Waiting is not easy but worth it: the online TSP on the line revisited

TL;DR

This paper investigates the online traveling salesman problem on the real line, introducing randomized algorithms that outperform deterministic ones and establishing bounds for their effectiveness, including against a restricted adversary.

Contribution

It presents the first randomized zealous algorithm that surpasses the optimal deterministic algorithm and introduces a natural waiting scheme achieving optimal bounds.

Findings

Randomized zealous algorithm improves deterministic performance.

A natural waiting scheme achieves the proposed lower bound.

Randomized algorithms outperform deterministic ones against a fair adversary.

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.

Tables

Table 1. Table 1 : Overview of the lower bound and upper bound results for the competitive ratio of deterministic and randomized algorithms for the OLTSPL

Adversary	Online	Lower Bound		Upper Bound
	Algorithm	Deterministic	Randomized	Deterministic	Randomized
Fair	Zealous	1.6 [7]	$\frac{4}{3}\approx 1.33$	1.75 [1]	1.625
	Non-zealous	$\frac{5+\sqrt{57}}{8}\approx 1.57$ [7]	$\frac{1+\sqrt{17}}{4}\approx 1.28$	$\frac{5+\sqrt{57}}{8}\approx 1.57$ [11]	$\frac{9+\sqrt{177}}{16}\approx 1.39$
General	Zealous	1.75 [7]	1.5	1.75 [1]	1.625
	Non-zealous	$\frac{9+\sqrt{17}}{8}\approx 1.64$ [1]	1.5	$\frac{9+\sqrt{17}}{8}\approx 1.64$ [5]	1.5

Equations

$$\alpha$$

$$=\frac{1}{3x}(p_{1r}\times (4x+2y)+p_{1l}\times (4x-2y))$$

$$\ge \frac{4x-2y}{3x}$$

$${\alpha }_{T+2x}\ge \frac{T+4x+(2-2(n+1)y)}{T+3x+2y}=1+\frac{x+(2-2n)y}{T+3x+2y}$$

$${\alpha }_{T+2x}\ge \frac{T+4x-2(n+1)y}{T+3x}=1+\frac{x+(2-2n)y}{T+3x}$$

$${\alpha }_{T+2x}\ge \frac{T+4x+2y}{T+3x+2y}=1+\frac{x}{T+3x+2y}$$

$${\alpha }_{T+2x}\ge \frac{T+4x}{T+3x}=1+\frac{x}{T+3x}$$

$$\alpha \ge \frac{T+2x}{T+x}\ge \frac{\alpha (2x+2y)+2x}{\alpha (2x+2y)+x}=1+\frac{x}{\alpha (2x+2y)+x}$$

$${\mathsf{E}(\mathsf{ALG})}_{{\sigma }_{t_n}}=t_n$$

$$⋮$$

$$+[\frac{1}{2^{n-1}}\times \frac{1}{2}\times (2p_x+2p_y-{s(t_n)}_{2^{n-1}})+\frac{1}{2^{n-1}}\times \frac{1}{2}\times (2p_x+2p_y+{s(t_n)}_{2^{n-1}})]$$

$$=t_n$$

$$\alpha =\frac{\mathsf{E}(\mathsf{ALG})}{\mathsf{OPT}}\le \frac{t_n+2p_x+2p_y}{t_n+p_x+2p_y}=1+\frac{p_x}{t_n+p_x+2p_y}\le 1+\frac{p_x}{2x_{<t_n}^{\prime }+2y_{<t_n}^{\prime }}\le 1+\frac{x_{<t_n}^{\prime }}{2x_{<t_n}^{\prime }+2y_{<t_n}^{\prime }}\le 1.5$$

$${\mathsf{E}(\mathsf{ALG})}_{{\sigma }_{t_n}}=t_n$$

$$+[\frac{1}{2^{n-1}}\times 1\times (2P_x-{s(t_n)}_2)]$$

$$⋮$$

$$+[\frac{1}{2^{n-1}}\times \frac{1}{2}\times (2P_x+2y_{<t_n}-{s(t_n)}_{2^{n-1}-1})+\frac{1}{2^{n-1}}\times \frac{1}{2}\times (2p_x+2y_{<t_n}+{s(t_n)}_{2^{n-1}-1})]$$

$$+[\frac{1}{2^{n-1}}\times 1\times (2p_x-{s(t_n)}_{2^{n-1}})]$$

$$=t_n$$

$$\le t_n$$

$$=t_n$$

$$\alpha$$

$$\alpha =\frac{\mathsf{E}(\mathsf{ALG})}{\mathsf{OPT}}\le \frac{t_n+2p_x+2p_y}{t_n+d(x_{<t_n}^{\prime },p_x)+p_x+2p_y}\le 1+\frac{p_x}{t_n+d(x_{<t_n}^{\prime },p_x)+p_x+2p_y}\le 1+\frac{p_x}{2p_x+2p_y}\le 1.5$$

$$\alpha =\frac{\mathsf{E}(\mathsf{ALG})}{\mathsf{OPT}}\le \frac{t_n+2p_x+\frac{3}{2}{y}_{<t_n}}{t_n+d(x_{<t_n}^{\prime },p_x)+p_x}\le 1+\frac{p_x+\frac{3}{2}{y}_{<t_n}}{t_n+d(x_{<t_n}^{\prime },p_x)+p_x}\le 1+\frac{p_x+\frac{3}{2}{y}_{<t_n}^{\prime }}{2p_x+2y_{<t_n}^{\prime }}\le \frac{13}{8}=1.625$$

