| Summary: | 碩士 === 國立清華大學 === 工業工程與工程管理學系 === 103 === The traveling salesman problem (TSP) is a well-studied combinatorial optimization problem. The problem requests for visiting cities all completely known and returning to the origin. In this paper, we consider its online version, called the online traveling salesman problem (OLTSP). The difference between TSP and OLTSP is that requests arrive at arbitrary time and no advance information about the requests is known a priori. The salesman moves at unit speed to serve all requests arrived online and goes back to a designated origin. The objective of the OLTSP is to find a route for the salesman that finishes his work as quickly as possible.
In this paper, we refer to the concept of fair adversary proposed by [Blom et al. INFORMS Journal on Computing, (2001), 13(2), pp. 138-148] and determine how to use waiting strategy properly. We consider two cases: the real line and the boundary of unit square, respectively. For the 1D space, i.e., the real line, we prove that the PQR algorithm presented by [Ausiello et al. Algorithmica, (2001), 29(4), pp. 560-581] has a better 5/3-competitive ratio against fair adversary. We also show that for any randomized algorithms, the lower bound is at least 4/3. For 2D space, i.e., the boundary of unit square, we provide a 2-competitive randomized algorithm against fair adversary, which can be improved to 1.7808, by using the waiting strategy. This result surpasses the deterministic lower bound of the 2D OLTSP.
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