**2. Preliminaries**

which is optimal in both cases with complete and limited communication on O-D edge-disjoint graphs. Randomized online strategies for the multi-agent *k*-CTP are investigated in [10], where lower bounds on the expected competitive ratio

together with optimal randomized online strategies on O-D edge-disjoint graphs are

Xu and Zhang [11] focused on a real-time rescue routing problem from a source node to an emergency spot in the presence of online blockages. They analyzed the problem with the objective to make all the rescuers arrive at the emergency spot with minimum total cost. They studied the problem in two scenarios, without communication and with complete communication. They investigated both of the scenarios on the grid networks and general networks, respectively. They showed that the consideration of both the grid network and the rescuers' communication

In the literature, the common unknown information in the *k*-CTP variants is the

As discussed at the beginning of this section, it is possible to identify the potential locations of the blocked edges in the graph in many real-world instances, such as in the urban traffic and post-disaster response. We introduce a new variation of the *k*-CTP with at most *k* number of uncertain edges with given locations and unknown traveling costs. We call this new problem the online *k*-Canadian Traveler Problem with uncertain edges. We consider both single-agent and multi-agent versions of this problem. In the multi-agent version of the problem, we analyze the problem with two different objectives, where the agents aim to ensure the first and the last arrival of the agents at D with minimum travel cost, respectively. The main contri-

1.We introduce new variations of the online *k*-CTP which find applications in real-world problems, namely, the S-*k*-CTP-U, the M-*k*-CTP-U-f, and the M-

2.We provide a tight lower bound on the competitive ratio of deterministic online strategies for the S-*k*-CTP-U and introduce an optimal deterministic

3.We derive lower bounds on the competitive ratio of deterministic online

The rest of this chapter is organized as follows. In Section 2, we describe the assumptions and give preliminaries. In Section 3, we analyze the single-agent version of the problem and provide a tight lower bound and an optimal strategy to this problem. In Section 4, we suggest lower bounds on the competitive ratio for the

strategies for the M-*k*-CTP-U-f and the M-*k*-CTP-U-l.

multi-agent versions of the problem. Finally, we conclude in Section 5.

locations of the blocked edges in the graph. In fact, in all of the versions of the online *k*-CTP, all of the edges are equally likely to be blocked, and the agents have to explore the blockages in the graph to identify a route from the source node to the destination node with minimum total travel cost. However, in many real-world instances, assuming that all of the edges are equally likely to be congested or blocked ignores valuable information. In other words, there might exist many edges in the graph in which the agent is assured that they are not blocked before she starts her travel. Hence, considering all of the edges to be blocked with equal chance is not

a realistic assumption in some of the real-world applications of the *k*-CTP.

proposed for the cases with limited and complete communication.

can significantly improve the rescue efficiency.

*Probability, Combinatorics and Control*

butions of our study are detailed below:

*k*-CTP-U-l.

online strategy.

**248**

**1.4 Our contributions**

We consider the single-agent and the multi-agent problems defined in Section 1.1 with the following assumptions [1]:


We apply the following symbols and definitions to describe our results. We call the O-D paths which contain uncertain edges *uncertain paths* and which do not have uncertain edges *deterministic paths*. Let *Di* denote the shortest deterministic path at the *i*th stage and *di* ð Þ *i* ¼ 1*;* 2*;* …*; k* denote its corresponding cost. If there are more than one shortest deterministic path at the *i*th stage, one of them can be selected as *Di* arbitrarily. Note that at any stage of the problem there exists at least one deterministic O-D path based on Assumption 2.

We define the *optimistic cost of the O-D path* as the cost of the O-D path after setting the costs of the unvisited uncertain edges on it equal to 0. The *optimistic shortest O-D path* at the *i*th stage of the problem is denoted by *πi*, which corresponds to the shortest O-D path after setting the costs of the remaining uncertain edges equal to 0. We denote its corresponding cost by *pi* ð Þ *i* ¼ 1*;* 2*;* …*; k* . That is, *π*<sup>1</sup> is the optimistic shortest O-D path at the initial stage of the problem. We denote the shortest path after the status of all of the uncertain edges is explored by *π<sup>k</sup>*þ1, i.e., *<sup>π</sup><sup>k</sup>*þ<sup>1</sup> is the offline optimum and *pk*þ<sup>1</sup> is its corresponding cost.