$$\alpha =\frac{\mathsf{E}(\mathsf{ALG})}{\mathsf{OPT}}\le \frac{t_n+2p_x+2p_y}{t_n+p_x+2p_y}\le 1+\frac{p_x}{t_n+p_x+2p_y}\le 1+\frac{p_x}{2p_x+2p_y}\le 1.5$$

$$\alpha =\frac{\mathsf{E}(\mathsf{ALG})}{\mathsf{OPT}}\le \frac{t_n+2p_x+\frac{3}{2}{y}_{<t_n}}{t_n+p_x}\le 1+\frac{p_x+\frac{3}{2}{y}_{<t_n}}{t_n+p_x}\le 1+\frac{p_x+\frac{3}{2}{y}_{<t_n}^{\prime }}{2p_x+2y_{<t_n}^{\prime }}\le \frac{13}{8}=1.625$$

$$W\ge (\alpha -1)t+(\alpha -2)p_x+s(t)$$

$$W$$

$$\ge (2{\alpha }^2-2\alpha )x_{<t}^{\prime }+(\alpha -2)p_x+(2{\alpha }^2-2\alpha )y_{<t}^{\prime }$$

$$\begin{array}{rl}W=& \frac{1}{2}((\alpha -1)t+(\alpha -2)p_x-2y_{<t}^{\prime }+s(t))+\frac{1}{2}((\alpha -1)t+(\alpha -2)p_x-2y_{<t}^{\prime }-s(t))\\ \ge & (\alpha -1)t+(\alpha -2)p_x-2y_{<t}^{\prime }\end{array}$$

$$W+(\alpha -1)W^{\prime }$$

$$\ge (2{\alpha }^2-2\alpha )x_{<t}^{\prime }-p_x+(2{\alpha }^2-4\alpha )y_{<t}^{\prime }\because p_x>s(t)$$

$$W=$$

$$+(1-p)((2{\alpha }^2-2\alpha )x_{<t}^{\prime }-p_x+(2{\alpha }^2-4\alpha )y_{<t}^{\prime })$$

$$\ge (2{\alpha }^2-2\alpha )x_{<t}^{\prime }-(\frac{3}{2}-\frac{1}{2}\alpha )p_x+(2{\alpha }^2-3\alpha )y_{<t}^{\prime }$$

$$\ge (2{\alpha }^2-\frac{3}{2}\alpha -\frac{3}{2})x_{<t}^{\prime }+(2{\alpha }^2-3\alpha )y_{<t}^{\prime }\because x_{<t}^{\prime }\ge p_x$$

$$\ge (4{\alpha }^2-\frac{9}{2}\alpha -\frac{3}{2})y_{<t}^{\prime }$$

$$W\ge (\alpha -1)t+(\alpha -2)p_x+s(t)+\alpha d(x_{<t}^{\prime },p_x)$$

Full text

Waiting is not easy but worth it: the online TSP on the line revisited

Pei-Chuan Chen

  

Erik D. Demaine

  

Chung-Shou Liao

  

Hao-Ting Wei

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.

1 Introduction

Imagine a robot or an automatic guided vehicle (AGV) being deployed in a row of storage shelves in a logistics company’s smart warehouse, e.g., Amazon Kiva robots deployed in their fulfillment centers. The robot is moving back and forth along the aisle, and attempts to grab parcels from the shelves according to customers’ order. However, customers’ purchase requests arrive in an online fashion. That is, the information of an online request, including its release time and the location of its parcel, only becomes known upon its arrival. The objective is to devise an efficient schedule for the robot, moving from a start point, finishing all online requests and going back to the start point as early as possible. The real-world problem can be directly referred to as the online traveling salesman problem on the real line (OLTSPL). The salesman is walking on the real line, and the input of the OLTSPL is a sequence of online requests, appeared over time on the real line. The salesman aims to begin at the origin, and serve all the requests and return to the origin such that the completion time is minimized.

The performance of an online algorithm is usually measured by competitive analysis [6, 8, 14]. Precisely, the quality of an online (randomized) algorithm $A$ for the OLTSPL is measured by the worst case ratio, called competitive ratio, which is defined to be the fraction between the (expected) output of the algorithm $A$ and the result of an offline strategy derived by an oblivious adversary that is aware of the whole input sequence of requests in any instances. That is, an online (randomized) algorithm $A$ is called $\alpha$-competitive if for any instances, the (expected) outcome of the algorithm $A$ is at most $\alpha$ times the offline optimum. In this study, we refer to [7, 11] and also consider online algorithms against a fair adversary. An adversary is called fair if the salesman cannot leave the convex region of the origin and the positions of all the currently released requests. The adversary with more reasonable power may allow an online model to have algorithms with better competitive ratios.

The online problem poses two key challenges. One is that decisions have to be made immediately for each request without being able to know future requests when designing an online algorithm based on currently partial information. Moreover, it is even impossible to know the total number of requests, i.e., which one is the last request. That makes the problem more difficult. The other involves waiting strategies for online routing. Obviously, if the salesman would wait until all the information about requests become clear, the strategy could result in a big waste of time. Though, for deterministic algorithms, Lipmann [11], Blom et al. [7] and Bjelde et al. [5] showed the merits of proper waiting, which is helpful to the competitive performance of their algorithms. In this paper, we prove that waiting also helps randomized algorithms to obtain a better competitive ratio against both the fair adversary and the general adversary.

