求解TSP问题的多级归约算法

TSP(traveling salesman problem)问题是最经典的NP-hard组合优化问题之一.长期以来,人们一直在寻求快速、高效的近似算法,以便在合理的计算时间内解决大规模问题.由于对较大规模的问题,目前的近似算法尚不能在较短的时间内给出高质量的解,因此提出了多重归约算法.该算法的基本原理是通过对TSP问题的局部最优解与全局最优解之间关系的分析,发现对局部最优解的简单的相交操作能以很高的概率得到全局最优解的部分解.利用这些部分解可以大大缩小原问题的搜索空间,同时也不会降低搜索的性能.这就是所谓的归约原理.再通过多次归约使问题的规模降到足够小,然后对这个较小规模的实例直接用已有的算法求解,最后通过相反的次序拼接部分解,最终得到一个合法的解.在TSPLIB(traveling salesman problem library)中,典型实例上的实验结果表明,此算法在求解质量和求解速度上与目前已知的算法相比有较大的改进.

A Multilevel Reduction Algorithm to TSP

The TSP (traveling salesman problem) is one of the typical NP-hard problems in combinatorial optimization problem. The fast and effective approximate algorithms are needed to solve the large-scale problem in reasonable computing time. The known approximate algorithm can not give a good enough tour for the larger instance in reasonable time. So an algorithm called multilevel reduction algorithm is proposed, which is based on the analysis of the relation of local optimal tour and global optimal tour of the TSP. The partial tour of the global optimal tour is obtained by a very high probability through simply intersecting the local optimal tours. Using these partial tours it could contract the searching space of the original problem, at the same time it did not cut the searching capability down, this is the so-called reduction theory. And then the scale of the instance could be reduced small enough by multi-reduction. Finally it could solve the small-scaled instance using the known best algorithm and get a good enough tour by putting the partial tours together. The experimental results on TSPLIB (traveling salesman problem library) show that the algorithm could almost get optimal tour every time for instance in reasonable time and thus outperformed the known best ones in the quality of solutions and the running time.