Negative Learning Ant Colony Optimization for MaxSAT

Recently, a new negative learning variant of ant colony optimization (ACO) has been used to successfully tackle a range of combinatorial optimization problems. For providing stronger evidence of the general applicability of negative learning ACO, we investigate how it can be adapted to solve the Maximum Satisfiability problem (MaxSAT). The structure of MaxSAT is different from the problems considered to date and there exists only a few ACO approaches for MaxSAT. In this paper, we describe three negative learning ACO variants. They differ in the way in which sub-instances are solved at each algorithm iteration to provide negative feedback to the main ACO algorithm. In addition to using IBM ILOG CPLEX, two of these variants use existing MaxSAT solvers for this purpose. The experimental results show that the proposed negative learning ACO variants significantly outperform the baseline ACO as well as IBM ILOG CPLEX and the two MaxSAT solvers. This result is of special interest because it shows that negative learning ACO can be used to improve over the results of existing solvers by internally using them to solve smaller sub-instances.

Ground State Properties of the Diluted Sherrington-Kirkpatrick Spin Glass

We present a numerical study of ground states of the dilute versions of the Sherrington-Kirkpatrick (SK) mean-field spin glass. In contrast to so-called "sparse" mean-field spin glasses that have been studied widely on random networks of finite (average or regular) degree, the networks studied here are randomly bond diluted to an overall density p, such that the average degree diverges as ∼pN with the system size N. Ground state energies are obtained with high accuracy for random instances over a wide range of fixed p. Since this is an NP-hard combinatorial problem, we employ the extremal optimization heuristic to that end. We find that the exponent describing the finite-size corrections ω varies continuously with p, a somewhat surprising result, as one would not expect that gradual bond dilution would change the T=0 universality class of a statistical model. For p→1, the familiar result of ω(p=1)≈2/3 for the SK model is obtained.

Continuous extremal optimization for Lennard-Jones clusters

We explore a general-purpose heuristic algorithm for finding high-quality solutions to continuous optimization problems. The method, called continuous extremal optimization (CEO), can be considered as an extension of extremal optimization and consists of two components, one which is responsible for global searching and the other which is responsible for local searching. The CEO's performance proves competitive with some more elaborate stochastic optimization procedures such as simulated annealing, genetic algorithms, and so on. We demonstrate it on a well-known continuous optimization problem: the Lennard-Jones cluster optimization problem.

Extremal optimization for graph partitioning

Extremal optimization is a new general-purpose method for approximating solutions to hard optimization problems. We study the method in detail by way of the computationally hard (NP-hard) graph partitioning problem. We discuss the scaling behavior of extremal optimization, focusing on the convergence of the average run as a function of run time and system size. The method has a single free parameter, which we determine numerically and justify using a simple argument. On random graphs, our numerical results demonstrate that extremal optimization maintains consistent accuracy for increasing system sizes, with an approximation error decreasing over run time roughly as a power law t(-0.4). On geometrically structured graphs, the scaling of results from the average run suggests that these are far from optimal with large fluctuations between individual trials. But when only the best runs are considered, results consistent with theoretical arguments are recovered.

Optimization with extremal dynamics

We explore a new general-purpose heuristic for finding high-quality solutions to hard discrete optimization problems. The method, called extremal optimization, is inspired by self-organized criticality, a concept introduced to describe emergent complexity in physical systems. Extremal optimization successively updates extremely undesirable variables of a single suboptimal solution, assigning them new, random values. Large fluctuations ensue, efficiently exploring many local optima. We use extremal optimization to elucidate the phase transition in the 3-coloring problem, and we provide independent confirmation of previously reported extrapolations for the ground-state energy of +/-J spin glasses in d = 3 and 4.