Prior work. There has been a considerable amount of research about the online traveling salesman problem (OLTSP) in the literature. Here we focus on the previous studies for the OLTSP on the line. For the related work and variants of the OLTSP, readers may refer to the papers [1, 2, 3, 4, 9, 10, 12, 13]. For the OLTSPL, Ausiello et al. [1] obtained a lower bound of $\frac{9+\sqrt{17}}{8}$ and proposed a 1.75-competitive zealous algorithm against the general adversary. Blom et al. [7] first discussed the concept of a fair adversary as well as zealous algorithms. They obtained a lower bound of 1.6 for any zealous algorithms against the fair adversary. For zealous algorithms against the general adversary, they derived another lower bound of 1.75, which shows that Ausiello et al.’s zealous algorithm is optimal. They also proved a lower bound of $\frac{5+\sqrt{57}}{8}$ for any non-zealous algorithms against the fair adversary. In addition, they also considered the OLTSP on the positive real line and obtained some tight bounds. Later, Lipmann [11] presented a non-zealous algorithm against the fair adversary for the OLTSPL to meet Blom et al.’s lower bound. Recently, Bjelde et al. [5] proposed an optimal $\frac{9+\sqrt{17}}{8}$-competitive algorithm against the general adversary. Note that all the above algorithms are deterministic. Table 1 shows the latest results for the OLTSPL.

Our results. We believe that this is the fist study on randomized algorithms for the OLTSPL. Here is the summary of our key contribution. First we have proved lower bounds for any randomized algorithms against both the fair adversary and the general adversary in the OLTSPL (as shown in Table 1). For the general adversary, we have developed a randomized zealous algorithm with a competitive ratio of 1.625, which surpasses the deterministic lower bound and improves the optimal deterministic algorithm. Furthermore, we have presented a randomized non-zealous 1.5-competitive algorithm that optimally achieves the proposed lower bound for randomized algorithms. For the fair adversary, the non-zealous algorithm derives a better competitive ratio of $\frac{9+\sqrt{177}}{16}$, which also improves the optimal deterministic algorithm. We remark that the proposed lower bounds for any randomized algorithms in the OLTSPL are the same as those for any deterministic algorithms in the OLTSP on the positive real line [7], but the worst-case examples we use need more observations. We will talk about more details later.

The remainder of this paper is organized as follows. In Section 2, we introduce some notation and preliminaries. In Section 3, we present lower bounds for randomized zealous/non-zealous algorithms against the fair/general adversaries. In Section 4, we develop online randomized zealous/non-zealous algorithms against the fair/general adversaries, each of which improves the best deterministic algorithms. Section 5 contains the concluding remarks.

2 Preliminaries

First we give some notation and definitions. Assume the online salesman starts at the origin [math] and moves with unit speed along the real line. Let $s(t)$ denote the position of the salesman at time $t\geq 0$ and $s(0)=0$. We denote a sequence of requests released at time $t$ by $\sigma_{t}=(t,P_{t})$ in which $P_{t}$ represents the set of requests. We let $\sigma_{\leq t}$ denote the subsequence of requests in $\sigma$ released up to time $t$. Similarly, let $\sigma_{<t}$ be the subsequence of $\sigma$ comprising the requests with release time strictly earlier than $t$.

As mentioned above, we use competitive ratio to evaluate the performance of an online algorithm for the problem.

Note that the offline adversary has entire information at time 0 about all requests in $\sigma$, while the online salesman has prior information neither about the release time and location of all requests nor about the total number of requests.

Here is the formal definition of competitive ratio. Let $\mathsf{E(ALG)}_{\sigma}$ denote the expected completion time of the online salesman moved by a randomized $\mathsf{ALG}$ algorithm on the input sequence of requests $\sigma$. Let $\mathsf{OPT}$ denote the offline optimum cost. An online randomized algorithm $\mathsf{ALG}$ for the problem is $\alpha$-competitive if there exists a constant $\alpha$ such that for every sequence of requests $\sigma$, $\mathsf{E(ALG)}_{\sigma}\leq\alpha\cdot\mathsf{OPT}_{\sigma}+c$, where $c$ is a constant. Note that when a new request arrives at time $t$, an online algorithm for the OLTSPL must immediately determine the behavior of the salesman at the moment $t$ as a function of all the requests in $\sigma_{\leq t}$.

In this study, we particularly refer to Bolm et al. [7] who first presented the concept of fair adversary in the OLTSPL, which gives a limit on the power of the offline adversary. They also discussed zealous algorithms concerning the fair adversary. We show the formal definitions in the following.

Definition 2.1** (Fair Adversary [7]).**

An offline adversary for the OLTSP in the Euclidean space $(R^{n},||.||)$ is fair, if at any time $t$, the position of the salesman operated by the adversary is within the convex hull of the origin and the requested points from $\sigma_{<t}$. That is to say, it must locate within the range of requested points from $\sigma_{<t}$.

We remark that the concept of a fair adversary can be extended to metric space or more general hyperplane if the fair idea can be applied.

Definition 2.2** (Zealous Algorithm [7]).**

An online algorithm A for the OLTSP is called zealous, if it satisfies the following two conditions:

• •

If there are still unserved requests, then the online salesman changes its direction only if a new request becomes known, or the salesman is either lying in the origin or standing at a request that has just been served.

• •

At any time, when there are unserved requests, the salesman operated by A either moves at unit speed towards an unserved request or the origin.

Next we are going to consider the lower bound of each scenario.

3 Lower Bound of Randomized Algorithms for OLTSPL

In this section, we present the lower bounds for any randomized algorithms under different scenarios, as shown in Table 1. First we consider the lower bounds for randomized zealous algorithms.

Theorem 3.1**.**

Any randomized zealous $\alpha$-competitive algorithm for the OLTSPL against the fair adversary has $\alpha\geq 4/3$.

