在組合最佳化的研究領域，旅行銷售員問題是一個經典且廣泛討論的題目。旅行銷售員要到所有城市服務需求，從原點出發，經過每一個城市恰好一次，再回到原點。問題為找出一個最短的路徑。在此篇論文中，我們考慮此經典問題的變形，即時規劃版本。即時規劃版本與傳統的旅行銷售員問題最大的不同在於，在最一開始銷售員並不知道所有的需求資訊並且當銷售員走在一條既定的最短路徑上時，新的需求會隨著時間出現而被銷售員得知。此在即時規劃版本，我們的問題為去服務所有會隨著時間出現的所有需求，並盡可能的縮短完成時間。
在此篇論文中我們參考了由[Blom et al. INFORMS Journal on Computing, (2001), 13(2), pp. 138-148] 所提出的公平策略，並且如何有效的使用等待策略。我們主要專注在實數線以及方格子邊的探討。在一維度的實數線上，我們證明了由[Ausiello et al. Algorithmica, (2001), 29(4), pp. 560-581] 所提出的決定性演算法PQR在公平策略下有一個更優良的5/3的競爭比率。此外我們也證明了在公平策略下對於所有隨機性演算法的最低下限為4/3競爭比率。在二維度的方格子邊上我們提出了一個在公平策略下為2的競爭比率的隨機性演算法。更進一步地，當我們在隨機性演算法中加入了等待策略，我們得到一個更低的競爭比率1.7808，這個結果低於了對所有確定性演算法在方格子邊的最低下限。

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.

摘要
Abstract
誌謝
Contents
List of Figures and Tables
1 Introduction
1.1 Some Variants
2 Preliminaries
2.1 Problem Definition and Notation
2.2 Fair Adversary, Zealous Algorithm and Boundary
of Unit Square
2.3 Previous work
3 Online Algorithm on the Real Line
3.1 Possibly Queue Request Algorithm on the Real Line,
PQR
3.2 Possibly Queue Request Algorithm against Fair
Adversary
3.3 The Lower bound for any Randomized Zealous
Algorithms against Fair Adversary
4 Online Randomized Zealous Algorithm against Fair
Adversary on the Boundary of Unit Square
4.1 Online Zealous Algorithm Plan-at-Furthest-with
Randomization, PFR
4.2 Optimization with Waiting Strategy
5 Concluding Remarks
References

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