On Submodular Search and Machine Scheduling

Fokkink, Robbert; Lidbetter, Thomas; Végh, László A.

doi:10.1287/moor.2018.0978

Cited by 21 publications

(14 citation statements)

References 51 publications

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“…This can be seen by defining a cost function f on subsets S of [n] in PREC such that f (S) is the cost of all the elements of [n] in the precedence-closure of S. Then, under this cost function, optimal strategies for SUB will correspond to optimal strategies in PREC. Fokkink et al (2016b) also consider the best response problem for SUB, and they prove that this problem can be solved approximately, within a factor of 2, generalizing the analogous result in the scheduling literature. It follows that our algorithms can be used to calculate 2-approximate strategies for the players in SUB, and so we obtain our next new result for search games.…”

Section: The Submodular Search Gamementioning

confidence: 64%

“…The Hider's strategy set and the payoff function remain unchanged. We call this game PREC, and while it has not been studied before in the form we define it in here, a more general version of it was studied in Fokkink et al (2016b), as we will discuss in the next subsection.…”

Section: Searching In Boxes With Precedence Constraintsmentioning

confidence: 99%

“…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%

See 2 more Smart Citations

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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Section: The Submodular Search Gamementioning

confidence: 64%

Section: Searching In Boxes With Precedence Constraintsmentioning

confidence: 99%

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

Hellerstein

Lidbetter

Pirutinsky

2019

Operations Research

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“…A discrete version of this problem was considered in Fokkink et al (2019), and a search game with multiple targets was solved in Lidbetter (2013). Finally, these problems could all be generalized by considering asymmetric (or windy) networks, for which the time to traverse an arc depends on the direction of travel.…”

Section: Resultsmentioning

confidence: 99%

Search and Delivery Man Problems: When are depth-first paths optimal?

Alpern

Lidbetter

2020

European Journal of Operational Research

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Let h be a probability measure on a network Q, viewed either as the location of a hidden object to be found or as the continuous distribution of customers receiving packages. We wish to find a trajectory starting from a specified root, or depot O that minimizes the expected search or delivery time. We call such a trajectory optimal. When Q is a tree, we ask for which h there is an optimal trajectory that is depth-first, and we find sufficient conditions and in some cases necessary and sufficient conditions on h. A consequence of our analysis is a determination of the optimal depot location in the Delivery Man Problem, correcting an error in the literature.A person, or more generally a target, is lost in a network. We know the lengths of the arcs of the network, and we have a belief about the probability distribution (or hiding distribution) according to which the target is hidden. We wish to choose a unit speed trajectory (or search) starting from a given point (the root) with the aim of minimizing the expected search time:that is the expected time to find the target. The target might be a child lost in a cave, as in the recent rescue in Thailand; or it might be a leak in a pipeline. The Searcher could be an underwater diver or an unmanned aerial vehicle (UAV), respectively. We call a search that minimizes the expected search time optimal. The target does not have to be at a node; it can be anywhere in the interior of any arc. This paper considers the problem when the network is a tree.

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“…Alpern and Lidbetter (2013) describe a polynomial algorithm for the expanding search problem when the underlying graph is a tree, and Tan et al (2019) study both exact and approximation algorithms when there are multiple searchers, again for the special case of trees. Fokkink et al (2019), finally, generalize the algorithm of Alpern and Lidbetter (2013) to submodular cost and supermodular weight functions. This generalization, however, does not work for general graphs since the length necessary to search a set of vertices is, in general, not a submodular function of that set.…”

Section: Introductionmentioning

confidence: 99%

Exact and Approximation Algorithms for the Expanding Search Problem

Hermans

Leus

Matuschke

2022

INFORMS Journal on Computing

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Suppose a target is hidden in one of the vertices of an edge-weighted graph according to a known probability distribution. Starting from a fixed root node, an expanding search visits the vertices sequentially until it finds the target, where the next vertex can be reached from any of the previously visited vertices. That is, the time to reach the next vertex equals the shortest-path distance from the set of all previously visited vertices. The expanding search problem then asks for a sequence of the nodes, so as to minimize the expected time to finding the target. This problem has numerous applications, such as searching for hidden explosives, mining coal, and disaster relief. In this paper, we develop exact algorithms and heuristics, including a branch-and-cut procedure, a greedy algorithm with a constant-factor approximation guarantee, and a local search procedure based on a spanning-tree neighborhood. Computational experiments show that our branch-and-cut procedure outperforms existing methods for instances with nonuniform probability distributions and that both our heuristics compute near-optimal solutions with little computational effort. Summary of Contribution: This paper studies new algorithms for the expanding search problem, which asks to search a graph for a target hidden in one of the nodes according to a known probability distribution. This problem has applications such as searching for hidden explosives, mining coal, and disaster relief. We propose several new algorithms, including a branch-and-cut procedure, a greedy algorithm, and a local search procedure; and we analyze their performance both experimentally and theoretically. Our analysis shows that the algorithms improve on the performance of existing methods and establishes the first constant-factor approximation guarantee for this problem.

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On Submodular Search and Machine Scheduling

Cited by 21 publications

References 51 publications

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

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

Search and Delivery Man Problems: When are depth-first paths optimal?

Exact and Approximation Algorithms for the Expanding Search Problem

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