**1. Introduction**

The online *k*-Canadian Traveler Problem (*k*-CTP) is a well-known navigation problem within the field of combinatorial optimization. In the online *k*-CTP, the objective is to reach a destination in a network within minimum travel time under uncertainty of some information. Uncertain information is revealed, while one or more travelers (agents) discover the information during their travels. In the *k*-CTP and its variants studied in the literature, uncertainty is on the locations of blocked edges in the input graph. That is, it is known that there are at most *k* blocked edges, but their locations are not known. In this study, we consider new variations of the *k*-CTP where a known set of edges have unknown (uncertain) travel times (costs). To the best of our knowledge, this variant of the *k*-CTP with given locations of edges that have unknown traveling costs has not been studied yet in the literature.

Uncertainty in travel times arises in various situations, such as following a disaster or in daily urban traffic systems. After a disaster, uncertainty in travel times arises

due to both damage on road segments and traffic congestion on some parts of the road network. We typically know which roads are likely to have damage and to be congested, but the actual travel times can be estimated more accurately when we observe the situation right on the spot. Regarding urban traffic systems, problematic road segments can be detected beforehand since in most current traffic management systems, data indicating locations with high accident frequency are available, but it is difficult to predict the time of occurrence or the intensity of the accident accurately. Also, we usually know where there is a high likelihood of heavy traffic, but travel times show variability. Moreover, nowadays navigation applications indicate which locations have heavy traffic, but the travel times are still not known with certainty, and the situation evolves dynamically as we reach the locations themselves.

information, which is called the *offline optimum* solution. An offline strategy is to solve the same problem as an online strategy, except that all information about the problem inputs is revealed to an offline strategy from the beginning. An optimal offline strategy is the optimal strategy in the presence of complete input information which produces the offline optimum solution. To analyze the performance of online strategies, *competitive ratio* has been introduced in [1] and used by many researchers. The competitive ratio is the maximum ratio of the cost of the online strategy to the cost of the offline strategy over all instances of the problem. In our problems, the costs of the uncertain edges are known in the offline counterparts.

*New Variations of the Online* k*-Canadian Traveler Problem: Uncertain Costs at Known Locations*

Next, we discuss related work in the literature. Then, we state our contributions

We focus on studies on the *k*-CTP which are conducted from the online optimization and the competitive analysis perspective, since these are the most related works to our survey. First, we review the literature for the single-agent variants.

The CTP was defined first in [2]. Papadimitriou and Yannakakis [2] proved that devising an online strategy with a bounded competitive ratio is PSPACE-complete for the CTP. Bar-Noy and Schieber [3] also considered the CTP and its variants. They introduced the *k*-CTP, where an upper bound *k* on the number of blocked edges is given as input. They showed that for arbitrary *k*, the problem of designing an online strategy that guarantees the minimum travel cost is PSPACE-complete. Westphal [4] considered the *k*-CTP from the competitive ratio perspective. He showed the lower bounds of 2*k* þ 1 and *k* þ 1 on the competitive ratio of deterministic and randomized online strategies, respectively. He also presented an optimal deterministic online strategy for the *k*-CTP which is called the *backtrack* strategy. Xu et al. [5] also considered the *k*-CTP and presented two online strategies, the *greedy* and the *comparison* strategy, and proved competitive ratios of 2*<sup>k</sup>*þ<sup>1</sup> � 1 and 2*k* þ 1, respectively, for these strategies. Bender and Westphal [6] presented a randomized online strategy for the *k*-CTP which meets the lower bound of *k* þ 1 in special cases. Shiri and Salman [7] modified the strategy given in [7] and proposed an optimal randomized online strategy for the *k*-CTP on O-D edge-disjoint graphs.

