Probabilistic analysis for metric optimization problems has mostly been conducted on random Euclidean instances, but little is known about metric instances drawn from distributions other than the Euclidean.
This motivates our study of random metric instances for optimization problems obtained as follows: Every edge of a complete graph gets a weight drawn independently at random. The length of an edge is then the length of a shortest path (with respect to the weights drawn) that connects its two endpoints.
We prove structural properties of the random shortest path metrics generated in this way. Our main structural contribution is the construction of a good clustering. Then we apply these findings to analyze the approximation ratios of heuristics for matching, the traveling salesman problem (TSP), and the k-center problem, as well as the running-time of the 2-opt heuristic for the TSP. The bounds that we obtain are considerably better than the respective worst-case bounds. This suggests that random shortest path metrics are easy instances, similar to random Euclidean instances, albeit for completely different structural reasons.
|Title of host publication||Proceedings of the 38th International Symposium on Mathematical Foundations of Computer Science (MFCS 2013)|
|Editors||K. Chatterjee, J. Sgall|
|Place of Publication||Berlin|
|Number of pages||12|
|Publication status||Published - 2013|
|Name||Lecture Notes in Computer Science|
- Metric spaces
- Traveling Salesman Problem
- Random shortest path
- Approximation algorithms
- First-passage percolation