The search value of a set

Fokkink, Robbert; Kikuta, Ken; Ramsey, David

doi:10.1007/s10479-016-2252-y

Cited by 5 publications

(8 citation statements)

References 8 publications

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“…We emphasize that the 2-approximation found in [13] was obtained independently by [35]. We also note that the 2-approximation result generalizes a similar result from [19], which says that any search strategy is a 2approximation for the equilibrium search strategy in the submodular search game. Definition 1.…”

Section: The Submodular Search Problemsupporting

confidence: 70%

“…Recall the definition of the submodular base polyhedron B(f ) in (4). We apply Theorem 1 to settle a question from [19].…”

Section: The Submodular Search Gamementioning

confidence: 99%

“…Thus, g(A) = x(A) := j∈A x j . We consider the finite zero-sum game between a Searcher and a cost maximizing Hider, introduced in [19]. A pure strategy for the Searcher is a permutation π of S and a pure strategy for the Hider is an element j ∈ S; the payoff is f (S π j ).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Submodular Search and Machine Scheduling

2019

Self Cite

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Suppose some objects are hidden in a finite set S of hiding places which must be examined one-by-one. The cost of searching subsets of S is given by a submodular function and the probability that all objects are contained in a subset is given by a supermodular function. We seek an ordering of S that finds all the objects in minimal expected cost. This problem is NP-hard and we give an efficient combinatorial 2-approximation algorithm, generalizing analogous results in scheduling theory. We also give a new scheduling application 1|prec| w A h(C A ), where a set of jobs must be ordered subject to precedence constraints to minimize the weighted sum of some concave function h of the completion times of subsets of jobs. We go on to give better approximations for submodular functions with low total curvature and we give a full solution when the problem is what we call series-parallel decomposable. Next, we consider a zero-sum game between a cost-maximizing Hider and a cost-minimizing Searcher. We prove that the equilibrium mixed strategies for the Hider are in the base polyhedron of the cost function, suitably scaled, and we solve the game in the seriesparallel decomposable case, giving approximately optimal strategies in other cases.

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Section: The Submodular Search Problemsupporting

confidence: 70%

“…Recall the definition of the submodular base polyhedron B(f ) in (4). We apply Theorem 1 to settle a question from [19].…”

Section: The Submodular Search Gamementioning

confidence: 99%

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On Submodular Search and Machine Scheduling

2019

Self Cite

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“…This game was introduced by Fokkink et al (2016a), and further studied by Fokkink et al (2016b). As in the game, BOX, the Hider's strategy set is [n] and the Searcher's strategy set is all permutations π of [n].…”

Section: The Submodular Search Gamementioning

confidence: 99%

“…An example of this is the classic search game for an immobile Hider on a network studied in Gal (1979) and Gal (2000). More recent examples can be found in Alpern (2016), Alpern andLidbetter (2015, 2013a,b), Angelopoulos et al (2016), Kikuta (2013, 2015), Fokkink et al (2016a) and Lin and Singham (2016). Many of these games are infinite, in the sense that one or both of the players has a strategy set of infinite cardinality, but in this paper we restrict ourselves to finite games, which lend themselves better to an algorithmic approach.…”

Section: Introductionmentioning

confidence: 99%

Solving Zero-Sum Games Using Best-Response Oracles with Applications to Search Games

Hellerstein

Lidbetter

Pirutinsky

2019

Operations Research

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We present efficient algorithms for computing optimal or approximately optimal strategies in a zero-sum game for which Player I has n pure strategies and Player II has an arbitrary number of pure strategies. We assume that for any given mixed strategy of Player I, a best response or "approximate" best response of Player II can be found by an oracle in time polynomial in n. We then show how our algorithms may be applied to several search games with applications to security and counter-terrorism. We evaluate our main algorithm experimentally on a prototypical search game. Our results show it performs well compared to an existing, well-known algorithm for solving zero-sum games that can also be used to solve search games, given a best response oracle.

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Approximate solutions for expanding search games on general networks

Alpern¹,

Lidbetter

2018

Ann Oper Res

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We study the classical problem introduced by R. Isaacs and S. Gal of minimizing the time to find a hidden point H on a network Q moving from a known starting point. Rather than adopting the traditional continuous unit speed path paradigm, we use the "expanding search" paradigm recently introduced by the authors. Here the regions S (t) that have been searched by time t are increasing from the starting point and have total length t. Roughly speaking the search follows a sequence of arcs a i such that each one starts at some point of an earlier one. This type of search is often carried out by real life search teams in the hunt for missing persons, escaped convicts, terrorists or lost airplanes. The paper which introduced this type of search solved the adversarial problem (where H is hidden to take a long time to find) for the cases where Q is a tree or is 2-arc-connected. This paper solves the game on some additional families of networks. However the main contribution is to give strategy classes which can be used on any network and have expected search times which are within a factor close to 1 of the value of the game (minimax search time). We identify cases where our strategies are in fact optimal.

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The search value of a set

Cited by 5 publications

References 8 publications

On Submodular Search and Machine Scheduling

On Submodular Search and Machine Scheduling

Solving Zero-Sum Games Using Best-Response Oracles with Applications to Search Games

Approximate solutions for expanding search games on general networks

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