A generalization of the *k*-CTP with multiple agents was first considered by Zhang et al. in [8]. They analyzed the multi-agent *k*-CTP in two scenarios, with limited and complete communication. They proposed lower bounds of 2 *<sup>k</sup>*�<sup>1</sup>

� � <sup>þ</sup> 1 on the competitive ratio of deterministic online strategies for the cases with limited and complete communication, respectively. Note that in the proposed lower bounds *L* denotes the total number of agents and *L*<sup>1</sup> denotes the number of agents who benefit from complete communication. They also proposed an optimal deterministic online strategy when there are two agents in the graph. Shiri and Salman [9] also investigated the multi-agent *k*-CTP. They provided an updated lower bound on the competitive ratio of deterministic online strategies for the case with limited communication. They also presented a deterministic online strategy

*L*1 j k <sup>þ</sup> <sup>1</sup>

Hence, the offline problems reduce to the shortest path problem.

Next, we discuss the relevant studies on the multi-agent versions.

to the defined problems later on in this section.

*DOI: http://dx.doi.org/10.5772/intechopen.88741*

*1.3.1 Single-agent* k*-CTP and variants*

*1.3.2 Multi-agent* k*-CTP and variants*

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

**247**

**1.3 Previous studies**

In many real-world emergency operations, including response to disasters and daily medical or fire emergencies, operations managers must give dispatching decisions urgently under uncertain travel times. Therefore, it is useful to develop online strategies beforehand. For example, for effective disaster response, these strategies can be adopted before the disaster so that they can be implemented in the shortest time after the disaster. Likewise, when traveling in traffic, in order to reach the desired destination in the shortest time, we need a strategy defined on a network which answers the following questions: when to arrive at the end-node of an uncertain edge to learn its travel cost and when to avoid visiting it; when the travel time of an uncertain edge is learned, whether to take it or change the travel route; and if there exists a route to the destination without any uncertain edges, whether to take it or not. In this chapter, we focus on both developing effective online strategies that answer these questions and analyzing their performances theoretically to reveal their worst-case behavior. We next define our problem and its variants formally.

#### **1.1 The online** *k***-CTP with uncertain edges**

Let *G* ¼ ð Þ *V; E; k* denote an undirected graph with O as the source and D as the destination in which the costs of *k* edges with given locations in the graph are unknown and a traveling agent can only discover their costs when she reaches an end-node of them. The costs of the remaining edges are known and deterministic. We call the edges with unknown costs *uncertain edges* and the edges with known costs *deterministic edges*. The objective is to provide an online strategy such that the traveling agent who is located at O initially receives *G* ¼ ð Þ *V; E; k* and the known costs as input and targets to reach D with minimum total travel cost under uncertainty. Since the problem is a new variation of the *k*-CTP, we call this problem the single-agent *k*-CTP with uncertain edges, in short the S-*k*-CTP-U. We also study the multi-agent version of this problem where there are *L* agents, who are initially located at O. We assume that the agents have the capability to transmit their location and edge cost information to the other agents in real time. We consider the multi-agent version of the problem with two different objectives, where the traveling agents follow an online strategy to ensure that the time when (1) the *first* agent and (2) the *last* agent arrive at D is minimum. We call these problems the M-*k*-CTP-U-f and the M-*k*-CTP-U-l, respectively. In real-life applications mentioned before, e.g., disaster response, the objective of the M-*k*-CTP-U-f is applicable when search-and-rescue teams try to reach a target in the shortest time, whereas the objective of the M-*k*-CTP-U-l is applicable when a convoy of *k* vehicles delivers aid to a point of distribution.

#### **1.2 Competitive analysis**

The key concept in analyzing an online strategy is to compare a solution produced by the online strategy with the best possible solution under complete

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

information, which is called the *offline optimum* solution. An offline strategy is to solve the same problem as an online strategy, except that all information about the problem inputs is revealed to an offline strategy from the beginning. An optimal offline strategy is the optimal strategy in the presence of complete input information which produces the offline optimum solution. To analyze the performance of online strategies, *competitive ratio* has been introduced in [1] and used by many researchers. The competitive ratio is the maximum ratio of the cost of the online strategy to the cost of the offline strategy over all instances of the problem. In our problems, the costs of the uncertain edges are known in the offline counterparts. Hence, the offline problems reduce to the shortest path problem.

