Deterministic annealing techniques for a discrete-time neural-network updating in a block-sequential mode

A global stability criterion for two constituent parameters of the discrete-time neural network updating in a block-sequential mode is derived, and two deterministic annealing techniques incorporating its stability condition are studied. One technique concerns reducing the decay rate of the membrane potential gradually toward zero; the other relates to increasing the neuron gain gradually toward infinity while updating the neuron states iteratively. It is shown that the deterministic annealing for parallel or partial-parallel updating can be successfully accomplished without falling into sustained oscillations by properly controlling the decay rate of the membrane potential as well as the neuron gain. It is also demonstrated that near optimal solutions are obtained for parallel, partial-parallel, and sequential updating by the suitable selection of the two constituent parameters.

Local routing algorithms based on Potts neural networks

A feedback neural approach to static communication routing in asymmetric networks is presented, where a mean field formulation of the Bellman-Ford method for the single unicast problem is used as a common platform for developing algorithms for multiple unicast, multicast and multiple multicast problems. The appealing locality and update philosophy of the Bellman-Ford algorithm is inherited. For all problem types the objective is to minimize a total connection cost, defined as the sum of the individual costs of the involved arcs, subject to capacity constraints. The methods are evaluated for synthetic problem instances by comparing to exact solutions for cases where these are accessible, and else with approximate results from simple heuristics. In general, the quality of the results are better than those of the heuristics. Furthermore, the computational demands are modest, even when the distributed nature of the the approach is unexploited numerically.

A fast neural-network algorithm for VLSI cell placement

Cell placement is an important phase of current VLSI circuit design styles such as standard cell, gate array, and Field Programmable Gate Array (FPGA). Although nondeterministic algorithms such as Simulated Annealing (SA) were successful in solving this problem, they are known to be slow. In this paper, a neural network algorithm is proposed that produces solutions as good as SA in substantially less time. This algorithm is based on Mean Field Annealing (MFA) technique, which was successfully applied to various combinatorial optimization problems. A MFA formulation for the cell placement problem is derived which can easily be applied to all VLSI design styles. To demonstrate that the proposed algorithm is applicable in practice, a detailed formulation for the FPGA design style is derived, and the layouts of several benchmark circuits are generated. The performance of the proposed cell placement algorithm is evaluated in comparison with commercial automated circuit design software Xilinx Automatic Place and Route (APR) which uses SA technique. Performance evaluation is conducted using ACM/SIGDA Design Automation benchmark circuits. Experimental results indicate that the proposed MFA algorithm produces comparable results with APR. However, MFA is almost 20 times faster than APR on the average.

A Potts Neuron Approach to Communication Routing

A feedback neural network approach to communication routing problems is developed, with emphasis on multiple shortest path problems, with several requests for transmissions between distinct start and end nodes. The basic ingredients are a set of Potts neurons for each request, with interactions designed to minimize path lengths and prevent overloading of network arcs. The topological nature of the problem is conveniently handled using a propagator matrix approach. Although the constraints are global, the algorithmic steps are based entirely on local information, facilitating distributed implementations. In the polynomially solvable single-request case, the approach reduces to a fuzzy version of the Bellman-Ford algorithm. The method is evaluated for synthetic problems of varying sizes and load levels, by comparing to exact solutions from a branch-and-bound method, or to approximate solutions from a simple heuristic. With very few exceptions, the Potts approach gives high-quality legal solutions. The computational demand scales merely as the product of the numbers of requests, nodes, and arcs.

Airline Crew Scheduling with Potts Neurons

A Potts feedback neural network approach for finding good solutions to resource allocation problems with a nonfixed topology is presented. As a target application, the airline crew scheduling problem is chosen. The topological complication is handled by means of a propagator defined in terms of Potts neurons. The approach is tested on artificial random problems tuned to resemble real-world conditions. Very good results are obtained for a variety of problem sizes. The computer time demand for the approach only grows like (number of flights)3. A realistic problem typically is solved within minutes, partly due to a prior reduction of the problem size, based on an analysis of the local arrival and departure structure at the single airports.

A Study of the Mean Field Approach to Knapsack Problems

The mean field theory approach to knapsack problems is extended to multiple knapsacks and generalized assignment problems with Potts mean field equations governing the dynamics. Numerical tests against "state of the art" conventional algorithms shows good performance for the mean field approach. The inherently parallelism of the mean field equations makes them suitable for direct implementations in microchips. It is demonstrated numerically that the performance is essentially not affected when only a limited number of bits is used in the mean field equations. Also, a hybrid algorithm with linear programming and mean field components is showed to further improve the performance for the difficult homogeneous N x M knapsack problem. Copyright 1997 Elsevier Science Ltd. All Rights Reserved.

Neural Networks for Optimization Problems with Inequality Constraints: The Knapsack Problem

A strategy for finding approximate solutions to discrete optimization problems with inequality constraints using mean field neural networks is presented. The constraints x ≤ 0 are encoded by x⊖(x) terms in the energy function. A careful treatment of the mean field approximation for the self-coupling parts of the energy is crucial, and results in an essentially parameter-free algorithm. This methodology is extensively tested on the knapsack problem of size up to 103 items. The algorithm scales like NM for problems with N items and M constraints. Comparisons are made with an exact branch and bound algorithm when this is computationally possible (N ≤ 30). The quality of the neural network solutions consistently lies above 95% of the optimal ones at a significantly lower CPU expense. For the larger problem sizes the algorithm is compared with simulated annealing and a modified linear programming approach. For "nonhomogeneous" problems these produce good solutions, whereas for the more difficult "homogeneous" problems the neural approach is a winner with respect to solution quality and/or CPU time consumption. The approach is of course also applicable to other problems of similar structure, like set covering.