Multi-Reduction Method for Traveling Salesman Problem: A Dimension Reduction Approach - Pr, Study Guides, Projects, Research of Mathematics

Download Multi-Reduction Method for Traveling Salesman Problem: A Dimension Reduction Approach - Pr and more Study Guides, Projects, Research Mathematics in PDF only on Docsity! Multi Reduction Method for Traveling Salesman Problem Fei Xue AMSC 662 Presentation Dec 7,2004 UMCP Intro. to TSP Traveling Salesman Problem (TSP) Given a set of fixed cities, find the shortest circuit (opt sol.) that passes each city once Most famous NP-complete Problem Num. of sol. & time cost are surprisingly great 4.23×1016321 sol. to 5000-city TSP vs. 1087 elementary particles in the universe 92 years on single Intel Xeon 2.8GHz Processor for the 24,978-City Sweden TSP MR Overview Reduction/retrieval in one level Original TSP Sub-opt-1 Sub-opt-2 Sub-opt-k Reduced TSP Common partial circuits (sub)opt of reduced TSP (sub)opt of original TSP CLK, LKH 1. Find common partial circuits 2. SA 3. Retrieval …… Performance Avg. Performance Reported 0.04-0.07% above the opt. for 5 TSP with 2-6k cities. Method to find opt NOT reported I don’t believe the reported time cost Best Performance No report yet. Why don’t people use it to find better sub-optimal, or the optimal? If all sub-optimal used contain a ‘wrong’ edge, we can never find the optimal Illustration of step1 Search for common partial circuits Heap sortSub-opt-A Sorted sub-opt-A 3 5 7 1 2 6 8 4 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 4 5 1 8 2 6 3 7 4 7 5 3 8 2 6 1 8 3 2 1 7 5 6 4 In sub-opt-B, search each element in sorted sub-opt-A and find its index in sub-opt-A Sub-opt-B 7-5-3 2-6and are common partial circuits of the two sub-opt 3,2,1 are consecutive numbers and so are 5,6 Time Complexity—O(nlogn) Experiment Results Summary TSP Name pr1002 rl1889 pr2392 pcb3038 pcb3038 fl3795 Sub-opt-1 259048 316562 378206 137767 137698 28788 Sub-opt-2 259125 316580 378243 137785 137699 28813 Sub-opt-MR 259045 316536 378175 137765 137698 28801 Method CLK CLK CLK CLK LKH LKH Above OPT OPT OPT 0.0378% 0.0516% F F Pros and Cons Strengths Easier to implement Convenient to make it distributed Attractive average performance Limitations (empiricism) 2 exc. sub-opt often share same ‘wrong’ edges 1 exc. & 1 avg. sub-opt seldom work Retrieved circuit from 2 avg. sub-opt seldom beat an exc. one Future work Try other methods than SA Time cost of SA—the bottleneck to further the research of multi-level reduction LK variant is most promising choice Look for empirical choice of sub-opt Any statistical relationship among # of sub- opt, their lengths and the retrieved one?