**4. Multi-agent** *k***-CTP with uncertain edges**

In this section, we study the M-*k*-CTP-U-f and the M-*k*-CTP-U-l. Note that *L* denotes the number of agents in the graph in these problems. We assume that there is no distinction between the *L* agents and all of the agents benefit from complete communication in the sense that they can transmit their locations and explored uncertain edges' cost information to the other agents in real time. By considering an instance of O-D edge-disjoint graphs, we derive lower bounds on the competitive ratio of deterministic online strategies to the M-*k*-CTP-U-f and the M-*k*-CTP-U-l.

Theorem 1.3 For the M-*k*-CTP-U-f and the M-*k*-CTP-U-l, there is no deterministic online strategy with competitive ratio less than min *d*1*=p*1*;* 2 *<sup>k</sup> L* <sup>þ</sup> <sup>1</sup> and min *d*1*=p*1*;* 2 *<sup>k</sup> L* <sup>þ</sup> <sup>1</sup> , respectively.

**Proof**. We again consider the special graph in **Figure 1**. In this case, any deterministic strategy corresponds to a permutation which describes in which order the uncertain paths and *D*<sup>1</sup> (not necessarily all of them) are going to be selected by the agents. For all of these strategies, consider the adverse instance which is defined in the proof of Theorem 1.1. Note that the agents will not reach D via uncertain paths unless the costs of all of the uncertain edges are specified. Before we present the rest of our proof, we need to propose the following lemma.

Lemma 1.4 In the adverse instance, the competitive ratio of the strategies in which the arrivals of the agents at D is via the uncertain paths is at least 2 *<sup>k</sup> L* <sup>þ</sup> 1 and 2 *<sup>k</sup> L* <sup>þ</sup> 1, for the M-*k*-CTP-U-f and the M-*k*-CTP-U-l, respectively.

**Proof**. Note that the agents will not reach D via the uncertain paths unless the costs of all of the uncertain edges are specified since we are considering the adverse instance. Now we present our proof for each claim separately.

*New Variations of the Online* k*-Canadian Traveler Problem: Uncertain Costs at Known Locations DOI: http://dx.doi.org/10.5772/intechopen.88741*


Note that since we are considering the arrivals of the agents at D via the uncertain paths, the performance of the strategies will not be improved if one or more agents take *D*1. The proof is complete.

Now, we present the rest of our proof for each problem separately:

	- Case 1. *<sup>d</sup>*<sup>1</sup> *p*1 ≥2 *<sup>k</sup> L* <sup>þ</sup> 1. In this case, the competitive ratio of the strategies in which the first arrival of the agents at D is via *D*<sup>1</sup> is at least *<sup>d</sup>*<sup>1</sup> *p*1 , which is greater than or equal to min *d*1*=p*1*;* 2 *<sup>k</sup> L* <sup>þ</sup> <sup>1</sup> . The competitive ratio of deterministic strategies in which the first arrival of the agents at D is via the uncertain paths would be at least 2 *<sup>k</sup> L* <sup>þ</sup> 1, which matches the proposed lower bound of the problem.
	- Case 2. *<sup>d</sup>*<sup>1</sup> *p*1 < 2 *<sup>k</sup> L* <sup>þ</sup> 1. In this case, the competitive ratio of deterministic strategies in which the first arrival of the agents at D is via the uncertain paths would be at least 2 *<sup>k</sup> L* <sup>þ</sup> 1, which is greater than the proposed lower bound of min *d*1*=p*1*;* 2 *<sup>k</sup> L* <sup>þ</sup> <sup>1</sup> . The competitive ratio of the strategies in which the first arrival of the agents at D is via *D*<sup>1</sup> is at least *<sup>d</sup>*<sup>1</sup> *p*1 , which matches the proposed lower bound of the problem.
	- Case 1. *<sup>d</sup>*<sup>1</sup> *p*1 ≥2 *<sup>k</sup> L* <sup>þ</sup> 1. In this case, the competitive ratio of the strategies in which the last arrival of the agents at D is via *D*<sup>1</sup> is at least *<sup>d</sup>*<sup>1</sup> *p*1 which is greater than or equal to min *d*1*=p*1*;* 2 *<sup>k</sup> L* <sup>þ</sup> <sup>1</sup> . The competitive ratio of deterministic strategies in which the last arrival of the agents at D is via the uncertain edges would be at least 2 *<sup>k</sup> L* <sup>þ</sup> 1, which matches the proposed lower bound of the problem.
	- Case 2. *<sup>d</sup>*<sup>1</sup> *p*1 < 2 *<sup>k</sup> L* <sup>þ</sup> 1. In this case, the competitive ratio of deterministic strategies in which the last arrival of the agents at D is via the uncertain

<sup>2</sup>*<sup>k</sup>* � <sup>1</sup> <sup>¼</sup> 3. Since <sup>11</sup>

*Wilmington as the destination node.*

*Probability, Combinatorics and Control*

**Figure 2.**

the described scenario.

