Improved lower bounds for the universal and a priori TSP

Authors:
Igor Gorodezky;Robert D. Kleinberg;David B. Shmoys;Gwen Spencer
Affiliations:
Center for Applied Mathematics, Cornell University, Ithaca, NY and Palantir Technologies, Palo Alto, CA;Dept. of Computer Science, Cornell University, Ithaca, NY;School of ORIE and Dept. of Computer Science, Cornell University, Ithaca, NY;School of ORIE, Cornell University, Ithaca, NY
Venue:
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Year:
2010

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Abstract

We consider two partial-information generalizations of the metric traveling salesman problem (TSP) in which the task is to produce a total ordering of a given metric space that performs well for a subset of the space that is not known in advance. In the universal TSP, the subset is chosen adversarially, and in the a priori TSP it is chosen probabilistically. Both the universal and a priori TSP have been studied since the mid-80's, starting with the work of Bartholdi & Platzman and Jaillet, respectively. We prove a lower bound of Ω(log n) for the universal TSP by bounding the competitive ratio of shortest-path metrics on Ramanujan graphs, which improves on the previous best bound of Hajiaghayi, Kleinberg & Leighton, who showed that the competitive ratio of the n × n grid is Ω(√6log n/log log n). Furthermore, we show that for a large class of combinatorial optimization problems that includes TSP, a bound for the universal problem implies a matching bound on the approximation ratio achievable by deterministic algorithms for the corresponding black-box a priori problem. As a consequence, our lower bound of Ω(log n) for the universal TSP implies a matching lower bound for the black-box a priori TSP.