On the b-Partite Random Asymmetric Traveling Salesman Problem and Its Assignment Relaxation

Authors:
Andreas Baltz;Tomasz Schoen;Anand Srivastav
Affiliations:
-;-;-
Venue:
APPROX '01/RANDOM '01 Proceedings of the 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and 5th International Workshop on Randomization and Approximation Techniques in Computer Science: Approximation, Randomization and Combinatorial Optimization
Year:
2001

Citing 2
Cited 0

When is the Assignment Bound Tight for the Asymmetric Traveling-Salesman Problem?

SIAM Journal on Computing
The probabilistic relationship between the assignment and asymmetric traveling salesman problems

SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms

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Abstract

We study the relationship between the value of optimal solutions to the random asymmetric b-partite traveling salesman problem and its assignment relaxation. In particular we prove that given a bn×bn weight matrix W = (wij) such that each finite entry has probability pn of being zero, the optimal values bATSP(W) and AP(W) are equal (almost surely), whenever npn tends to infinity with n. On the other hand, if npn tends to some constant c then P[bATSP(W) ≠ AP(W)] Ɛ 0, and for npn → 0, P[bATSP(W) ≠ AP(W)] → 1 (a.s.). This generalizes results of Frieze, Karp and Reed (1995) for the ordinary asymmetric TSP.