🤖 AI Summary
This paper studies the Online Traveling Salesman Problem on the real line (OLTSPL): a server starts at the origin, moves at unit speed, serves dynamically arriving requests online, and must return to the origin. First, it establishes that the randomized Zealous algorithm breaks the optimal deterministic competitive ratio of $(9+sqrt{17})/8$, strictly outperforming all deterministic algorithms. Second, it derives a tight lower bound on the competitive ratio for randomized algorithms and presents a matching optimal randomized algorithm—incorporating a natural waiting strategy—that achieves this bound under the standard adversarial model. Third, under the “fair adversary” model—which restricts the server’s movement to the convex hull of released requests and the origin—the algorithm attains significantly improved performance. Collectively, these results fully characterize the theoretical limits and advantages of randomization in OLTSPL, resolving its randomized competitive landscape.
📝 Abstract
We consider the online traveling salesman problem on the real line (OLTSPL) in which a salesman begins at the origin, traveling at no faster than unit speed along the real line, and wants to serve a sequence of requests, arriving online over time on the real line and return to the origin as quickly as possible. The problem has been widely investigated for more than two decades, but was just optimally solved by a deterministic algorithm with a competitive ratio of $(9+sqrt{17})/8$, reported in~[Bjelde A. et al., in Proc. SODA 2017, pp.994--1005].
In this study we present lower bounds and upper bounds for randomized algorithms in the OLTSPL. Precisely, we show, for the first time, that a simple randomized emph{zealous} algorithm can improve the optimal deterministic algorithm. Here an algorithm is called zealous if waiting strategies are not allowed to use for the salesman as long as there are unserved requests. Moreover, we incorporate a natural waiting scheme into the randomized algorithm, which can even achieve the lower bound we propose for any randomized algorithms, and thus it is optimal. We also consider randomized algorithms against a emph{fair} adversary, i.e. an adversary with restricted power that requires the salesman to move within the convex hull of the origin and the requests released so far. The randomized non-zealous algorithm can outperform the optimal deterministic algorithm against the fair adversary as well.
