本篇我們討論的主題為實線上之即時銷售員問題。銷售員由原點出發，並在實數線上以單位速度移動，目標是能越快處理完一連串的在實線上之即時需求後回到原點。這個問題已經被廣泛的討論逾二十年，截至目前的最佳結果僅是由 [Bjelde A. et al., in Proc. SODA 2017, pp.994–1005]提出的(9+√17)/8的競爭比率之確定性演算法。
在此研究裡，我們提出即時銷售員問題的隨機演算法之最低下限及最高上限。在這裡首次由我們發現隨機貪婪演算法有較確定性演算法佳的表現。貪婪演算法，意指在有尚未處理的需求時，不允許即時演算法有任何等待時間。若我們將等待策略與隨機演算法結合，則能使上限和任意隨機演算法之下限一致，也就是最佳化。另外，我們也討論在公平策略下的隨機演算法。公平策略亦即在探討最佳解時，銷售員只能在截至目前所出現之需求集合的凸包內移動。而隨機非貪婪演算法的表現，也依然優於最佳確定性演算法。

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+√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 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 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.

摘要…………………………………………………………………………………………………………………..….....I

Abstract………………………………………………………………………………………………………..………..II

誌謝…………………………………………………………………………………………………………..……III

Contents…………………………………………………………………………………………………………..……IV

List of Figures and Tables…………………………………………………………………………………….…V

1 Introduction………………………………………………………………….………………………………………1


2 Preliminaries………………………………………………………………..………………………………………4


3 Lower Bound of Randomized Algorithms for OLTSPL……………………………………….6


4 Upper Bound of Randomized Algorithms for OLTSPL……………………………………..12

4.1 Randomized Zealous Algorithm for OLTSPL……………………..……………………12
4.2 Randomized Non-Zealous Algorithm for OLTSPL……………………..………………16

5 OLTSP on the Cycles………………………………………………………………………………………….22

6 Concluding Remarks………………………………………………………………………………………….29

References………………….………………………………………………………………………………………….30

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