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On the approximation ratio of the Random Chinese Postman Tour for network search

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  • Lidbetter, Thomas

Abstract

We consider a classic search problem first proposed by Gal in which a Searcher randomizes between unit speed paths on a network, aiming to find a hidden point in minimal expected time in the worst case. This can be viewed as a zero-sum game between the Searcher and a time maximizing Hider. It is a natural model for many search problems such as search and rescue operations; the search for an enemy, a bomb or weapons in a military context; or predator–prey search. A Chinese Postman Tour (CPT) is a minimal time tour of the network that searches all the arcs and a Random Chinese Postman Tour (RCPT) is an equiprobable choice of any given CPT and its reverse. The full class of networks for which a RCPT is optimal is known, but otherwise little is known about the solution of the game except in some special cases that have complicated optimal strategies that would be impractical to implement. The question of how well a RCPT or any other search strategy performs for general networks has never been analyzed. We show that a RCPT has an approximation ratio of 4/3: that is, the maximum expected time it takes to find a point on the network is no greater than 4/3 times that of the optimal search strategy. We then examine the performance of a RCPT in a related search game recently proposed by Alpern in which the Searcher must return to his starting point after finding the Hider.

Suggested Citation

  • Lidbetter, Thomas, 2017. "On the approximation ratio of the Random Chinese Postman Tour for network search," European Journal of Operational Research, Elsevier, vol. 263(3), pages 782-788.
    • Handle: RePEc:eee:ejores:v:263:y:2017:i:3:p:782-788
    DOI: 10.1016/j.ejor.2017.06.004
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    References listed on IDEAS

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    1. Steve Alpern & Thomas Lidbetter, 2013. "Mining Coal or Finding Terrorists: The Expanding Search Paradigm," Operations Research, INFORMS, vol. 61(2), pages 265-279, April.
    2. Alpern, Steven & Morton, Alec & Papadaki, Katerina, 2011. "Patrolling games," LSE Research Online Documents on Economics 32210, London School of Economics and Political Science, LSE Library.
    3. Steve Alpern, 2011. "A New Approach to Gal’s Theory of Search Games on Weakly Eulerian Networks," Dynamic Games and Applications, Springer, vol. 1(2), pages 209-219, June.
    4. Steve Alpern & Thomas Lidbetter, 2015. "Optimal Trade-Off Between Speed and Acuity When Searching for a Small Object," Operations Research, INFORMS, vol. 63(1), pages 122-133, February.
    5. Alpern, Steven & Lidbetter, Thomas, 2015. "Optimal trade-off between speed and acuity when searching for a small object," LSE Research Online Documents on Economics 61504, London School of Economics and Political Science, LSE Library.
    6. Steve Alpern, 2011. "Find-and-Fetch Search on a Tree," Operations Research, INFORMS, vol. 59(5), pages 1258-1268, October.
    7. Steve Alpern & Vic Baston & Shmuel Gal, 2008. "Network search games with immobile hider, without a designated searcher starting point," International Journal of Game Theory, Springer;Game Theory Society, vol. 37(2), pages 281-302, June.
    8. Steve Alpern & Alec Morton & Katerina Papadaki, 2011. "Patrolling Games," Operations Research, INFORMS, vol. 59(5), pages 1246-1257, October.
    9. Vic Baston & Kensaku Kikuta, 2015. "Search games on a network with travelling and search costs," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(2), pages 347-365, May.
    10. Shmuel Gal, 2001. "On the optimality of a simple strategy for searching graphs," International Journal of Game Theory, Springer;Game Theory Society, vol. 29(4), pages 533-542.
    11. Reijnierse, J H & Potters, J A M, 1993. "Search Games with Immobile Hider," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(4), pages 385-394.
    12. Zoroa, N. & Fernández-Sáez, M.J. & Zoroa, P., 2012. "Patrolling a perimeter," European Journal of Operational Research, Elsevier, vol. 222(3), pages 571-582.
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    Cited by:

    1. Baston, Vic & Kikuta, Kensaku, 2019. "A search problem on a bipartite network," European Journal of Operational Research, Elsevier, vol. 277(1), pages 227-237.

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