Next, we discuss related work in the literature. Then, we state our contributions to the defined problems later on in this section.

#### **1.3 Previous studies**

due to both damage on road segments and traffic congestion on some parts of the road network. We typically know which roads are likely to have damage and to be congested, but the actual travel times can be estimated more accurately when we observe the situation right on the spot. Regarding urban traffic systems, problematic road segments can be detected beforehand since in most current traffic management systems, data indicating locations with high accident frequency are available, but it is difficult to predict the time of occurrence or the intensity of the accident accurately. Also, we usually know where there is a high likelihood of heavy traffic, but travel times show variability. Moreover, nowadays navigation applications indicate which locations have heavy traffic, but the travel times are still not known with certainty,

and the situation evolves dynamically as we reach the locations themselves.

worst-case behavior. We next define our problem and its variants formally.

destination in which the costs of *k* edges with given locations in the graph are unknown and a traveling agent can only discover their costs when she reaches an end-node of them. The costs of the remaining edges are known and deterministic. We call the edges with unknown costs *uncertain edges* and the edges with known costs *deterministic edges*. The objective is to provide an online strategy such that the traveling agent who is located at O initially receives *G* ¼ ð Þ *V; E; k* and the known costs as input and targets to reach D with minimum total travel cost under uncertainty. Since the problem is a new variation of the *k*-CTP, we call this problem the single-agent *k*-CTP with uncertain edges, in short the S-*k*-CTP-U. We also study the multi-agent version of this problem where there are *L* agents, who are initially located at O. We assume that the agents have the capability to transmit their location and edge cost information to the other agents in real time. We consider the multi-agent version of the problem with two different objectives, where the traveling agents follow an online strategy to ensure that the time when (1) the *first* agent and (2) the *last* agent arrive at D is minimum. We call these problems the M-*k*-CTP-U-f and the M-*k*-CTP-U-l, respectively. In real-life applications mentioned before, e.g., disaster response, the objective of the M-*k*-CTP-U-f is applicable when search-and-rescue teams try to reach a target in the shortest time, whereas the objective of the M-*k*-CTP-U-l is applicable when a convoy of *k* vehicles delivers aid to a point of distribution.

Let *G* ¼ ð Þ *V; E; k* denote an undirected graph with O as the source and D as the

The key concept in analyzing an online strategy is to compare a solution pro-

duced by the online strategy with the best possible solution under complete

**1.1 The online** *k***-CTP with uncertain edges**

*Probability, Combinatorics and Control*

**1.2 Competitive analysis**

**246**

In many real-world emergency operations, including response to disasters and daily medical or fire emergencies, operations managers must give dispatching decisions urgently under uncertain travel times. Therefore, it is useful to develop online strategies beforehand. For example, for effective disaster response, these strategies can be adopted before the disaster so that they can be implemented in the shortest time after the disaster. Likewise, when traveling in traffic, in order to reach the desired destination in the shortest time, we need a strategy defined on a network which answers the following questions: when to arrive at the end-node of an uncertain edge to learn its travel cost and when to avoid visiting it; when the travel time of an uncertain edge is learned, whether to take it or change the travel route; and if there exists a route to the destination without any uncertain edges, whether to take it or not. In this chapter, we focus on both developing effective online strategies that answer these questions and analyzing their performances theoretically to reveal their

We focus on studies on the *k*-CTP which are conducted from the online optimization and the competitive analysis perspective, since these are the most related works to our survey. First, we review the literature for the single-agent variants. Next, we discuss the relevant studies on the multi-agent versions.

### *1.3.1 Single-agent* k*-CTP and variants*

The CTP was defined first in [2]. Papadimitriou and Yannakakis [2] proved that devising an online strategy with a bounded competitive ratio is PSPACE-complete for the CTP. Bar-Noy and Schieber [3] also considered the CTP and its variants. They introduced the *k*-CTP, where an upper bound *k* on the number of blocked edges is given as input. They showed that for arbitrary *k*, the problem of designing an online strategy that guarantees the minimum travel cost is PSPACE-complete.