M-*k*-CTP-U-l.

min *d*1*=p*1*;* 2 *<sup>k</sup>*

following lemma.

2 *<sup>k</sup> L*

**254**

*L*

<sup>þ</sup> <sup>1</sup> , respectively.

(2,6), i.e., *<sup>c</sup>*<sup>1</sup> <sup>¼</sup> 3. Then she checks if *<sup>p</sup>*1þ*c*<sup>1</sup>

**4. Multi-agent** *k***-CTP with uncertain edges**

<sup>3</sup> >3, the strategy enters step 2. Next, the agent takes the

<sup>&</sup>lt; <sup>2</sup>*<sup>k</sup>* � 1. Since <sup>6</sup>

<sup>6</sup> < 3, the agent takes edge

*L* <sup>þ</sup> <sup>1</sup> and

> *L* <sup>þ</sup> 1 and

shortest optimistic path *π*<sup>1</sup> and arrives at node 2 after traversing edge (1,2). We assume that the costs of the uncertain edges (2,6) and (5,6) are 3 and 2,

*A scenario from the Gulf Coast area of the United States network with Atlanta as the source node and*

*p*2

(2,6) to arrive at node 6 and the strategy ends. Note that the cost of the offline optimum is 6. Therefore, the competitive ratio of the pessimistic strategy is one in

respectively. When the agent arrives at node 2, she learns the traveling time of edge

In this section, we study the M-*k*-CTP-U-f and the M-*k*-CTP-U-l. Note that *L* denotes the number of agents in the graph in these problems. We assume that there is no distinction between the *L* agents and all of the agents benefit from complete communication in the sense that they can transmit their locations and explored uncertain edges' cost information to the other agents in real time. By considering

Theorem 1.3 For the M-*k*-CTP-U-f and the M-*k*-CTP-U-l, there is no determin-

**Proof**. We again consider the special graph in **Figure 1**. In this case, any deterministic strategy corresponds to a permutation which describes in which order the uncertain paths and *D*<sup>1</sup> (not necessarily all of them) are going to be selected by the agents. For all of these strategies, consider the adverse instance which is defined in the proof of Theorem 1.1. Note that the agents will not reach D via uncertain paths unless the costs of all of the uncertain edges are specified. Before we present the rest of our proof, we need to propose the

Lemma 1.4 In the adverse instance, the competitive ratio of the strategies in

**Proof**. Note that the agents will not reach D via the uncertain paths unless the costs of all of the uncertain edges are specified since we are considering the adverse

which the arrivals of the agents at D is via the uncertain paths is at least 2 *<sup>k</sup>*

<sup>þ</sup> 1, for the M-*k*-CTP-U-f and the M-*k*-CTP-U-l, respectively.

instance. Now we present our proof for each claim separately.

an instance of O-D edge-disjoint graphs, we derive lower bounds on the competitive ratio of deterministic online strategies to the M-*k*-CTP-U-f and the

istic online strategy with competitive ratio less than min *d*1*=p*1*;* 2 *<sup>k</sup>*

paths would be at least 2 *<sup>k</sup> L* <sup>þ</sup> 1 which is greater than the proposed lower bound of min *d*1*=p*1*;* 2 *<sup>k</sup> L* <sup>þ</sup> <sup>1</sup> . The competitive ratio of the strategies in which the last arrival of the agents at D is via *D*<sup>1</sup> is at least *<sup>d</sup>*<sup>1</sup> *p*1 which matches the proposed lower bound of the problem.

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We just proved that the competitive ratio of deterministic strategies in the adverse instance is not better than min *d*1*=p*1*;* 2 *<sup>k</sup> L* <sup>þ</sup> <sup>1</sup> and min *d*1*=p*1*;* 2 *<sup>k</sup> L* <sup>þ</sup> <sup>1</sup> , for the M-*k*-CTP-U-f and the M-*k*-CTP-U-l, respectively. Hence, we conclude that the competitive ratio of the problems cannot be better than the proposed lower bounds.
