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Probabilistic physical search on general graphs: approximations and heuristics

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Abstract

We consider an agent seeking to obtain an item, potentially available at different locations in a physical environment. The traveling costs between locations are known in advance, but there is only probabilistic knowledge regarding the possible prices of the item at any given location. Given such a setting, the problem is to find a plan that maximizes the probability of acquiring the good while minimizing both travel and purchase costs. Sample applications include agents in search-and-rescue or exploration missions, e.g., a rover on Mars seeking to mine a specific mineral. These probabilistic physical search problems have been previously studied, but we present the first approximation and heuristic algorithms for solving such problems on general graphs. We establish an interesting connection between these problems and classical graph-search problems, which led us to provide the approximation algorithms and hardness of approximation results for our settings. We further suggest several heuristics for practical use, and demonstrate their effectiveness with simulation on a real graph structure.

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Notes

  1. Although the lengths of edges in the Deadline-TSP problem are integers, and in the Max-Probability problem they are not necessarily so, we note that they do not play any role in the optimization process.

  2. http://www.mapcruzin.com/download-mexico-canada-us-shapefile/us-highways.zip.

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Acknowledgements

This research was partially supported by the Israel Science Foundation (Grant No. 1488/14). The authors would like to thank Max Kleb, Gilad Tsmadia, Sefi Erlich and Elyasaf Schwer for the implementation of the heuristics and the simulation environment.

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  1. Department of Computer Science, Ariel University, Ariel, Israel

    Noam Hazon & Mira Gonen

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Appendix

Appendix

For analyzing the hardness of approximation of the Min-Budget problem, we were tempted to investigate the dual of the Deadline-TSP problem, which we term Min-Length-Deadline-TSP, as follows:

Definition 3

Given a weighted graph \(G=(V,E)\) on n nodes, with a start node a, a prize function \(\pi :V\rightarrow Z^+\cup \{0\}\), deadlines \(D:V\rightarrow Z^+\cup \{0\}\), a length function \(\ell :E\rightarrow Z^+\cup \{0\}\), and a prize quota q, find a path starting at a that minimizes the total length while collecting a total prize of at least q, where a path starting at a collects the prize \(\pi (v)\) at node v if it reaches v before D(v).

That is, in this problem there is a known quota for the prize, and the target function is finding a minimum length path that meets this prize quota. We show a strong hardness of approximation result for the Min-Length-Deadline-TSP.

Theorem 5

The Min-Length-Deadline-TSP problem is hard to approximate within any constant ratio.

Proof

We reduce the decision version of the TSP problem with triangle inequality to the Min-Length-Deadline-TSP problem. In the decision version of the TSP problem we are given a parameter W and a weighted graph, and we need to find if there is a TSP path with weight at most W. We define an instance of the Min-Length-Deadline-TSP with the same graph, where the deadlines of all nodes equal W, all nodes have a unit prize, and \(q=n\) (the prize quota equals the number of nodes). Any feasible solution to the Min-Length-Deadline-TSP problem, even an approximated one, must traverse all nodes before their deadline, which implies a feasible TSP path with a weight at most W. This cannot be achieved in polynomial time if \(P\ne NP\). \(\square \)

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Hazon, N., Gonen, M. Probabilistic physical search on general graphs: approximations and heuristics. Auton Agent Multi-Agent Syst 34, 1 (2020). https://doi.org/10.1007/s10458-019-09423-z

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