Asymptotic experimental analysis for the Held-Karp traveling salesman bound
- AT&T Bell Laboratories, Murray Hill, NJ (United States)
- Amherst College, MA (United States)
- Silicon Graphics, Mountain View, CA (United States)
The Held-Karp (HK) lower bound is the solution to the linear programming relaxation of the standard integer Programming formulation of the traveling salesman problem (TSP). For numbers of cities N up to 30,000 or more it can be computed exactly using the Simplex method and appropriate separation algorithms, and for N up to a million good approximations can be obtained via iterative Lagrangean relaxation techniques first suggested by Held and Karp. In this paper, we consider three applications of our ability to compute/approximate this bound. First, we provide empirical evidence in support of using the HK bound as a stand-in for the optimal tour length when evaluating the quality of near-optimal tours. We show that for a wide variety of randomly generated instance types the optimal tour length averages less than 0.8% over the HK bound, and even for the real-world instances in TSPLIB the gap is almost always less than 2%. Moreover, our data indicates that the HK bound can provide substantial {open_quotes}variance reduction{close_quotes} in experimental studies involving randomly generated instances. Second, we estimate the expected HK bound as a function of N for a variety of random instance types, based on extensive computations. For example, for random Euclidean instances it is known that the ratio of the Held-Karp bound to {radical}N approaches a constant C{sub HK}, and we estimate both that constant and the rate of convergence to it. Finally, we combine this information with our earlier results on expected HK gaps to obtain estimates for expected optimal tour lengths. For random Euclidean instances, we conclude that C{sub OPT}, the limiting ratio of the optimal tour length to {radical}N, is .7124 +- .0002, thus invalidating the commonly cited estimates of .749 and .765 and undermining many claims of good heuristic performance based on those estimates. For random distance matrices, the expected optimal tour length appears to be about 2.042.
- OSTI ID:
- 416817
- Report Number(s):
- CONF-960121-; TRN: 96:005887-0040
- Resource Relation:
- Conference: 7. annual ACM-SIAM symposium on discrete algorithms, Atlanta, GA (United States), 28-30 Jan 1996; Other Information: PBD: 1996; Related Information: Is Part Of Proceedings of the seventh annual ACM-SIAM symposium on discrete algorithms; PB: 596 p.
- Country of Publication:
- United States
- Language:
- English
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