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234:-hard. Another variant is to make no assumption on the distribution but require that each realization with non-zero probability be explicitly stated (such as “Probability 0.1 of edge set { {3,4},{1,2} }, probability 0.2 of...”). This is called the Distribution Stochastic Shortest Path Problem (d-SSPPR or R-SSPPR) and is
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An additional parameter is how new knowledge is being discovered on the realization. In traditional variants of CTP, the agent uncovers the exact weight (or status) of an edge upon reaching an adjacent vertex. A new variant was recently suggested where an agent also has the ability to perform remote
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The fourth and final parameter is how the graph changes over time. In CTP and SSPPR, the realization is fixed but not known. In the
Stochastic Shortest Path Problem with Recourse and Resets or the Expected Shortest Path problem, a new realization is chosen from the distribution after each step taken
206:
There are primarily five parameters distinguishing the number of variants of the
Canadian Traveller Problem. The first parameter is how to value the walk produced by a policy for a given instance and realization. In the Stochastic Shortest Path Problem with Recourse, the goal is simply to minimize
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The third parameter is restricted to the stochastic versions and is about what assumptions we can make about the distribution of the realizations and how the distribution is represented in the input. In the
Stochastic Canadian Traveller Problem and in the Edge-independent Stochastic Shortest Path
230:
Problem (i-SSPPR), each uncertain edge (or cost) has an associated probability of being in the realization and the event that an edge is in the graph is independent of which other edges are in the realization. Even though this is a considerable simplification, the problem is still
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by the policy. This problem can be solved in polynomial time by reducing it to a Markov decision process with polynomial horizon. The Markov generalization, where the realization of the graph may influence the next realization, is known to be much harder.
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1577:. If the goal is never reached, we say that we have an infinite cost. If the goal is reached, we define the cost of the walk as the sum of the costs of all of the edges traversed, with cardinality.
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The second parameter is how to evaluate a policy with respect to different realizations consistent with the instance under consideration. In the
Canadian Traveller Problem, one wishes to study the
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the cost of the walk (defined as the sum over all edges of the cost of the edge times the number of times that edge was taken). For the
Canadian Traveller Problem, the task is to minimize the
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We define the variant studied in the paper from 1989. That is, the goal is to minimize the competitive ratio in the worst case. It is necessary that we begin by introducing certain terms.
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Despite the age of the problem and its many potential applications, many natural questions still remain open. Is there a constant-factor approximation or is the problem
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in 1989 and a number of variants of the problem have been studied since. The name supposedly originates from conversations of the authors who learned of a difficulty
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238:. The first variant is harder than the second because the former can represent in logarithmic space some distributions that the latter represents in linear space.
1824:. It was also shown that finding an optimal path in the case where each edge has an associated probability of being in the graph (i-SSPPR) is a PSPACE-easy but
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Given an instance and policy for the instance, every realization produces its own (deterministic) walk in the graph. Note that the walk is not necessarily a
109:. In other words, a "traveller" on a given point on the graph cannot see the full graph, rather only adjacent nodes or a certain "realization restriction."
260:
187:. The CTP task is to compute the expected cost of the optimal policies. To compute an actual description of an optimal policy may be a harder problem.
1828:-hard problem. It was an open problem to bridge this gap, but since then both the directed and undirected versions were shown to be PSPACE-hard.
1855:, communication networks, and routing. A variant of the problem has been studied for robot navigation with probabilistic landmark recognition.
1523:). When we take a step in the graph, the edges incident to our new location become known to us. Those edges that are in the graph are added to
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Consider a given graph and the family of undirected graphs that can be constructed by adding one or more edges from a given set. Formally, let
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sensing from any location on the realization. In this variant, the task is to minimize the travel cost plus the cost of sensing operations.
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version, where each edge is associated with a probability of independently being in the graph, has been given considerable attention in
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183:, of how the hidden graph may look. Given an instance, a description of how to follow the instance in the best way is called a
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Briggs, Amy J.; Detweiler, Carrick; Scharstein, Daniel (2004). "Expected shortest paths for landmark-based robot navigation".
1812:, where the players compete over the cost of their respective paths and the edge set is chosen by the second player (nature).
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637:. This is called the off-line problem because an algorithm for such a problem would have complete information of the graph.
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1867:-hard? Is it i-SSPPR #P-complete? An even more fundamental question has been left unanswered: is there a polynomial-size
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of the walk; i.e., to minimize the number of times longer the produced walk is to the shortest path in the realization.
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since the best strategy may be to, e.g., visit every vertex of a cycle and return to the start. This differs from the
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of an optimal policy, setting aside for a moment the time necessary to compute the description?
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In other words, we evaluate the policy based on the edges we currently know are in the graph (
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drivers had: traveling a network of cities with snowfall randomly blocking roads. The
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367:{\displaystyle {\mathcal {G}}(V,E,F)=\{(V,E+F')|F'\subseteq F\},E\cap F=\emptyset }
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The original paper analysed the complexity of the problem and reported it to be
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under the name "the
Stochastic Shortest Path Problem with Recourse" (SSPPR).
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of the graph family. Furthermore, let W be an associated cost matrix where
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C.H. Papadimitriou; M. Yannakakis (1989). "Shortest paths without a map".