Proof 3.2**.**

We give the following instance: initially, $\sigma_{0}=(0,\{x,-y\})$, where $x$ and $-y$ lie on the positive and negative sides of the real line, respectively. Let $y=\epsilon$, where $\epsilon$ is a sufficiently small constant. At this moment, for any randomized zealous algorithm, there are two cases:

• 1

If the online salesman chooses going right at time $t=0$, then the salesman is lying at the origin at time $t=2x$ with a remaining request $-y$.

• 2

If the online salesman chooses going left at time $t=0$, then the salesman is lying at the position $2y$ at time $t=2x$ with no remaining requests.

Next, a new request arrives at position $x$ at time $2x$, i.e. $\sigma_{2x}=(2x,\{x\})$. Assume the online salesman went right at time $t=0$; that is, he is lying at the origin at time $2x$ and has to serve two requests: one at $-y$ and the other at $x$. At the moment, for any randomized zealous algorithm, irrespective of whether the salesman chooses going right or left, he will finish serving the two requests and return to the origin at time $t=4x+2y$.

On the other hand, if the online salesman went left at time $t=0$, he is lying at $2y$ at time $2x$ and has to serve only the new request at $x$. The salesman thus gets to serve $x$ and back to the origin at time $t=4x-2y$.

The optimal strategy of the offline adversary is to move left at time $t=0$ and serve $-y$ then $x$, but he waits at the position $x$ until time $t=2x$. Next, when he is beginning to return to the origin, he also serves the new request at $x$ and the total completion time is $2x+x=3x$. We let $p_{k,r}$ and $p_{k,l}$ denote the probability of choosing going right and left at the $k^{th}$ iteration, respectively. Therefore, the competitive ratio is:

[TABLE]

When the value of $y$ is approaching to a very small constant, the ratio is at most $\frac{4}{3}$. Based on the above result, we have the following observations:

Observation 1**.**

The lower bound of any randomized zealous algorithms is derived when letting $y=\epsilon$, where $\epsilon\approx 0$.

Observation 2**.**

If there are unserved requests in both the negative side and the positive part, the lower bound for any randomized zealous algorithms appears in the case that the online salesman gets to serve $-y$ first and then $x$, where $x$ and $-y$ represent the currently rightmost and leftmost requests, respectively, and $|x|>|-y|$.

Next, we further prove that similar future requests cannot increase the lower bound. Let a new sequence of similar requests $\sigma_{2kx}=(2kx,\{x_{2kx},-y_{2kx}\})$ be released at time $2kx$, where $k=1,\ 2,\ldots,n$, $n\in\mathbb{Z}^{+}$, i.e. a pair of requests arriving at $x_{2kx}$ and $-y_{2kx}$ and $|x_{2kx}|>|-y_{2kx}|$. According to Observation 2, the online salesman chose going left every time at $2kx$, and it thus lies at position $2ny$ at time $2nx$. For the next request released at time $T=2nx$, we divide it into two cases:

• •

*Similarly $\sigma_{T}=(T,\{x,-y\})$, and we let $T\geq 2x,\ y=\epsilon$. *

The best way for the online salesman is to serve $x$, then $-y$, and go back to the origin. Thus we can derive that the online cost is at least $T+(x-2ny)+x+2y=T+2x+(2-2n)y$. For the offline adversary, the optimal strategy should be able to reach $x$ at time $T$ so that the offline cost is $T+x+2y$. Therefore, the competitive ratio is at least $\alpha_{T}=\frac{T+2x+(2-2n)y}{T+x+2y}=1+\frac{x-2ny}{T+x+2y}$.

If there is one more new request $\sigma_{T+2x}=(T+2x,\{x,-y\})$ arriving, we have the ratio:

[TABLE]

Obviously, the value of $\alpha_{T+2x}$ is not larger than $\alpha_{T}$ and it cannot be used to increase the lower bound. If there is another new request $\sigma_{T+2x}=(T+2x,\{x\})$ arriving, we derive the ratio:

[TABLE]

*Again, the value of $\alpha_{T+2x}$ is not bigger than $\alpha_{T}$.

• •

*Assume $\sigma_{T}=(T,\{x\})$ and $T\geq 2x$. *

The online salesman just gets to serve $x$ and then back to the origin. The online cost is at least $T+(x-2ny)+x=T+2x-2ny$; For the offline adversary, the offline cost is $T+x$. Therefore, the competitive ratio is at least $\alpha_{T}=\frac{T+2x-2ny}{T+x}=1+\frac{x-2ny}{T+x}$.

Similarly, if there is one more new request $\sigma_{T+2x}=(T+2x,\{x,-y\})$ arriving, we have the ratio:

[TABLE]

Obviously, the value of $\alpha_{T+2x}$ is not bigger than $\alpha_{T}$ and the request cannot increase the lower bound. If there is another new request $\sigma_{T+2x}=(T+2x,\{x\})$ arriving, we derive the ratio:

[TABLE]

Again, the lower bound cannot increase by using the request. Hence, based on the above cases, we show that the lower bound for any randomized zealous algorithms is at least $4/3$ and it cannot be increased by such future requests.

We remark that if the last requests arrive in both the negative side and the positive part, the online cost as well as the offline (optimal) cost increases so that the competitive ratio actually decreases. The observation is helpful to the design of the worst cases.

Observation 3**.**

*The lower bound for any randomized zealous algorithms appears in the worst case in which the last requests arrive only on the positive side of the origin. *

Next, we consider the lower bound for the general adversary case.

