1,402 research outputs found
Hide-and-seek games on a network, using combinatorial search paths
This paper introduces a new search paradigm to hide-and-seek games on networks. The Hider locates at any point on any arc. The Searcher adopts a âcombinatorialâ path when searching the network: a sequence of arcs, each adjacent to the last, and traced out at unit speed. In previous literature the Searcher was allowed âsimple motion,â any unit speed path, including ones that turn around inside an arc. The new approach more closely models real problems such as search for improvised explosive devices using vehicles that can only turn around at particular locations on a road. The search game is zero sum, with the time taken to find the Hider as the payoff.
Using a lemma giving an upper bound for the expected search time on a semi Eulerian network, we solve the search game on a network Q3 consisting of two nodes connected by three arcs of arbitrary lengths. When two Q3 networks with unit length arcs are linked by two small central arcs incident at the start node, one of these arcs must be traversed at least three times in an optimal search. This property holds for both combinatorial paths and simple motion paths, and the latter makes it a counterexample to a conjecture of Gal, which said that two traversals were always sufficient
Search for an Immobile Hider on a Stochastic Network
Harry hides on an edge of a graph and does not move from there. Sally,
starting from a known origin, tries to find him as soon as she can. Harry's
goal is to be found as late as possible. At any given time, each edge of the
graph is either active or inactive, independently of the other edges, with a
known probability of being active. This situation can be modeled as a zero-sum
two-person stochastic game. We show that the game has a value and we provide
upper and lower bounds for this value. Finally, by generalizing optimal
strategies of the deterministic case, we provide more refined results for trees
and Eulerian graphs.Comment: 28 pages, 9 figure
Hide-and-seek and other search games
In the game of hide-and-seek played between two players, a Hider picks a hiding place and a Searcher tries to find him in the least possible time. Since Isaacs had the idea of formulating this mathematically as a zero-sum game almost fifty years ago in his book, Differential Games, the
theory of search games has been studied and developed extensively. In the classic model of search games on networks, first formalised by Gal in 1979, a Hider strategy is a point on the network and a Searcher strategy is a
constant speed path starting from a designated point of the network. The Searcher wishes to minimise the time to find the Hider (the payoff), and the Hider wishes to maximise it. Gal solved this game for certain classes of networks: that is, he found optimal strategies and the payoff assuming
best play on both sides. Here we study new formulations of search games, starting with a model proposed by Alpern where the speed of the Searcher depends on which direction he is traveling. We give a solution of this game on a class of networks called trees, generalising Gal's work. We also show how the game relates to another new model of search studied by Baston and Kikuta, where the Searcher must pay extra search costs to search the network's nodes (or vertices). We go on to study another new model of search called expanding search, which models coal mining. We solve
this game on trees and also study the related problem where the Hider's strategy is known to the Searcher. We extend the expanding search game to consider what happens if there are several hidden objects and solve this game for certain classes of networks. Finally we study a game in which a
squirrel hides nuts from a pilferer
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-HĂŒbner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro PezzĂ©, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
Optimal trade-off between speed and acuity when searching for a small object
A Searcher seeks to find a stationary Hider located at some point H (not necessarily a node) on a given network Q. The Searcher can move along the network from a given starting point at unit speed, but to actually find the Hider she must pass it while moving at a fixed slower speed (which may depend on the arc). In this âbimodal search game,â the payoff is the first time the Searcher passes the Hider while moving at her slow speed. This game models the search for a small or well hidden object (e.g., a contact lens, improvised explosive device, predator search for camouflaged prey). We define a bimodal Chinese postman tour as a tour of minimum time ÎŽ which traverses every point of every arc at least once in the slow mode. For trees and weakly Eulerian networks (networks containing a number of disjoint Eulerian cycles connected in a tree-like fashion) the value of the bimodal search game is ÎŽ/2. For trees, the optimal Hider strategy has full support on the network. This differs from traditional search games, where it is optimal for him to hide only at leaf nodes. We then consider the notion of a lucky Searcher who can also detect the Hider with a positive probability q even when passing him at her fast speed. This paper has particular importance for demining problems
Network Inspection for Detecting Strategic Attacks
This article studies a problem of strategic network inspection, in which a
defender (agency) is tasked with detecting the presence of multiple attacks in
the network. An inspection strategy entails monitoring the network components,
possibly in a randomized manner, using a given number of detectors. We
formulate the network inspection problem as a large-scale
bilevel optimization problem, in which the defender seeks to determine an
inspection strategy with minimum number of detectors that ensures a target
expected detection rate under worst-case attacks. We show that optimal
solutions of can be obtained from the equilibria of a
large-scale zero-sum game. Our equilibrium analysis involves both
game-theoretic and combinatorial arguments, and leads to a computationally
tractable approach to solve . Firstly, we construct an
approximate solution by utilizing solutions of minimum set cover (MSC) and
maximum set packing (MSP) problems, and evaluate its detection performance. In
fact, this construction generalizes some of the known results in network
security games. Secondly, we leverage properties of the optimal detection rate
to iteratively refine our MSC/MSP-based solution through a column generation
procedure. Computational results on benchmark water networks demonstrate the
scalability, performance, and operational feasibility of our approach. The
results indicate that utilities can achieve a high level of protection in
large-scale networks by strategically positioning a small number of detectors
Path Planning Problems with Side Observations---When Colonels Play Hide-and-Seek
International audienceResource allocation games such as the famous Colonel Blotto (CB) and Hide-and-Seek (HS) games are often used to model a large variety of practical problems, but only in their oneshot versions. Indeed, due to their extremely large strategy space, it remains an open question how one can efficiently learn in these games. In this work, we show that the online CB and HS games can be cast as path planning problems with side-observations (SOPPP): at each stage, a learner chooses a path on a directed acyclic graph and suffers the sum of losses that are adversarially assigned to the corresponding edges; and she then receives semi-bandit feedback with sideobservations (i.e., she observes the losses on the chosen edges plus some others). We propose a novel algorithm, EXP3-OE, the first-of-its-kind with guaranteed efficient running time for SOPPP without requiring any auxiliary oracle. We provide an expected-regret bound of EXP3-OE in SOPPP matching the order of the best benchmark in the literature. Moreover, we introduce additional assumptions on the observability model under which we can further improve the regret bounds of EXP3-OE. We illustrate the benefit of using EXP3-OE in SOPPP by applying it to the online CB and HS games
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