: The first one is algorithm dependent and marks the dynamical transition from linear to exponential solution times. The second one gives the maximum computational complexity, and is found exactly at the threshold where the system undergoes an algorithm-independent phase transition in its solvability. Analytical results are corroborated by numerical simulations.

Number of guards needed by a museum: a phase transition in vertex covering of random graphs.

In this Letter we study the NP-complete vertex cover problem on finite connectivity random graphs. When the allowed size of the cover set is decreased, a discontinuous transition in solvability and typical-case complexity occurs. This transition is characterized by means of exact numerical simulations as well as by analytical replica calculations. The replica symmetric phase diagram is in excellent agreement with numerical findings up to average connectivity e, where replica symmetry becomes locally unstable.