Theorem 3.3**.**

Any randomized zealous $\alpha$-competitive algorithm for the OLTSPL against the general adversary has $\alpha\geq 1.5$.

Proof 3.4**.**

Assume there are $n$ requests and $\sigma_{0}=(0,\{x,-y\})$. According to Observation 3, we let a new request $\sigma_{2x}=(2x,\{2x\})$ be released at time $2x$, which results in the largest gap between the online salesman and the offline adversary. We then devise the following instance of requests: $\sigma_{2\times 3^{k}x}=(2\times 3^{k}x,\{2\times 3^{k}x,-y\})$, where $0\leq k\leq n-3,\ k\in\mathbb{Z}$, and the last request appears when $k=n-2$. Similarly we let $y=\epsilon$ by Observation 1. For the online salesman, the best strategy is to choose going left for the first $n-1$ requests and going right for the last request. Therefore, the online cost is at most $2\times 3^{n-2}x+(2\times 3^{n-2}x-2(n-1)y)+2\times 3^{n-2}x=6\times 3^{n-2}x-2(n-1)y$.

On the other hand, the offline optimal cost is at least $2(2\times 3^{n-2}x)+2y$, because the offline adversary should be able to reach the position of $(2\times 3^{n-2})x$ when the last request is released. As a result, we derive the competitive ratio $\alpha\geq\frac{6\times 3^{n-2}x-2(n-1)y}{4\times 3^{n-2}x+2y}\approx\frac{6\times 3^{n-2}x}{4\times 3^{n-2}x}=1.5$. The proof is complete.

In the following, we consider randomized non-zealous algorithms; that is, waiting strategies can be allowed to use in the online algorithms.

Theorem 3.5**.**

Any randomized $\alpha$-competitive algorithm for the OLTSPL against the fair adversary has $\alpha\geq\frac{1+\sqrt{17}}{4}$.

Proof 3.6**.**

Suppose there is an $\alpha$-competitive randomized algorithm. Let $\sigma_{0}=(0,\{x,-y\})$. We consider the time that after the algorithm had served $\sigma_{0}$ and returned to the origin, and let $T$ be the minimum time among all the routes that the algorithm had randomly selected. Because the algorithm is $\alpha$-competitive, we have $T\leq\alpha\cdot\mathsf{OPT}_{\sigma_{0}}$. Obviously, $\mathsf{OPT}_{\sigma_{0}}=2x+2y$ and it implies that $T\leq\alpha\cdot(2x+2y)$. Let the next request $\sigma_{T}=(T,\{x\})$ be released at time $T$. The online cost is at least $T+2x$; and the offline optimum is $T+x$ because the offline adversary should be able to arrive at $x$ at time $T$. Therefore, the competitive ratio is:

[TABLE]

We can derive that $\alpha\geq\frac{1+\sqrt{17}}{4}$.

Theorem 3.7**.**

Any randomized $\alpha$-competitive algorithm for the OLTSPL against the general adversary has $\alpha\geq 1.5$.

Proof 3.8**.**

Suppose there have been $n$ released requests. Consider the time after ALG had served all the $n$ requests and returned to the origin, and let $T$ be the minimum time among all the routes ALG had randomly chosen.

Let the next request $\sigma_{T}=(T,\{T\})$ be released at time $T$.

Because the offline adversary is not restricted by a fair adversary, it should be able to reach position $T$ at time $T$. Thus, the competitive ratio $\frac{\mathsf{E(ALG)}_{\sigma}}{\mathsf{OPT}_{\sigma}}\geq\frac{T+2T}{T+T}=1.5$. The remaining proof that shows that future requests cannot help increase the bound can be provided in a similar way.

In the next section, we are going to consider the upper bound of each scenario.

4 Upper Bound of Randomized Algorithms for OLTSPL

In the following we first propose a randomized zealous algorithm against both the fair and general adversaries. Then we devise a randomized non-zealous algorithm with a simple waiting strategy, which improves the optimal deterministic algorithms.

4.1 Randomized Zealous Algorithm for OLTSPL

We first define some notation. Suppose a set of new requests $\sigma_{t}=(t,P_{t})$ arrives at time $t$. Let $p_{x}>0$ denote the rightmost request in $P_{t}$ and $-p_{y}\leq 0$ denote the leftmost request in $P_{t}$ at time $t$. We also let $x_{<t}$ and $-y_{<t}$ be the furthest unserved requests on the positive and negative sides, respectively, before time $t$ in $\sigma_{<t}$. Without loss of generality, assume $x_{<t}\geq y_{<t}$. Let $x^{\prime}_{<t}$ and $-y^{\prime}_{<t}$ be the furthest requests ever presented on the positive and negative sides, respectively, before time $t$. Similarly, assume $x^{\prime}_{<t}\geq y^{\prime}_{<t}$. Algorithm RZ presents a simple randomized strategy for the online request $\sigma_{t}$ at time $t$. The online salesman may change his direction when the furthest unserved request changes on at least one of the two sides of the salesman. The salesman greedily gets to serve the unserved requests on one side if there are no requests on the other side. Otherwise, the salesman has equal probabilities to serve the unserved requests on the two sides. Obviously, it is a zealous algorithm.

Theorem 4.1**.**

Algorithm RZ is $1.625$-competitive against the fair adversary for the OLTSPL.

