PERFORMANCE OF NEURAL ALGORITHMS FOR MAXIMUM-CUT PROBLEMS

1992 ◽  
Vol 02 (04) ◽  
pp. 389-395 ◽  
Author(s):  
KIICHI URAHAMA
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problems ◽  
Set Partitioning ◽  
Combinatorial Optimization Problems ◽  
Maximum Cut ◽  
Cut Problems

The author previously developed a new neural algorithm effective for set-partitioning combinatorial optimization problems by extending the logistic transformation used in the Hopfield algorithm into its multivariable version. In this letter the performance of the algorithm is theoretically evaluated and it is proved that this algorithm is 1/p-approximate for p-partitioning maximum-cut problems.

CONSTRAINED POTTS MEAN FIELD SYSTEMS AND THEIR ELECTRONIC IMPLEMENTATION

1994 ◽  
Vol 05 (03) ◽  
pp. 229-239 ◽  
Author(s):  
KIICHI URAHAMA ◽  
TADASHI YAMADA
Keyword(s):  
Combinatorial Optimization ◽  
Present Method ◽  
Electronic Circuit ◽  
Optimization Problems ◽  
Mean Field ◽  
Local Optimum ◽  
Combinatorial Optimization Problems ◽  
Maximum Cut ◽  
Cut Problems ◽  
Additional Constraints

The Potts mean field approach for solving combinatorial optimization problems subject to winner-takes-all constraints is extended for problems subject to additional constraints. Extra variables corresponding to the Lagrange multipliers are incorporated into the Potts formulation for the additional constraints to be satisfied. The extended Potts equations are solved by using constrained gradient descent differential systems. This gradient system is proven theoretically to always produce a legal local optimum solution of the constrained combinatorial optimization problems. An analog electronic circuit implementing the present method is designed on the basis of the previous Potts electronic circuit. The performance of the present method is theoretically evaluated for the constrained maximum cut problems. The lower bound of the cut size obtained with the present method is proven to be the same as that of the basic Potts scheme for the unconstrained maximum cut problems.

PERFORMANCE OF THE RELAXATION ALGORITHM FOR MAXIMUM-CUT PROBLEMS

1996 ◽  
Vol 06 (04) ◽  
pp. 375-384
Author(s):  
KIICHI URAHAMA ◽  
HIROSHI NISHIYUKI
Keyword(s):  
Graph Partitioning ◽  
Optimization Problems ◽  
Set Partitioning ◽  
Performance Guarantee ◽  
Relaxation Algorithm ◽  
Combinatorial Optimization Problems ◽  
Maximum Cut ◽  
Computational Speed ◽  
Cut Problems ◽  
Present Algorithm

A relaxation algorithm is presented for solving a class of combinatorial optimization problems called set-partitioning tasks. The convergence property of the presented algorithm is investigated theoretically. A performance guarantee is derived theoretically for the present algorithm applied to an NP-hard example problem called the maximum-cut graph partitioning. The experimental examination of its performance manifests its superiority in computational speed to the conventional gradient method.

GUIDING CONSTRUCTIVE SEARCH WITH STATISTICAL INSTANCE-BASE LEARNING

2002 ◽  
Vol 11 (02) ◽  
pp. 247-266 ◽  
Author(s):  
ORESTIS TELELIS ◽  
PANAGIOTIS STAMATOPOULOS
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problems ◽  
Statistical Information ◽  
Combinatorial Problems ◽  
Set Partitioning ◽  
Greedy Heuristics ◽  
High Quality ◽  
Search Spaces ◽  
Combinatorial Optimization Problems ◽  
Real World Applications

Several real world applications involve solving combinatorial optimization problems. Commonly, existing heuristic approaches are designed to address specific difficulties of the underlying problem and are applicable only within its framework. We suspect, however, that search spaces of combinatorial problems are rich in intuitive statistical and numerical information, which could be exploited heuristically in a generic manner, towards achievement of optimized solutions. Our work presents such a heuristic methodology, which can be adequately configured for several types of optimization problems. Experimental results are discussed, concerning two widely used problem models, namely the Set Partitioning and the Kanpsack problems. It is shown that, by gathering statistical information upon previously found solutions to the problems, the heuristic is able to incrementally adapt its behaviour and reach high quality solutions, exceeding the ones obtained by commonly used greedy heuristics.

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