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

Approximation and complexity of multi-target graph search and the Canadian traveler problem

In the Canadian traveler problem, we are given an edge weighted graph with two specified vertices s and t and a probability distribution over the edges that tells which edges are present. The goal is to minimize the expected length of a walk from s to t. However, we only get to know whether an edge is active the moment we visit one of its incident vertices. Under the assumption that the edges are active independently, we show NP-hardness on series-parallel graphs and give results on the adaptivity gap. We further show that this problem is NP-hard on disjoint-path graphs and cactus graphs when the distribution is given by a list of scenarios. We also consider a special case called the multi-target graph search problem. In this problem, we are given a probability distribution over subsets of vertices. The distribution specifies which set of vertices has targets. The goal is to minimize the expected length of the walk until finding a target. For the

Humanitarian Logistics – the First Week

Decisions made on material flow during the first week of a natural disaster are critical for victims. Currently, decision makers appears to be making important choices based on experience and intuition with little or no support from quantitative approaches because they do not exist. This research proposes a paradigm and offers two supporting models that will assist decision makers regarding the routing of materials during the first week of a disaster. It explicitly includes information regarding the victims’ needs and the degree to which routes are available in a quantitative way that allows updating as information improves. The paradigm involves the use of information gap theory adapted to the this situation for deciding on the types of supplies to send and the Canadian traveler problem for making decisions on the routes to take

AO* and penalty based algorithms for the Canadian traveler problem

Tezin basılısı İstanbul Şehir Üniversitesi Kütüphanesi'ndedir.The Canadian Traveler Problem (CTP) is a challenging path planning problem on stochastic graphs where some edges are blocked with certain probabilities and status of edges can be disambiguated only upon reaching an end vertex. The goal is to devise a traversal policy that results in the shortest expected traversal length between a given starting vertex and a termination vertex. The organization of this thesis is as follows: In the first chapter we define CTP and its variant SOSP and present an extensive literature review related to these problems. In the second chapter, we introduce an optimal algorithm for the problem, based on an MDP formulation which is a new improvement on AO* search that takes advantage of the special problem structure in CTP. The new algorithm is called CAO*, which stands for AO* with Caching. CAO* uses a caching mechanism and makes use of admissible upper bounds for dynamic state-space pruning. CAO* is not polynomial-time, but it can dramatically shorten the execution time needed to find an exact solution for moderately sized instances. We present computational experiments on a realistic variant of the problem involving an actual maritime minefield data set. In the third chapter, we introduce a simple, yet fast and effective penalty-based heuristic for CTP that can be used in an online fashion. We present computational experiments involving real-world and synthetic data that suggest our algorithm finds near-optimal policies in very short execution times. Another efficient method for sub-optimally solving CTP, rollout-based algorithms, have also been shown to provide high quality policies for CTP. In the final chapter, we com- pare the two algorithmic frameworks via computational experiments involving Delaunay and grid graphs using one specific penalty-based algorithm and four rollout-based algo- rithms. Our results indicate that the penalty-based algorithm executes several orders of magnitude faster than rollout-based ones while also providing better policies, suggest- ing that penalty-based algorithms stand as a prominent candidate for fast and efficient sub-optimal solution of CTP.Declaration of Authorship ii Abstract iii Öz iv Acknowledgments v List of Figures viii List of Tables ix Abbreviations x 1 Introduction 1 1.1 Overview .................................... 1 1.2 The Canadian Traveler Problem ........................ 1 1.2.1 The Discrete Stochastic Obstacle Scene Problem .......... 2 1.3 Literature Review ................................ 3 1.4 Organization of the Thesis ........................... 4 2 An AO* Based Exact Algorithm for the Canadian Traveler Problem 5 2.1 Introduction ................................... 5 2.2 MDP and POMDP Formulations ....................... 6 2.2.1 MDP Formulation and The Bellman Equation ............ 7 2.2.2 Deterministic POMDP Formulation ................. 9 2.3 The CAO* Algorithm ............................. 11 2.3.1 AO Trees ................................ 11 2.3.2 The AO* Algorithm .......................... 14 2.3.3 The CAO* Algorithm ......................... 16 2.4 Computational Experiments .......................... 19 2.4.1 The BAO* and PAO* Algorithms ................... 19 2.4.2 Experimental Setup .......................... 21 2.4.3 Simulation Environment A ...................... 21 2.4.4 Simulation Environment B ....................... 22 2.4.5 Simulation Environment C....................... 24 2.4.6 Simulation Environment D ...................... 25 2.5 Summary and Conclusions ........................... 26 3 A Fast and Effective Online Algorithm for the Canadian Traveler Prob- lem 29 3.1 Introduction ................................... 29 3.2 The DT Algorithm ............................... 30 3.3 Computational Experiments .......................... 32 3.3.1 Environment 1 ............................. 32 3.3.2 Environment 2 ............................. 34 3.4 Conclusions and Future Research ....................... 34 3.4.1 Conclusions ............................... 34 3.4.2 Limitations and Future Research ................... 35 4 A Comparison of Penalty and Rollout-Based Policies for the Canadian Traveler Problem 36 4.1 Introduction ................................... 36 4.2 Algorithms for CTP .............................. 37 4.2.1 Optimism (OMT) ........................... 37 4.2.2 Hindsight Optimization (HOP) .................... 38 4.2.3 Optimistic Rollout (ORO) ....................... 39 4.2.4 Blind UCT (UCTB) .......................... 39 4.2.5 Optimistic UCT (UCTO) ....................... 40 4.3 Computational Experiments .......................... 41 4.3.1 Delaunay Graph Results ........................ 43 4.3.2 Grid Graph Results .......................... 45 4.4 Conclusions and Future Research ....................... 46 4.4.1 Conclusions ............................... 46 4.4.2 Limitations and Future Research ................... 46 A Problem Instances in Simulation Environments C and D 48 Bibliography 5

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

The Complexity of Graph Exploration Games

The graph exploration problem asks a searcher to explore an unknown graph. This problem can be interpreted as the online version of the Traveling Salesman Problem. The treasure hunt problem is the corresponding online version of the shortest s-t-path problem. It asks the searcher to find a specific vertex in an unknown graph at which a treasure is hidden. Recently, the analysis of the impact of a priori knowledge is of interest. In graph problems, one form of a priori knowledge is a map of the graph. We survey the graph exploration and treasure hunt problem with an unlabeled map, which is an isomorphic copy of the graph, that is provided to the searcher. We formulate decision variants of both problems by interpreting the online problems as a game between the online algorithm (the searcher) and the adversary. The map, however, is not controllable by the adversary. The question is, whether the searcher is able to explore the graph fully or find the treasure for all possible decisions of the adversary. We prove the PSPACE-completeness of these games, whereby we analyze the variations which ask for the mere existence of a tour through the graph or path to the treasure and the variations that include costs. Additionally, we analyze the complexity of related problems that ask for a tour in the graph or a s-t path