Proof 4.2**.**

Assume there are $n$ requests. Let $\sigma_{t_{k}}=(t_{k},P_{t_{k}})$, where $1\leq k\leq n$, $k\in\mathbb{Z}$, be the $k^{th}$ request. When a new set of requests that may replace the currently furthest unserved requests arrives, the randomized algorithm has at most two options to choose. Thus, when the last request $\sigma_{t_{n}}=(t_{n},P_{t_{n}})$ is released, we let ${s(t_{k})}_{i}$ denote the possible position of the online salesman at time $t_{k}$, $-y^{\prime}_{<t_{k}}\leq{s(t_{k})}_{i}\leq x^{\prime}_{<t_{k}}$, and $1\leq i\leq 2^{k-1}$. That is, there are totally $2^{k-1}$ possible events at time $t_{k}$, and the probability of each event is $\frac{1}{2^{k-1}}$. Next, we divide the proof into four cases, depending on the relative position of $x_{<t_{n}}$, $x^{\prime}_{<t_{n}}$ and the new request $\sigma_{t_{n}}$. Note that it is unnecessary to consider the case $p_{x}\leq x_{<t_{n}}$, since the furthest unserved request does not change and thus the online salesman keeps following the remaining schedule and serves $x_{<t_{n}}$ then $p_{x}$ on his way back to the origin.

• •

*Case 1: $x_{<t_{n}}<p_{x}\leq x^{\prime}_{<t_{n}}$ and $p_{y}>0$ *

*We consider the online cost after the last request is released at time $t_{n}$. Therefore, the online expected cost is:

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[TABLE]

*For the fair adversary, the optimal cost $\mathsf{OPT}$ is at least $t_{n}+p_{x}+2p_{y}\geq 2x^{\prime}_{<t_{n}}+2y^{\prime}_{<t_{n}}$. Then we can derive the competitive ratio:

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[TABLE]

• •

*Case 2: $x_{<t_{n}}<p_{x}\leq x^{\prime}_{<t_{n}}$ and *$p_{y}=0$

In this case, if the online salesman chose going left at time $t_{n-1}$, he could choose going right at time $t_{n}$ because there are no requests on the negative side of the origin.

*Therefore, we can derive the online cost:

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[TABLE]

*For the fair adversary, the optimal cost $\mathsf{OPT}$ is at least $t_{n}+p_{x}\geq 2x^{\prime}_{<t_{n}}+2y^{\prime}_{<t_{n}}$. Then we can derive the competitive ratio:

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[TABLE]

• •

*Case 3: $x_{<t_{n}}\leq x^{\prime}_{<t_{n}}<p_{x}$ and $p_{y}>0$ *

*The online cost is the same as that in Case 1, i.e. $\mathsf{E(ALG)}\leq t_{n}+2p_{x}+2p_{y}$. For the fair adversary, the furthest position it can reach at time $t_{n}$ is $x^{\prime}_{<t_{n}}$. Thus, $\mathsf{OPT}\geq t_{n}+d(x^{\prime}_{<t_{n}},p_{x})+p_{x}+2p_{y}\geq 2p_{x}+2p_{y}$, where $d(x^{\prime}_{<t_{n}},p_{x})$ denotes the distance between $x^{\prime}_{<t_{n}}$ and $p_{x}$. We can derive the competitive ratio as follows:

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[TABLE]

• •

*Case 4: $x_{<t_{n}}\leq x^{\prime}_{<t_{n}}<p_{x}$ and $p_{y}=0$ *

*The online cost is the same as that in Case 2, i.e. $\mathsf{E(ALG)}\leq t_{n}+2p_{x}+\frac{3}{2}y_{<t_{n}}$. The furthest position the fair adversary can reach at time $t_{n}$ is $x^{\prime}_{<t_{n}}$. Thus, in this case $\mathsf{OPT}\geq t_{n}+d(x^{\prime}_{<t_{n}},P_{x})+p_{x}\geq 2p_{x}+2y^{\prime}_{<t_{n}}$. We can derive the competitive ratio as follows:

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[TABLE]

Theorem 4.3**.**

Algorithm RZ is also 1.625-competitive against the general adversary for the OLTSPL.

Proof 4.4**.**

The proof is similar to that of Theorem 4.1. However, all we need to consider is the case $p_{x}>x^{\prime}_{<t_{n}}$ because the general adversary does not make any difference from a fair adversary when $p_{x}\leq x^{\prime}_{<t_{n}}$.

• •

*Case 1: $x_{<t_{n}}\leq x^{\prime}_{<t_{n}}<p_{x}$ and $p_{y}>0$ *

*The online cost is the same as that in Case 1 of Theorem 4.1, i.e. $\mathsf{E(ALG)}\leq t_{n}+2p_{x}+2p_{y}$. For the general adversary, it can reach $p_{x}$ at time $t_{n}$. Thus, $\mathsf{OPT}\geq t_{n}+p_{x}+2p_{y}\geq 2p_{x}+2p_{y}$. We can derive the competitive ratio as follows:

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[TABLE]

• •

*Case 2: $x_{<t_{n}}\leq x^{\prime}_{<t_{n}}<p_{x}$ and $p_{y}=0$ *

*The online cost is the same as that in Case 2 of Theorem 4.1, i.e. $\mathsf{E(ALG)}\leq t_{n}+2p_{x}+\frac{3}{2}y_{<t_{n}}$. The general adversary can reach $p_{x}$ at time $t_{n}$. Thus, $\mathsf{OPT}\geq t_{n}+p_{x}\geq 2p_{x}+2y^{\prime}_{<t_{n}}$. We can derive the competitive ratio as follows:

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[TABLE]

4.2 Randomized Non-Zealous Algorithm for OLTSPL

We incorporate a simple waiting strategy into the RZ algorithm. When the online salesman reaches one furthest request, the salesman decides to wait for a moment and see if he could make a better decision for future requests. Precisely, at time $t$, if needed, we set the waiting time to be $W:=\alpha\mathsf{OPT}_{\sigma_{\leq t}}-C_{t}-t$, where $C_{t}$ denotes the cost of serving the remaining unserved requests in $\sigma_{\leq t}$ at time $t$ as well as going back to the origin. Later we will prove that $\alpha=\frac{9+\sqrt{177}}{16}$ against the fair adversary and $\alpha=1.5$ against the general adversary.