Westphal [4] considered the *k*-CTP from the competitive ratio perspective. He showed the lower bounds of 2*k* þ 1 and *k* þ 1 on the competitive ratio of deterministic and randomized online strategies, respectively. He also presented an optimal deterministic online strategy for the *k*-CTP which is called the *backtrack* strategy. Xu et al. [5] also considered the *k*-CTP and presented two online strategies, the *greedy* and the *comparison* strategy, and proved competitive ratios of 2*<sup>k</sup>*þ<sup>1</sup> � 1 and 2*k* þ 1, respectively, for these strategies. Bender and Westphal [6] presented a randomized online strategy for the *k*-CTP which meets the lower bound of *k* þ 1 in special cases. Shiri and Salman [7] modified the strategy given in [7] and proposed an optimal randomized online strategy for the *k*-CTP on O-D edge-disjoint graphs.

#### *1.3.2 Multi-agent* k*-CTP and variants*

A generalization of the *k*-CTP with multiple agents was first considered by Zhang et al. in [8]. They analyzed the multi-agent *k*-CTP in two scenarios, with limited and complete communication. They proposed lower bounds of 2 *<sup>k</sup>*�<sup>1</sup> *L*1 j k <sup>þ</sup> <sup>1</sup> and 2 *<sup>k</sup> L* � � <sup>þ</sup> 1 on the competitive ratio of deterministic online strategies for the cases with limited and complete communication, respectively. Note that in the proposed lower bounds *L* denotes the total number of agents and *L*<sup>1</sup> denotes the number of agents who benefit from complete communication. They also proposed an optimal deterministic online strategy when there are two agents in the graph. Shiri and Salman [9] also investigated the multi-agent *k*-CTP. They provided an updated lower bound on the competitive ratio of deterministic online strategies for the case with limited communication. They also presented a deterministic online strategy

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 proposed for the cases with limited and complete communication.

**2. Preliminaries**

problem.

literature.

dependent.

edge.

common edge.

**249**

1.1 with the following assumptions [1]:

*DOI: http://dx.doi.org/10.5772/intechopen.88741*

ministic O-D path based on Assumption 2.

*<sup>π</sup><sup>k</sup>*þ<sup>1</sup> is the offline optimum and *pk*þ<sup>1</sup> is its corresponding cost.

**3. Single-agent** *k***-CTP with uncertain edges**

We consider the single-agent and the multi-agent problems defined in Section

*New Variations of the Online* k*-Canadian Traveler Problem: Uncertain Costs at Known Locations*

1.The agent(s) are initially located at O. We call this stage the *initial stage* of the

2. If any *k* edges are removed from the graph, there still exists a path between the source and the destination node. This is a standard assumption in the

uncertain edge with explored cost equal to *M* would be considered as a blocked

identified, *stages* of the problem. That is, there are *k* stages in the problem, i.e., stage 1 corresponds to the time period starting at the initial stage and ending at

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 deter-

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.,

In this section, we analyze the single-agent problem, namely, the S-*k*-CTP-U. We present a lower bound to this problem and prove its tightness by introducing a simple strategy. To suggest a lower bound on the competitive ratio of deterministic strategies, we need to analyze the performance of all of deterministic strategies on a special instance. Below, we propose our lower bound on the S-*k*-CTP-U, by analyzing an instance of O-D edge-disjoint graphs. Note that an O-D edge-disjoint graph is an undirected graph *G* with a given source node O and a destination node D, such that any two distinct O-D paths in *G* are edge-disjoint, that is, they do not have a

4.Once the cost of an uncertain edge is learned, it remains the same whenever the traveler visits that edge. In other words the cost is not assumed to be time-

5.We call the time periods in which the cost of a new uncertain edge is

the moment before the cost of the first uncertain edge is learned.

3.The cost of the uncertain edges can take any value between 0 and *M*. An

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 can significantly improve the rescue efficiency.

#### **1.4 Our contributions**

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

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 contributions of our study are detailed below:


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

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