Canadian Traveller Problem with Predictions

In this work, we consider the $k$-Canadian Traveller Problem ($k$-CTP) under the learning-augmented framework proposed by Lykouris & Vassilvitskii. $k$-CTP is a generalization of the shortest path problem, and involves a traveller who knows the entire graph in advance and wishes to find the shortest route from a source vertex $s$ to a destination vertex $t$, but discovers online that some edges (up to $k$) are blocked once reaching them. A potentially imperfect predictor gives us the number and the locations of the blocked edges. We present a deterministic and a randomized online algorithm for the learning-augmented $k$-CTP that achieve a tradeoff between consistency (quality of the solution when the prediction is correct) and robustness (quality of the solution when there are errors in the prediction). Moreover, we prove a matching lower bound for the deterministic case establishing that the tradeoff between consistency and robustness is optimal, and show a lower bound for the randomized algorithm. Finally, we prove several deterministic and randomized lower bounds on the competitive ratio of $k$-CTP depending on the prediction error, and complement them, in most cases, with matching upper bounds

The Covering Canadian Traveller Problem Revisited

In this paper, we consider the k-Covering Canadian Traveller Problem (k-CCTP), which can be seen as a variant of the Travelling Salesperson Problem. The goal of k-CCTP is finding the shortest tour for a traveller to visit a set of locations in a given graph and return to the origin. Crucially, unknown to the traveller, up to k edges of the graph are blocked and the traveller only discovers blocked edges online at one of their respective endpoints. The currently best known upper bound for k-CCTP is O(?k) which was shown in [Huang and Liao, ISAAC \u2712]. We improve this polynomial bound to a logarithmic one by presenting a deterministic O(log k)-competitive algorithm that runs in polynomial time. Further, we demonstrate the tightness of our analysis by giving a lower bound instance for our algorithm

Canadians Should Travel Randomly

We study online algorithms for the Canadian Traveller Problem (CTP) introduced by Papadimitriou and Yannakakis in 1991. In this problem, a traveller knows the entire road network in advance, and wishes to travel as quickly as possible from a source vertex s to a destination vertex t, but discovers online that some roads are blocked (e.g., by snow) once reaching them. It is PSPACE-complete to achieve a bounded competitive ratio for this problem. Furthermore, if at most k roads can be blocked, then the optimal competitive ratio for a deterministic online algorithm is 2k + 1, while the only randomized result known is a lower bound of k + 1. In this paper, we show for the first time that a polynomial time randomized algorithm can beat the best deterministic algorithms, surpassing the 2k + 1 lower bound by an o(1) factor. Moreover, we prove the randomized algorithm achieving a competitive ratio of (1 + [√2 over 2])k + 1 in pseudo-polynomial time. The proposed techniques can also be applied to implicitly represent multiple near-shortest s-t paths.NSC Grant 102-2221-E-007-075-MY3Japan Society for the Promotion of Science (KAKENHI 23240002

Efficient Routing for Disaster Scenarios in Uncertain Networks: A Computational Study of Adaptive Algorithms for the Stochastic Canadian Traveler Problem with Multiple Agents and Destinations

The primary objective of this research is to develop adaptive online algorithms for solving the Canadian Traveler Problem (CTP), which is a well-studied problem in the literature that has important applications in disaster scenarios. To this end, we propose two novel approaches, namely Maximum Likely Node (MLN) and Maximum Likely Path (MLP), to address the single-agent single-destination variant of the CTP. Our computational experiments demonstrate that the MLN and MLP algorithms together achieve new best-known solutions for 10,715 instances. In the context of disaster scenarios, the CTP can be extended to the multiple-agent multiple-destination variant, which we refer to as MAD-CTP. We propose two approaches, namely MAD-OMT and MAD-HOP, to solve this variant. We evaluate the performance of these algorithms on Delaunay and Euclidean graphs of varying sizes, ranging from 20 nodes with 49 edges to 500 nodes with 1500 edges. Our results demonstrate that MAD-HOP outperforms MAD-OMT by a considerable margin, achieving a replan time of under 9 seconds for all instances. Furthermore, we extend the existing state-of-the-art algorithm, UCT, which was previously shown by Eyerich et al. (2010) to be effective for solving the single-source single-destination variant of the CTP, to address the MAD-CTP problem. We compare the performance of UCT and MAD-HOP on a range of instances, and our results indicate that MAD-HOP offers better performance than UCT on most instances. In addition, UCT exhibited a very high replan time of around 10 minutes. The inferior results of UCT may be attributed to the number of rollouts used in the experiments but increasing the number of rollouts did not conclusively demonstrate whether UCT could outperform MAD-HOP. This may be due to the benefits obtained from using multiple agents, as MAD-HOP appears to benefit to a greater extent than UCT when information is shared among agents

