Biased landscapes for random Constraint Satisfaction Problems

The typical complexity of Constraint Satisfaction Problems (CSPs) can be investigated by means of random ensembles of instances. The latter exhibit many threshold phenomena besides their satisfiability phase transition, in particular a clustering or dynamic phase transition (related to the tree reconstruction problem) at which their typical solutions shatter into disconnected components. In this paper we study the evolution of this phenomenon under a bias that breaks the uniformity among solutions of one CSP instance, concentrating on the bicoloring of k-uniform random hypergraphs. We show that for small k the clustering transition can be delayed in this way to higher density of constraints, and that this strategy has a positive impact on the performances of Simulated Annealing algorithms. We characterize the modest gain that can be expected in the large k limit from the simple implementation of the biasing idea studied here. This paper contains also a contribution of a more methodological nature, made of a review and extension of the methods to determine numerically the discontinuous dynamic transition threshold.Comment: 32 pages, 16 figure

A Landscape Analysis of Constraint Satisfaction Problems

We discuss an analysis of Constraint Satisfaction problems, such as Sphere Packing, K-SAT and Graph Coloring, in terms of an effective energy landscape. Several intriguing geometrical properties of the solution space become in this light familiar in terms of the well-studied ones of rugged (glassy) energy landscapes. A `benchmark' algorithm naturally suggested by this construction finds solutions in polynomial time up to a point beyond the `clustering' and in some cases even the `thermodynamic' transitions. This point has a simple geometric meaning and can be in principle determined with standard Statistical Mechanical methods, thus pushing the analytic bound up to which problems are guaranteed to be easy. We illustrate this for the graph three and four-coloring problem. For Packing problems the present discussion allows to better characterize the `J-point', proposed as a systematic definition of Random Close Packing, and to place it in the context of other theories of glasses.Comment: 17 pages, 69 citations, 12 figure

Glassy Behavior and Jamming of a Random Walk Process for Sequentially Satisfying a Constraint Satisfaction Formula

Random $K$-satisfiability ($K$-SAT) is a model system for studying typical-case complexity of combinatorial optimization. Recent theoretical and simulation work revealed that the solution space of a random $K$-SAT formula has very rich structures, including the emergence of solution communities within single solution clusters. In this paper we investigate the influence of the solution space landscape to a simple stochastic local search process {\tt SEQSAT}, which satisfies a $K$-SAT formula in a sequential manner. Before satisfying each newly added clause, {\tt SEQSAT} walk randomly by single-spin flips in a solution cluster of the old subformula. This search process is efficient when the constraint density $\alpha$ of the satisfied subformula is less than certain value $\alpha_{cm}$; however it slows down considerably as $\alpha > \alpha_{cm}$ and finally reaches a jammed state at $\alpha \approx \alpha_{j}$. The glassy dynamical behavior of {\tt SEQSAT} for $\alpha \geq \alpha_{cm}$ probably is due to the entropic trapping of various communities in the solution cluster of the satisfied subformula. For random 3-SAT, the jamming transition point $\alpha_j$ is larger than the solution space clustering transition point $\alpha_d$, and its value can be predicted by a long-range frustration mean-field theory. For random $K$-SAT with $K\geq 4$, however, our simulation results indicate that $\alpha_j = \alpha_d$. The relevance of this work for understanding the dynamic properties of glassy systems is also discussed.Comment: 10 pages, 6 figures, 1 table, a mistake of numerical simulation corrected, and new results adde

Advantages of Unfair Quantum Ground-State Sampling

The debate around the potential superiority of quantum annealers over their classical counterparts has been ongoing since the inception of the field by Kadowaki and Nishimori close to two decades ago. Recent technological breakthroughs in the field, which have led to the manufacture of experimental prototypes of quantum annealing optimizers with sizes approaching the practical regime, have reignited this discussion. However, the demonstration of quantum annealing speedups remains to this day an elusive albeit coveted goal. Here, we examine the power of quantum annealers to provide a different type of quantum enhancement of practical relevance, namely, their ability to serve as useful samplers from the ground-state manifolds of combinatorial optimization problems. We study, both numerically by simulating ideal stoquastic and non-stoquastic quantum annealing processes, and experimentally, using a commercially available quantum annealing processor, the ability of quantum annealers to sample the ground-states of spin glasses differently than classical thermal samplers. We demonstrate that i) quantum annealers in general sample the ground-state manifolds of spin glasses very differently than thermal optimizers, ii) the nature of the quantum fluctuations driving the annealing process has a decisive effect on the final distribution over ground-states, and iii) the experimental quantum annealer samples ground-state manifolds significantly differently than thermal and ideal quantum annealers. We illustrate how quantum annealers may serve as powerful tools when complementing standard sampling algorithms.Comment: 13 pages, 11 figure