The Canadian Traveller Problem on unit-weighted and arbitrarily weighted outerplanar graphs

This paper studies the multi-robot reliable navigation problem in uncertain topological networks, which aims at maximizing the robot team's on-time arrival probabilities in the face of road network uncertainties. The uncertainty in these networks stems from the unknown edge traversability, which is only revealed to the robot upon its arrival at the edge's starting node. Existing approaches often struggle to adapt to real-time network topology changes, making them unsuitable for varying topological environments. To address the challenge, we reformulate the problem into a Partially Observable Markov Decision Process (POMDP) framework and introduce the Dynamic Adaptive Graph Embedding method to capture the evolving nature of the navigation task. We further enhance each robot's policy learning process by integrating deep reinforcement learning with Graph Attention Networks (GATs), leveraging self-attention to focus on critical graph features. The proposed approach, namely Multi-robot Adaptive Navigation via Graph Attention-based Reinforcement learning (MANGAR) employs the generalized policy gradient algorithm to optimize the robots' real-time decision-making process iteratively. We compare the performance of MANGAR with state-of-the-art reliable navigation algorithms as well as Canadian traveller problem solutions in a range of canonical transportation networks, demonstrating improved adaptability and performance in uncertain topological networks. Additionally, real-world experiments with two robots navigating within a self-constructed indoor environment with uncertain topological structures demonstrate MANGAR's practicality.

The influence of maximum (s,t)-cuts on the competitiveness of deterministic strategies for the Canadian Traveller Problem

This article studies the mean standard deviation (mean-std) Canadian traveller problem (CTP). Different from the canonical CTP, which aims at minimizing the traveller's expected travel time, while considering edge breakdown probabilities, we introduce the reliability version of CTP, which tries to find a routing policy with the minimal linear combination of the travel time's mean and standard deviation. With the recent development of internet-of-things (IoT) technology, the transportation network's edges' travel-time statistics, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> . <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</i> ., mean and standard deviation, as well as the traversal probabilities, are available to the end users. With those information, we propose a dual dynamic programming (DDP) method, which simultaneously estimates the first-order and the second-order moments of a given decision-list (DL) policy, and thereby makes improvements towards to the optimal one through the generalized policy iteration (GPI) scheme. We construct an open source benchmark environment to evaluate the performance of different mean-std CTP solutions, and show that the DDP method outperforms state of the arts in a range of transportation networks.

In this paper, we study online algorithms for the Canadian Traveller Problem defined by Papadimitriou and Yannakakis in 1991. This problem involves a traveller who 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. Achieving a bounded competitive ratio for the problem is PSPACE-complete. Furthermore, if at most k roads can be blocked, the optimal competitive ratio for a deterministic online algorithm is \(2k+1\), while the only randomized result known so far is a lower bound of \(k+1\). We show, for the first time, that a polynomial time randomized algorithm can outperform the best deterministic algorithms when there are at least two blockages, and surpass the lower bound of \(2k+1\) by an o(1) factor. Moreover, we prove that the randomized algorithm can achieve a competitive ratio of \(\big (1+ \frac{\sqrt{2}}{2} \big )k + \sqrt{2}\) in pseudo-polynomial time. The proposed techniques can also be exploited to implicitly represent multiple near-shortest s-t paths.

The Covering Canadian Traveller Problem

This study investigates a generalization of the Canadian Traveller Problem (CTP), which finds real applications in dynamic navigation systems used to avoid traffic congestion. Given a road network $$G=(V,E)$$ G = ( V , E ) in which there is a source $$s$$ s and a destination $$t$$ t in $$V$$ V , every edge $$e$$ e in $$E$$ E is associated with two possible distances: original $$d(e)$$ d ( e ) and jam $$d^+(e)$$ d + ( e ) . A traveller only finds out which one of the two distances of an edge upon reaching an end vertex incident to the edge. The objective is to derive an adaptive strategy for travelling from $$s$$ s to $$t$$ t so that the competitive ratio, which compares the distance traversed with that of the static $$s,t$$ s , t -shortest path in hindsight, is minimized. This problem was initiated by Papadimitriou and Yannakakis. They proved that it is PSPACE-complete to obtain an algorithm with a bounded competitive ratio. In this paper, we propose tight lower bounds of the problem when the number of "traffic jams" is a given constant $$k$$ k ; and we introduce a deterministic algorithm with a $$\mathrm{min}\{ r, 2k+1\}$$ min { r , 2 k + 1 } -ratio, which meets the proposed lower bound, where $$r$$ r is the worst-case performance ratio. We also consider the Recoverable CTP, where each blocked edge is associated with a recovery time to reopen. Finally, we discuss the uniform jam cost model, i.e., for every edge $$e$$ e , $$d^+(e) = d(e) + c$$ d + ( e ) = d ( e ) + c , for a constant $$c$$ c .

Probing-based anypath forwarding routing algorithms in wireless mesh networks

Abstract From the online point of view, we study the Canadian Traveller Problem (CTP), in which the traveller knows in advance the structure of the graph and the costs of all edges. However, some edges may fail and the traveller only observes that upon reaching an adjacent vertex of the blocked edge. The goal is to find the least-cost route from the source O to the destination D, more precisely, to find an adaptive strategy minimizing the competitive ratio, which compares the performance of this strategy with that of a hypothetical offline algorithm that knows the entire topology in advance. In this paper, we present two adaptive strategies—a greedy or myopic strategy and a comparison strategy combining the greedy strategy and the reposition strategy in which the traveller backtracks to the source every time when he/she sees a failed edge. We prove tight competitive ratios of 2 k+1−1 and 2k+1 respectively for the two strategies, where k is the number of failed edges in the graph. Finally, we propose an explanation of why the greedy strategy and the comparison strategy are usually preferred by drivers in an urban traffic environment, based on an argument related to the length of the second-shortest path in a grid graph.