The 0-1 integer linear programming feasibility problem is an important NP-complete problem. This paper proposes a continuous-time dynamical system for solving that problem without getting trapped in non-solution local minima. First, the problem is transformed to an easier form in linear time. Then, we propose an "impulse algorithm" to escape from local traps and show its performance is better than randomization for escaping traps. Second, we present the time-to-solution distribution of the impulse algorithm and compare it with exhaustive search to see its advantages. Third, we show that the fractional size of the basin of attraction of the global minimum is significantly larger than @math , the corresponding discrete probability for exhaustive search. Finally, we conduct a case study to show that the location of the basin is independent of different dimensions. These findings reveal a better way to solve the 0-1 integer linear programming feasibility problem continuously and show that its cost could be less than discrete methods in average cases.
If one reduces the 0-1 ILP feasibility problem to SAT, then the new problem can be solved continuously without getting caught in local traps. For example, Ercsey-Ravasz and Toroczkai added exponentially growing auxiliary terms to each constraint, so that the network is able to escape from traps and converge to the @math -SAT solution, possibly after a period of chaotic behavior @cite_1 . Moln 'ar and Ercsey-Ravasz designed an asymmetric continuous-time neural network to solve @math -SAT problems @cite_14 , and this kind of network could also be used to solve the MaxSAT problem with good analog performance @cite_9 . These methods inspired us to design a similar continuous-time dynamical system to solve the 0-1 ILP feasibility problem directly, since the reduction to 3-SAT would enlarge the size of the problem.
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Many real-life optimization problems can be formulated in Boolean logic as MaxSAT, a class of problems where the task is finding Boolean assignments to variables satisfying the maximum number of logical constraints. Since MaxSAT is NP-hard, no algorithm is known to efficiently solve these problems. Here we present a continuous-time analog solver for MaxSAT and show that the scaling of the escape rate, an invariant of the solver’s dynamics, can predict the maximum number of satisfiable constraints, often well before finding the optimal assignment. Simulating the solver, we illustrate its performance on MaxSAT competition problems, then apply it to two-color Ramsey number R(m, m) problems. Although it finds colorings without monochromatic 5-cliques of complete graphs on N ≤ 42 vertices, the best coloring for N = 43 has two monochromatic 5-cliques, supporting the conjecture that R(5, 5) = 43. This approach shows the potential of continuous-time analog dynamical systems as algorithms for discrete optimization. There has been a long history of using neural networks for combinatorial optimization and constraint satisfaction problems. Symmetric Hopfield networks and similar approaches use steepest descent dynamics, and they always converge to the closest local minimum of the energy landscape. For finding global minima additional parameter-sensitive techniques are used, such as classical simulated annealing or the so-called chaotic simulated annealing, which induces chaotic dynamics by addition of extra terms to the energy landscape. Here we show that asymmetric continuous-time neural networks can solve constraint satisfaction problems without getting trapped in non-solution attractors. We concentrate on a model solving Boolean satisfiability (k-SAT), which is a quintessential NP-complete problem. There is a one-to-one correspondence between the stable fixed points of the neural network and the k-SAT solutions and we present numerical evidence that limit cycles may also be avoided by appropriately choosing the parameters of the model. This optimal parameter region is fairly independent of the size and hardness of instances, this way parameters can be chosen independently of the properties of problems and no tuning is required during the dynamical process. The model is similar to cellular neural networks already used in CNN computers. On an analog device solving a SAT problem would take a single operation: the connection weights are determined by the k-SAT instance and starting from any initial condition the system searches until finding a solution. In this new approach transient chaotic behavior appears as a natural consequence of optimization hardness and not as an externally induced effect. Constraint-satisfaction problems are among the computationally hardest tasks: solutions are efficiently checkable, but no efficient algorithms are known to compute those solutions. Fresh insight might come from physics. A study mapping optimization hardness onto the phenomena of turbulence and chaos suggests that constraint-satisfaction problems can be tackled using analog devices.