Continuous numerical algorithm for a class of combinatorial optimization problems
It is well known that many optimization problems become very hard because discrete constraints of variables are introduced. For combinatorial optimization problems, almost present algorithms find optimal solution in a discrete set and are usually complicated (the complexity is exponential in time). We consider a class of combinatorial optimization problems including TSP, max-cut problem, k-coloring problem (4-coloring problem), etc.; all these problems are known as NP-complete. At first, a unifying 0-1 quadratic programming model is constructed to formulate above problems. This model`s constraints are very special and separable. For this model we have obtained an equivalence between the discrete model and its relaxed problem in the sense of global or local minimum; this equivalence guarantees that a 0-1 solution will be obtained in a simple constructing process by use of a continuous global or local minimum. Thus, those combinatorial optimization problems can be solved by being converted into a special non-convex quadratic programming. By use of some special properties of this model, a necessary and sufficient condition for local minimum of this model is given. Secondly, the well-known linear programming approximate algorithm is quoted and is verified to converge to one local minimum certainly (this algorithm is well known to converge only on K-T point but not local minimum certainly for the general case). The corresponding linear programming is very easy because the constraints are separable. Finally, several class examining problems are constructed to test the algorithm for all above combinatorial optimization problems, and sufficient computational tests including solving examining problems and comparing with other well-known algorithms are reported. The computational tests show the effectiveness of this algorithm. For example, a k-coloring (k {double_dagger} 4) problem with 1000 nodes can be easily solved on microcomputer (COMPAQ 386/25e) in 15 minutes.
- OSTI ID:
- 35877
- Report Number(s):
- CONF-9408161-; TRN: 94:009753-0139
- Resource Relation:
- Conference: 15. international symposium on mathematical programming, Ann Arbor, MI (United States), 15-19 Aug 1994; Other Information: PBD: 1994; Related Information: Is Part Of Mathematical programming: State of the art 1994; Birge, J.R.; Murty, K.G. [eds.]; PB: 312 p.
- Country of Publication:
- United States
- Language:
- English
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