After waiting for time $W$, the online salesman gets to serve the remaining requests, or returns to the origin if there are no unserved requests. Note that while the online salesman is waiting, if a new request that will change the furthest unserved request on one of the two sides is released, the salesman stops waiting and plans a new schedule (see Algorithm RNZ).

Theorem 4.5**.**

Algorithm RNZ is $\frac{9+\sqrt{177}}{16}$-competitive against the fair adversary for the OLTSPL.

Proof 4.6**.**

Given a new request $\sigma_{t}=(t,P_{t})$ at time $t$, we consider two cases, depending on the relative position of $x_{<t}$, $x^{\prime}_{<t}$ and $p_{x}$. Here we focus on only the case $p_{y}=0$ because the previous proofs reveal the fact that it is the worst case. In other words, when $p_{y}>0$, the additional cost for both the online salesman and the adversary leads to a smaller ratio instead. Moreover, for the same reason in the proof of Theorem 4.1, we skip the case of $p_{x}\leq x_{<t}$.

• •

*Case 1: $x_{<t}<p_{x}\leq x^{\prime}_{<t}$ and *$p_{y}=0$

Obviously, the optimal cost $\mathsf{OPT}_{\sigma_{\leq t}}$ is at least $t+p_{x}$. In addition, $x^{\prime}_{<t}$ has been served before time $t$. Otherwise, $x^{\prime}_{<t}=x_{<t}$ leads to a contradiction. In the following, we consider whether $-y^{\prime}_{<t}$ has been served or not.

• –

case 1.1: Both $x^{\prime}_{<t}$ and $-y^{\prime}_{<t}$ have been served before time $t$.

Since the online salesman has already served $-y^{\prime}_{<t}$, the worst case happens, similarly, when there are no requests on the left side of the origin. Hence we assume that $s(t)$ is on the positive side. That is, the online cost is $\alpha\mathsf{OPT}_{\sigma_{\leq t}}=t+(p_{x}-s(t))+W+p_{x}$; i.e., $C_{t}=(p_{x}-s(t))+p_{x}$. The waiting time is thus $W=\alpha\mathsf{OPT}_{\sigma_{\leq t}}-(t-s(t)+2p_{x})$, which implies (using $\mathsf{OPT}_{\sigma_{\leq t}}\geq t+p_{x}$):*

[TABLE]

Next, we let $t^{\prime}$ be the moment when the online salesman left $x^{\prime}_{<t}$ after serving $-y^{\prime}_{<t}$ and $x^{\prime}_{<t}$. Obviously, $t\geq t^{\prime}+d(x^{\prime}_{<t},s(t))=t^{\prime}+x^{\prime}_{<t}-s(t)$. At time $t^{\prime}$, the online cost is $\alpha\mathsf{OPT}_{\sigma_{\leq t^{\prime}}}=t^{\prime}+x^{\prime}_{<t}$ and the offline optimum is $\mathsf{OPT}_{\sigma_{\leq t^{\prime}}}\geq 2x^{\prime}_{<t}+2y^{\prime}_{<t}$. Hence $t^{\prime}+x^{\prime}_{<t}\geq\alpha(2x^{\prime}_{<t}+2y^{\prime}_{<t})$, which implies $t\geq 2\alpha x^{\prime}_{<t}+2\alpha y^{\prime}_{<t}-s(t)$. We combine this equation with (1) to obtain:111Note that the right-hand side of the inequality will result in a smaller value if $\alpha>2$. That implies a smaller lower bound for $W$. We thus derive a smaller value of $\alpha$ than $\frac{9+\sqrt{177}}{16}$ if we want to guarantee the waiting time $W\geq 0$. It leads to a contradiction.

[TABLE]

• –

case 1.2: Only $x^{\prime}_{<t}$ has been served before time $t$.

The online salesman may go either right or left at time $t$ since $-y^{\prime}_{<t}$ has not been served. Without loss of generality, suppose $s(t)$ is on the positive side. (The proof is similar when $s(t)$ is on the negative side.) We have $C_{t}=d(s(t),p_{x})+p_{x}+2y^{\prime}_{<t}$. If the online salesman chooses going right, the waiting time is $W=\alpha\mathsf{OPT}_{\sigma_{\leq t}}-(t-s(t)+2p_{x}+2y^{\prime}_{<t})$. By using $\mathsf{OPT}_{\sigma_{\leq t}}\geq t+p_{x}$, $W\geq(\alpha-1)t+(\alpha-2)p_{x}-2y^{\prime}_{<t}+s(t)$. Otherwise, if the online salesman chooses going left, then $C_{t}=s(t)+2p_{x}+2y^{\prime}_{<t}$ and we have $W\geq(\alpha-1)t+(\alpha-2)p_{x}-2y^{\prime}_{<t}-s(t)$. The probability of going right or left is equal, i.e. $\frac{1}{2}$, so

[TABLE]