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1999:
Dror Fried; Solomon Eyal
Shimony; Amit Benbassat; Cenny Wenner (2013).
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Exact
Algorithms for the Canadian Traveller Problem on Paths and Trees
2060:. International Joint Conference On Artificial Intelligence (IJCAI).
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For a given instance, there are a number of possibilities, or
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222:. For average case analysis, one must furthermore specify an
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The directed version of the stochastic problem is known in
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Zahy Bnaya; Ariel Felner; Solomon Eyal
Shimony (2009).
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as the
Stochastic Shortest Path Problem with Recourse.
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Papadimitriou and Yannakakis noted that this defines a
717:{\displaystyle ({\mathcal {P}}(E),{\mathcal {P}}(F),V)}
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2044:David Karger; Evdokia Nikolova (January 28, 2008).
1580:Finally, we define the Canadian traveller problem.
1274:{\displaystyle v_{i+1}=\pi (E_{i+1},F_{i+1},v_{i})}
629:be the cost of the shortest path in the graph from
382:as the edges that may be in the graph. We say that
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may be too technical for most readers to understand
2001:"Complexity of Canadian traveler problem variants"
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2051:(Report). Massachusetts Institute of Technology.
1096:{\displaystyle E_{i+1}=E_{i}\cup E_{B}(v_{i},V)}
528:its incident edges with respect to the edge set
1496:) and the edges we know might be in the graph (
2058:Canadian Traveller Problem with remote sensing
1716:{\displaystyle (V,B)\in {\mathcal {G}}(V,E,F)}
378:as the edges that must be in the graph and of
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1843:The problem is said to have applications in
1179:{\displaystyle F_{i+1}=F_{i}-E_{F}(v_{i},V)}
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660:to navigate such a graph is a mapping from
1930:International Journal of Robotics Research
1918:Fried, Shimony, Benbassat, and Wenner 2013
1909:Papadimitriou and Yannakakis, 1989, p. 148
580:{\displaystyle G\in {\mathcal {G}}(V,E,F)}
426:{\displaystyle G\in {\mathcal {G}}(V,E,F)}
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2085:Computational problems in graph theory
55:make it understandable to non-experts
7:
226:distribution over the realizations.
1990:. Proc. 16th ICALP. Vol. 372.
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1988:Lecture Notes in Computer Science
1458:{\displaystyle c(\pi ,B)=\infty }
770:{\displaystyle {\mathcal {P}}(X)}
536:. Furthermore, for a realization
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1972:Karger and Nikolova, 2008, p. 1
927:{\displaystyle v_{0}=s,E_{0}=E}
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1633:{\displaystyle (V,E,F,s,t,r)}
101:) is a generalization of the
2005:Theoretical Computer Science
1762:times the off-line optimal,
2080:Travelling salesman problem
1847:, transportation planning,
1013:{\displaystyle i=0,1,2,...}
27:Computational graph problem
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1797:{\displaystyle d_{B}(s,t)}
622:{\displaystyle d_{B}(s,t)}
521:{\displaystyle E_{B}(v,V)}
95:Canadian traveller problem
18:Canadian Traveller Problem
2028:10.1016/j.tcs.2013.03.016
1751:{\displaystyle c(\pi ,B)}
813:{\displaystyle c(\pi ,B)}
2075:PSPACE-complete problems
1952:10.1177/0278364904045467
1849:artificial intelligence
1316:{\displaystyle v_{T}=t}
960:{\displaystyle F_{0}=F}
871:{\displaystyle G=(V,B)}
640:We say that a strategy
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118:Christos Papadimitriou
1891:Shortest path problem
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1584:Given a CTP instance
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1570:{\displaystyle F_{i}}
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1994:. pp. 610–620.
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140:Problem description
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105:to graphs that are
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122:Mihalis Yannakakis
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2019:
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1859:Open problems
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168:February 2017
162:
158:
155:This section
153:
150:
146:
145:
139:
137:
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131:
127:
123:
119:
115:
110:
108:
104:
100:
96:
92:
88:
77:
74:
66:
56:
52:
46:
43:This article
41:
32:
31:
19:
2057:
2008:
2004:
1987:
1968:
1933:
1929:
1923:
1914:
1905:
1886:Hitting time
1868:
1862:
1842:
1839:Applications
1830:
1819:
1807:
1759:
1579:
1468:
1287:
878:as follows.
782:
777:denotes the
639:
634:
630:
533:
529:
483:
479:
477:
472:
468:
434:
379:
375:
256:
253:
244:
240:
228:
220:average case
213:
205:
189:
184:
181:realizations
180:
178:
165:
161:adding to it
156:
111:
98:
94:
91:graph theory
84:
69:
60:
44:
1869:description
1723:, the cost
435:realization
236:NP-complete
2069:Categories
1980:References
1816:Complexity
1290:such that
486:, we call
471:to vertex
216:worst case
130:stochastic
2018:1207.4710
1938:CiteSeerX
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547:∈
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362:∅
353:∩
338:⊆
63:June 2021
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779:powerset
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334:′
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224:a priori
202:Variants
126:Canadian
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