Asymptotic bounds on minimum number of disks required to hide a disk

We consider the problem of blocking all rays emanating from a closed unit disk with a minimum number of closed unit disks in the two-dimensional space, where the minimum distance from a disk to any other disk is given. We study the asymptotic behavior of the minimum number of disks as the minimum mutual distance approaches infinity. Using a regular ordering of disks on concentric circular rings we derive an upper bound and prove that the minimum number of disks required for blocking is quadratic in the minimum distance between the disks

Simulated annealing and Boltzmann machines

No abstract

Combinatorial optimization on a Boltzmann machine

We discuss the problem of solving (approximately) combinatorial optimization problems on a Boltzmann machine. It is shown for a number of combinatorial optimization problems how they can be mapped directly onto a Boltzmann machine by choosing appropriate connection patterns and connection strengths. In this way maximizing the consensus in the Boltzmann machine is equivalent to finding an optimal solution of the corresponding optimization problem. The approach is illustrated by numerical results obtained by applying the model of Boltzmann machines to randomly generated instances of the independent set, the max cut, and the graph coloring problem. For these instances the Boltzmann machine finds near-optimal solutions whose quality is comparable to that obtained with sequential simulated annealing algorithms. The advantage of the Boltzmann machine is the potential for carrying out operations in parallel. For the problems we have been investigating, this results in a considerable speedup over the sequential simulated annealing algorithms

Boltzmann machines as a model for parallel annealing

The potential of Boltzmann machines to cope with difficult combinatorial optimization problems is investigated. A discussion of various (parallel) models of Boltzmann machines is given based on the theory of Markov chains. A general strategy is presented for solving (approximately) combinatorial optimization problems with a Boltzmann machine. The strategy is illustrated by discussing the details for two different problems, namely MATCHING and GRAPH PARTITIONING

Simulated annealing and Boltzmann machines

No abstract

Boltzmann machines

No abstract

Scheduling periodic tasks with slack

We consider the problem of nonpreemptively scheduling periodic tasks on a minimum number of identical processors, assuming that some slack is allowed in the time between successive executions of a periodic task. We prove that the problem is NP-hard in the strong sense. Necessary and sufficient conditions are derived for scheduling two periodic tasks on a single processor, and for combining two periodic tasks into one larger task. Based on these results, we propose an approximation algorithm