Similarly, we let $t^{\prime}$ be the moment when the online salesman served $x^{\prime}_{<t}$ and just left. Again, $t\geq t^{\prime}+d(x^{\prime}_{<t},s(t))=t^{\prime}+x^{\prime}_{<t}-s(t)$. On the other hand, we let $W^{\prime}$ be the waiting time when the salesman stops at $-y^{\prime}_{<t}$. We have $t^{\prime}+x^{\prime}_{<t}+2y^{\prime}_{<t}+W^{\prime}=\alpha\mathsf{OPT}_{\sigma_{\leq t^{\prime}}}\geq\alpha(2x^{\prime}_{<t}+2y^{\prime}_{<t})$, Therefore, $t\geq 2\alpha x^{\prime}_{<t}+(2\alpha-2)y^{\prime}_{<t}-s(t)-W^{\prime}$. Combine this equation with (2) to obtain:

[TABLE]

We let the possibility of case 1.1 be $p$ and case 1.2 be $1-p$. Therefore, the expected waiting time is:

[TABLE]

Here we let $p=\frac{1}{2}$ to minimize the value of $W$. In order to guarantee the waiting time $W\geq 0$, we obtain $\alpha=\frac{9+\sqrt{177}}{16}\approx 1.39$.

• •

*Case 2: $x_{<t}\leq x^{\prime}_{<t}<p_{x}$ and *$p_{y}=0$

Due to the fair adversary, we have the optimal cost $\mathsf{OPT}_{\sigma_{\leq t}}\geq t+d(x^{\prime}_{<t},p_{x})+p_{x}$. Note that $t\geq x^{\prime}_{<t}+2y^{\prime}_{<t}$ if the adversary wants to reach $x^{\prime}_{<t}$ at time $t$. It thus implies that $x^{\prime}_{<t}$ has been served by the online salesman before time $t$. Otherwise, at least $-y^{\prime}_{<t}$ has been served by the salesman. He was then going right to serve other requests without randomness. Therefore, we divide the proof into two cases in a similar manner.

• –

case 2.1: Both $x^{\prime}_{<t}$ and $-y^{\prime}_{<t}$ have been served before time $t$.

The statement is similar, and inserting the new bound of the optimal cost yields:

[TABLE]

We also combine $t\geq 2\alpha x^{\prime}_{<t}+2\alpha y^{\prime}_{<t}-s(t)$ with (3) to obtain:

[TABLE]

• –

case 2.2: Only $x^{\prime}_{<t}$ has been served before time $t$.

By inserting the new bound of the optimal cost into the waiting time $W=\alpha\mathsf{OPT}_{\sigma_{\leq t}}-t-C_{t}$, where $C_{t}=(d(s(t),p_{x})+p_{x}+2y^{\prime}_{<t})$, we have the similar inequality for the two options of going right and left:

[TABLE]

Then, similarly we combine $t\geq 2\alpha x^{\prime}_{<t}+(2\alpha-2)y^{\prime}_{<t}-s(t)-W^{\prime}$ with (4) to yield:

[TABLE]

Again, we let the possibility of case 2.1 be $p$ and case 2.2 be $1-p$, and the expected waiting time is:

[TABLE]

Similarly, we let $p=\frac{1}{2}$ to minimize the value of $W$, and let $\alpha=\frac{9+\sqrt{177}}{16}\approx 1.39$ to can guarantee the waiting time $W\geq 0$.

The proof is complete.

Theorem 4.7**.**

Algorithm RNZ is 1.5-competitive against the general adversary for the OLTSPL.

Proof 4.8**.**

The proof is similar to Theorem 4.5. However, we only need to consider the case that $x_{<t}\leq x^{\prime}_{<t}<p_{x}$ and $p_{y}=0$, because in the other case the general adversary does not make any difference from the fair adversary. Note that $\mathsf{OPT}_{\sigma_{\leq t}}\geq t+p_{x}$ due to the ability of the general adversary.

• •

Case 1: Both $x^{\prime}_{<t}$ and $y^{\prime}_{<t}$ have been served before time $t$.

We incorporate $\mathsf{OPT}_{\sigma_{\leq t}}\geq t+p_{x}$ into Inequality (3) to yield: $W\geq(\alpha-1)t+(\alpha-2)p_{x}+s(t)$. In addition, we have $t\geq p_{x}+2y^{\prime}_{<t}$ if the offline adversary has to reach $p_{x}$ at time $t$. Therefore,*

[TABLE]

• •

Case 2: Only $x^{\prime}_{<t}$ has been served before time $t$.

Again, we incorporate $\mathsf{OPT}_{\sigma_{\leq t}}\geq t+p_{x}$ into Inequality (4) to yield: $W\geq(\alpha-1)t+(\alpha-2)p_{x}-2y^{\prime}_{<t}$. By inserting $t\geq p_{x}+2y^{\prime}_{<t}$ into it, we have

[TABLE]

Let the possibility of Case 1 be $p$ and Case 2 be $1-p$. The expected waiting time is

[TABLE]

We let $\alpha=\frac{3}{2}$ to guarantee the waiting time $W\geq 0$. The proof is complete.

5 Concluding Remarks

In this study we have shown the lower bounds for randomized algorithms for the OLTSPL and presented the optimal randomized non-zealous algorithm against the general adversary, which surpasses the deterministic lower bound [5]. The algorithm has also improved the optimal deterministic non-zealous algorithm against the fair adversary. For zealous algorithms, our simple randomized algorithm has beaten the best deterministic algorithm as well as the deterministic lower bound for the general adversary. However, there is still a gap between some lower bounds and upper bounds, especially for the fair adversary. It would be also worthwhile to extend the idea of the proposed randomized non-zealous algorithm for the OLTSP in general metric spaces, where Ausiello et al. [1] proved a lower bound of 2 and presented a 3-competitive zealous algorithm.