arXiv subjects

Martin Weigt

Publications and source records attributed to Martin Weigt.

66 records · Page 4Linked to original sources

arXiv2004

Approximation schemes for the dynamics of diluted spin models: the Ising ferromagnet on a Bethe lattice

We discuss analytical approximation schemes for the dynamics of diluted spin models. The original dynamics of the complete set of degrees of freedom is replaced by a hierarchy of equations including an increasing number of global observables, which can be closed approximately at different levels of the hierarchy. We illustrate this method on the simple example of the Ising ferromagnet on a Bethe lattice, investigating the first three possible closures, which are all exact in the long time limit, and which yield more and more accurate predictions for the finite-time behavior. We also investigate the critical region around the phase transition, and the behavior of two-time correlation functions. We finally underline the close relationship between this approach and the dynamical replica theory under the assumption of replica symmetry.

cond-mat.stat-mech↗

arXiv2003

Statistical mechanics of the vertex-cover problem

We review recent progress in the study of the vertex-cover problem (VC). VC belongs to the class of NP-complete graph theoretical problems, which plays a central role in theoretical computer science. On ensembles of random graphs, VC exhibits an coverable-uncoverable phase transition. Very close to this transition, depending on the solution algorithm, easy-hard transitions in the typical running time of the algorithms occur. We explain a statistical mechanics approach, which works by mapping VC to a hard-core lattice gas, and then applying techniques like the replica trick or the cavity approach. Using these methods, the phase diagram of VC could be obtained exactly for connectivities $c e$, the solution of VC exhibits full replica symmetry breaking. The statistical mechanics approach can also be used to study analytically the typical running time of simple complete and incomplete algorithms for VC. Finally, we describe recent results for VC when studied on other ensembles of finite- and infinite-dimensional graphs.

cond-mat.dis-nn↗

arXiv2003

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

Stochastic local search algorithms are frequently used to numerically solve hard combinatorial optimization or decision problems. We give numerical and approximate analytical descriptions of the dynamics of such algorithms applied to random satisfiability problems. We find two different dynamical regimes, depending on the number of constraints per variable: For low constraintness, the problems are solved efficiently, i.e. in linear time. For higher constraintness, the solution times become exponential. We observe that the dynamical behavior is characterized by a fast equilibration and fluctuations around this equilibrium. If the algorithm runs long enough, an exponentially rare fluctuation towards a solution appears.

cond-mat.stat-mech↗

arXiv2002

Computational complexity arising from degree correlations in networks

We apply a Bethe-Peierls approach to statistical-mechanics models defined on random networks of arbitrary degree distribution and arbitrary correlations between the degrees of neighboring vertices. Using the NP-hard optimization problem of finding minimal vertex covers on these graphs, we show that such correlations may lead to a qualitatively different solution structure as compared to uncorrelated networks. This results in a higher complexity of the network in a computational sense: Simple heuristic algorithms fail to find a minimal vertex cover in the highly correlated case, whereas uncorrelated networks seem to be simple from the point of view of combinatorial optimization.

cond-mat.dis-nn↗

arXiv2002

Towards finite-dimensional gelation

We consider the gelation of particles which are permanently connected by random crosslinks, drawn from an ensemble of finite-dimensional continuum percolation. To average over the randomness, we apply the replica trick, and interpret the replicated and crosslink-averaged model as an effective molecular fluid. A Mayer-cluster expansion for moments of the local static density fluctuations is set up. The simplest non-trivial contribution to this series leads back to mean-field theory. The central quantity of mean-field theory is the distribution of localization lengths, which we compute for all connectivities. The highly crosslinked gel is characterized by a one-to-one correspondence of connectivity and localization length. Taking into account higher contributions in the Mayer-cluster expansion, systematic corrections to mean-field can be included. The sol-gel transition shifts to a higher number of crosslinks per particle, as more compact structures are favored. The critical behavior of the model remains unchanged as long as finite truncations of the cluster expansion are considered. To complete the picture, we also discuss various geometrical properties of the crosslink network, e.g. connectivity correlations, and relate the studied crosslink ensemble to a wider class of ensembles, including the Deam-Edwards distribution.

cond-mat.soft↗

arXiv2002

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.

cond-mat.stat-mech↗

arXiv2001

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 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 $c\simeq 1.43$. The replica-symmetric solution breaks down at $c=e\simeq 2.72$. We give a simple one-step replica symmetry broken solution, and discuss the problems in interpretation and generalization of this solution.

cond-mat.dis-nn↗

arXiv2000

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

In this letter, 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.

cond-mat.dis-nn↗

arXiv2000

Statistical mechanics perspective on the phase transition in vertex covering 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 $c\ll 0.5$, 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 the critical backbone size $b_c(c)$ 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.

cond-mat.stat-mech↗

arXiv2000

The 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.

cond-mat.dis-nn↗

arXiv1999

A variational description of the ground state structure in random satisfiability problems

A variational approach to finite connectivity spin-glass-like models is developed and applied to describe the structure of optimal solutions in random satisfiability problems. Our variational scheme accurately reproduces the known replica symmetric results and also allows for the inclusion of replica symmetry breaking effects. For the 3-SAT problem, we find two transitions as the ratio $α$ of logical clauses per Boolean variables increases. At the first one $α_s \simeq 3.96$, a non-trivial organization of the solution space in geometrically separated clusters emerges. The multiplicity of these clusters as well as the typical distances between different solutions are calculated. At the second threshold $α_c \simeq 4.48$, satisfying assignments disappear and a finite fraction $B_0 \simeq 0.13$ of variables are overconstrained and take the same values in all optimal (though unsatisfying) assignments. These values have to be compared to $α_c \simeq 4.27, B_0 \simeq 0.4$ obtained from numerical experiments on small instances. Within the present variational approach, the SAT-UNSAT transition naturally appears as a mixture of a first and a second order transition. For the mixed $2+p$-SAT with $p<2/5$, the behavior is as expected much simpler: a unique smooth transition from SAT to UNSAT takes place at $α_c=1/(1-p)$.

cond-mat.dis-nn↗

arXiv1999

Entropy and typical properties of Nash equilibria in two-player games

We use techniques from the statistical mechanics of disordered systems to analyse the properties of Nash equilibria of bimatrix games with large random payoff matrices. By means of an annealed bound, we calculate their number and analyse the properties of typical Nash equilibria, which are exponentially dominant in number. We find that a randomly chosen equilibrium realizes almost always equal payoffs to either player. This value and the fraction of strategies played at an equilibrium point are calculated as a function of the correlation between the two payoff matrices. The picture is complemented by the calculation of the properties of Nash equilibria in pure strategies.

cond-mat.dis-nn↗