Complexity of Canadian traveler problem variants

Abstract

The Canadian traveler problem (CTP) is the problem of traversing a given graph, where some of the edges may be blocked -a state which is revealed only upon reaching an incident vertex. Originally stated by Papadimitriou and Yannakakis (1991), the adversarial version of CTP was shown to be PSPACE-complete, with the stochastic version shown to be #P-hard.We show that stochastic CTP is also PSPACE-complete: initially proving PSPACE-hardness for the dependent version of stochastic CTP, and proceeding with gadgets that allow us to extend the proof to the independent case.Since for disjoint-path graphs, CTP can be solved in polynomial time, we examine the complexity of the more general remote-sensing CTP, and show that it is NP-hard even for disjoint-path graphs.probability p(e). The actual state of each edge e ∈ E becomes known only upon reaching a vertex incident on e. Traversing an unblocked edge e incurs a non-negative cost equal to the weight of e. The problem is to find a policy π that minimizes the expected traversal cost C(π) from s to t.CTP formalizes a basic question of navigating in a partially known environment, which is a fundamental task for transportation, autonomous robotic systems, computer games, and more. Other variants of CTP have been introduced and analyzed in the research literature [2,3,4]. There has been a strong recent resurgence of interest in CTP, both theoretical [5,6] and empirical [7,8,9]. A preliminary alternative proof of Theorem 2 appears in an unpublished work by one of the authors [10].When originally introduced in [1], two variants were examined: the adversarial variant and the stochastic variant. The adversarial variant was shown to be PSPACE-complete by reduction from QSAT. For the stochastic version, membership in PSPACE was shown, however only #P-hardness was established by reduction from the st-reliability problem, leaving the question of PSPACE-hardness open. Apparently proving the stronger result requires some form of dependency between the edges, achieved "through the back door" in the adversarial variant. This paper settles the question, showing that CTP is indeed PSPACE-complete.Since the size of an optimal policy is potentially exponential in the size of the problem description, we in fact show that it is PSPACE-hard to find even the optimal first action at s.We begin with a variant of CTP with dependent directed edges, CTP-Dep, which allows for a simple proof of PSPACE-hardness by reduction from QSAT, before proceeding with the proof for the "standard" stochastic CTP. Although the latter result subsumes the former, proving the dependent CTP result first greatly simplifies the intuition behind the proof of the standard case.Another variant we explore is remote-sensing CTP, henceforth called Sensing-CTP, in which additional actions called remote-sensing actions are allowed. Each such action reveals, for a certain cost, the status of a nonincident edge. Recently it was shown [9] that stochastic CTP can be solved in low-order polynomial time on disjoint-path graphs. It was b...

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

“…Finally, we show that CTP in the scenario model on cactus graphs (graphs with the property that each edge is in at most one cycle) is also NP-hard. At this point, we would like to mention that the CTP in the scenario model has similar features with CTP with dependencies [10] and CTP with remote sensing [5,10]. In all these problems, it might be beneficial to explore the state of the graph before proceeding along a path.…”

“…They also showed that the problem is polynomially solvable if the number of scenarios is bounded by a constant. In the independent decision model, the decision version of CTP is PSPACE-complete [10] and it is #P-hard to compute the expected length [19,20]. It is even not possible to describe an optimal policy, unless PSPACE ⊆ P/poly [24].…”

Approximating the Canadian Traveller Problem with Online Randomization

“…Recently, Bender and Westphal proposed a randomized algorithm for special graphs in which all s-t paths are vertex-disjoint [3]. There were also some recent work on the k-CTP for special graphs [4,5,24] and the variants of the k-CTP [11,25,29].…”

Long-run Satisfaction of Path Properties

“…A crucial difference however with mean-payoff and other long-run properties is that, for long-run probabilities, the "weights" along a path are not fixed a priori, but do depend on the scheduler. In this aspect, there is some conceptual relation to dynamic Markov processes [34] where cost or transition probabilities depend on previously made decisions, or the stochastic variant of the Canadian traveler problem [25]. These problems, however, are concerned with finite-horizon objectives; moreover, their weights are affected by the past, whereas our "weights" (satisfaction probabilities) are induced by the future scheduler.…”