Efficient Routing for Disaster Scenarios in Uncertain Networks: A Computational Study of Adaptive Algorithms for the Stochastic Canadian Traveler Problem with Multiple Agents and Destinations

The primary objective of this research is to develop adaptive online algorithms for solving the Canadian Traveler Problem (CTP), which is a well-studied problem in the literature that has important applications in disaster scenarios. To this end, we propose two novel approaches, namely Maximum Likely Node (MLN) and Maximum Likely Path (MLP), to address the single-agent single-destination variant of the CTP. Our computational experiments demonstrate that the MLN and MLP algorithms together achieve new best-known solutions for 10,715 instances. In the context of disaster scenarios, the CTP can be extended to the multiple-agent multiple-destination variant, which we refer to as MAD-CTP. We propose two approaches, namely MAD-OMT and MAD-HOP, to solve this variant. We evaluate the performance of these algorithms on Delaunay and Euclidean graphs of varying sizes, ranging from 20 nodes with 49 edges to 500 nodes with 1500 edges. Our results demonstrate that MAD-HOP outperforms MAD-OMT by a considerable margin, achieving a replan time of under 9 seconds for all instances. Furthermore, we extend the existing state-of-the-art algorithm, UCT, which was previously shown by Eyerich et al. (2010) to be effective for solving the single-source single-destination variant of the CTP, to address the MAD-CTP problem. We compare the performance of UCT and MAD-HOP on a range of instances, and our results indicate that MAD-HOP offers better performance than UCT on most instances. In addition, UCT exhibited a very high replan time of around 10 minutes. The inferior results of UCT may be attributed to the number of rollouts used in the experiments but increasing the number of rollouts did not conclusively demonstrate whether UCT could outperform MAD-HOP. This may be due to the benefits obtained from using multiple agents, as MAD-HOP appears to benefit to a greater extent than UCT when information is shared among agents

New Variations of the Online <em>k</em>-Canadian Traveler Problem: Uncertain Costs at Known Locations

In this chapter, we study new variations of the online k-Canadian Traveler Problem (k-CTP) in which there is an input graph with a given source node O and a destination node D. For a specified set consisting of k edges, the edge costs are unknown (we call these uncertain edges). Costs of the remaining edges are known and given. The objective is to find an online strategy such that the traveling agent finds a route from O to D with minimum total travel cost. The agent learns the cost of an uncertain edge, when she arrives at one of its end-nodes and decides on her travel path based on the discovered cost. We call this problem the online k-Canadian Traveler Problem with uncertain edges. We analyze both the single-agent and the multi-agent versions of the problem. We propose a tight lower bound on the competitive ratio of deterministic online strategies together with an optimal online strategy for the single-agent version. We consider the multi-agent version with two different objectives. We suggest lower bounds on the competitive ratio of deterministic online strategies to these two problems

ℓ-CTP: Utilizing Multiple Agents to Find Efficient Routes in Disrupted Networks

Recent hurricane seasons have demonstrated the need for more effective methods of coping with flooding of roadways. A key complaint of logistics managers is the lack of knowledge when developing routes for vehicles attempting to navigate through areas which may be flooded. In particular, it can be difficult to re-route large vehicles upon encountering a flooded roadway. We utilize the Canadian Traveller’s Problem (CTP) to construct an online framework for utilizing multiple vehicles to discover low-cost paths through networks with failed edges unknown to one or more agents a priori. This thesis demonstrates the following results: first, we develop the ℓ-CTP framework to extend a theoretically validated set of path planning policies for a single agent in combination with the iterative penalty method, which incentivizes a group of ℓ \u3e 1 agents to explore dissimilar paths on a graph between a common origin and destination. Second, we carry out simulations on random graphs to determine the impact of the addition of agents on the path cost found. Through statistical analysis of graphs of multiple sizes, we validate our technique against prior work and demonstrate that path cost can be modeled as an exponential decay function on the number of agents. Finally, we demonstrate that our approach can scale to large graphs, and the results found on random graphs hold for a simulation of the Houston metro area during hurricane Harvey