A search problem on a bipartite network

Abstract A searcher starting from a given node in a connected network can move along the edges and, when at a node can (but does not have to) search that node; moving on an edge involves a travelling cost and searching a node involves a search cost. The searcher wants to find an immobile object located at one of the nodes other than the searcher’s starting node and, for each node, knows the probability that it is located there. The object is found if and only if the searcher is at the node where the object is hidden and searches there. How can the searcher find the object at minimum cost? We analyze this problem when the network is complete bipartite, where the set of nodes is partitioned to two. Necessary conditions for a searcher strategy to be optimal are obtained and optimal strategies for the case when one of the sets of the partition is a singleton are investigated. Optimal strategies are obtained for the case in which both of the partition sets have the property that all of its nodes have the same cost and assigned the same probability

[1]  Brian Gluss.  Approximately optimal one-dimensional search policies in which search costs vary through time , 1961 .

[2]  Rudolf Ahlswede,et al.  Search Problems , 1987 .

[3]  Steve Alpern,et al.  Optimizing periodic patrols against short attacks on the line and other networks , 2019, Eur. J. Oper. Res..

[4]  Shmuel Gal,et al.  The theory of search games and rendezvous , 2002, International series in operations research and management science.

[5]  Ingo Wegener,et al.  Discrete Sequential Search with Positive Switch Cost , 1982, Math. Oper. Res..

[7]  S. Alpern.  The Rendezvous Search Problem , 1995 .

[8]  L. Stone.  Theory of Optimal Search , 1975 .

[9]  Steve Alpern,et al.  Patrolling a Border , 2016, Oper. Res..

[10]  M. Degroot.  Optimal Statistical Decisions , 1970 .

[11]  Martin L. Puterman,et al.  Technical Note - Trading Off Quick versus Slow Actions in Optimal Search , 2015, Oper. Res..

[12]  K. Kikuta.  AN EXACTLY OPTIMAL STRATEGY FOR A SEARCH PROBLEM WITH TRAVELING COST , 1991 .

[13]  Steve Alpern,et al.  Mining Coal or Finding Terrorists: The Expanding Search Paradigm , 2013, Oper. Res..

[14]  Shmuel Gal,et al.  Search Games: A Review , 2013 .

[15]  Thomas Lidbetter,et al.  The solution to an open problem for a caching game , 2015 .

[16]  Kensaku Kikuta.  A ONE-DIMENSIONAL SEARCH WITH TRAVELING COST , 1990 .

[17]  Tom Lidbetter.  On the approximation ratio of the Random Chinese Postman Tour for network search , 2017, Eur. J. Oper. Res..

[18]  Steve Alpern,et al.  Optimal Trade-Off Between Speed and Acuity When Searching for a Small Object , 2015, Oper. Res..

[19]  Ingo Wegener.  Optimal Search With Positive Switch Cost is NP-Hard , 1985, Inf. Process. Lett..

[20]  Wanjiun Liao,et al.  On the Multichannel Rendezvous Problem: Fundamental Limits, Optimal Hopping Sequences, and Bounded Time-to-Rendezvous , 2015, Math. Oper. Res..

[21]  Kensaku Kikuta.  SEARCH PROBLEM WITH TWO LEVELS OF EXAMINATION COSTS , 2009 .

[22]  Kensaku Kikuta,et al.  Search games on a network with travelling and search costs , 2015, Int. J. Game Theory.

[23]  Steve Alpern,et al.  Searching a Variable Speed Network , 2014, Math. Oper. Res..

[24]  Pierre Leone,et al.  Rendezvous search with markers that can be dropped at chosen times , 2018 .

[25]  Pierre Leone,et al.  Search-and-Rescue Rendezvous , 2016, Eur. J. Oper. Res..

[26]  Steve Alpern.  Hide-and-Seek Games on a Network, Using Combinatorial Search Paths , 2017, Oper. Res..