Ground-state clusters of two-, three-, and four-dimensional +/-J Ising spin glasses

Replica symmetry and replica symmetry breaking for the traveling salesperson problem

We study the energy landscape of the traveling salesperson problem (TSP) using exact ground states and a novel linear programming approach to generate excited states with closely defined properties. We look at four different ensembles, notably the classic finite dimensional Euclidean TSP and the mean-field-like (1,2)-TSP, which has its origin directly in the mapping of the Hamiltonian circuit problem on the TSP. Our data supports previous conjectures that the Euclidean TSP does not show signatures of replica symmetry breaking neither in two nor in higher dimension. On the other hand the (1,2)-TSP exhibits some signature which does not exclude broken replica symmetry, making it a candidate for further studies in the future.

From near to eternity: Spin-glass planting, tiling puzzles, and constraint-satisfaction problems

We present a methodology for generating Ising Hamiltonians of tunable complexity and with a priori known ground states based on a decomposition of the model graph into edge-disjoint subgraphs. The idea is illustrated with a spin-glass model defined on a cubic lattice, where subproblems, whose couplers are restricted to the two values {-1,+1}, are specified on unit cubes and are parametrized by their local degeneracy. The construction is shown to be equivalent to a type of three-dimensional constraint-satisfaction problem known as the tiling puzzle. By varying the proportions of subproblem types, the Hamiltonian can span a dramatic range of typical computational complexity, from fairly easy to many orders of magnitude more difficult than prototypical bimodal and Gaussian spin glasses in three space dimensions. We corroborate this behavior via experiments with different algorithms and discuss generalizations and extensions to different types of graphs.

Ground-state phase-space structures of two-dimensional ±J spin glasses: A network approach

We illustrate a complex-network approach to study the phase spaces of spin glasses. By mapping the whole ground-state phase spaces of two-dimensional Edwards-Anderson bimodal (±J) spin glasses exactly into networks for analysis, we discovered various phase-space properties. The Gaussian connectivity distribution of the phase-space networks demonstrates that both the number of free spins and the visiting frequency of all microstates follow the Gaussian distribution. The spectra of phase-space networks are Gaussian, which is proven to be exact when the system is infinitely large. The phase-space networks exhibit community structures. By coarse graining to the community level, we constructed a network representing the entropy landscape of the ground state and discovered its scale-free property. The phase-space networks exhibit fractal structures, as a result of the rugged entropy landscape. Moreover, we show that the connectivity distribution, community structures, and fractal structures change drastically at the ferromagnetic-to-glass phase transition. These quantitative measurements of the ground states provide new insight into the study of spin glasses. The phase-space networks of spin glasses share a number of common features with those of lattice gases and geometrically frustrated spin systems and form a new class of complex networks with unique topology.

Clustering analysis of the ground-state structure of the vertex-cover problem

Vertex cover is one of the classical NP-complete problems in theoretical computer science. A vertex cover of a graph is a subset of vertices such that for each edge at least one of the two endpoints is contained in the subset. When studied on Erdo s-Re nyi random graphs (with connectivity c) one observes a threshold behavior: In the thermodynamic limit the size of the minimal vertex cover is independent of the specific graph. Recent analytical studies show that on the phase boundary, for small connectivities c<e , the system is replica symmetric, while for larger connectivities replica symmetry breaking occurs. This change coincides with a change of the typical running time of algorithms from polynomial to exponential. To understand the reasons for this behavior and to compare with the analytical results, we numerically analyze the structure of the solution landscape. For this purpose, we have also developed an algorithm, which allows the calculation of the backbone, without the need to enumerate all solutions. We study exact solutions found with a branch-and-bound algorithm as well as configurations obtained via a Monte Carlo simulation. We analyze the cluster structure of the solution landscape by direct clustering of the states, by analyzing the eigenvalue spectrum of correlation matrices and by using a hierarchical clustering method. All results are compatible with a change at c=e . For small connectivities, the solutions are collected in a finite small number of clusters, while the number of clusters diverges slowly with system size for larger connectivities and replica symmetry breaking, but not one-step replica symmetry breaking (1-RSB) occurs.