arXiv subjects

Thomas Lidbetter

Publications and source records attributed to Thomas Lidbetter.

28 records · Page 2Linked to original sources

arXiv2019

Competitive Search in a Network

We study the classic problem in which a Searcher must locate a hidden point, also called the Hider in a network, starting from a root point. The network may be either bounded or unbounded, thus generalizing well-known settings such as linear and star search. We distinguish between pathwise search, in which the Searcher follows a continuous unit-speed path until the Hider is reached, and expanding search, in which, at any point in time, the Searcher may restart from any previously reached point. The former has been the usual paradigm for studying search games, whereas the latter is a more recent paradigm that can model real-life settings such as hunting for a fugitive, demining a field, or search-and-rescue operations. We seek both deterministic and randomized search strategies that minimize the competitive ratio, namely the worst-case ratio of the Hider's discovery time, divided by the shortest path to it from the root. Concerning expanding search, we show that a simple search strategy that applies a "waterfilling" principle has optimal deterministic competitive ratio; in contrast, we show that the optimal randomized competitive ratio is attained by fairly complex strategies even in a very simple network of three arcs. Motivated by this observation, we present and analyze an expanding search strategy that is a 5/4 approximation of the randomized competitive ratio. Our approach is also applicable to pathwise search, for which we give a strategy that is a 5 approximation of the randomized competitive ratio, and which improves upon strategies derived from previous work.

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arXiv2018

Solving Zero-sum Games using Best Response Oracles with Applications to Search Games

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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arXiv2018

On Submodular Search and Machine Scheduling

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 $1|prec|\sum w_A h(C_A)$, 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 {\em subsets} of jobs. We go on to give better approximations for submodular functions with low {\em total curvature} and we give a full solution when the problem is what we call {\em 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 series-parallel decomposable case, giving approximately optimal strategies in other cases.

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arXiv2017

Searching for Multiple Objects in Multiple Locations

Many practical search problems concern the search for multiple hidden objects or agents, such as earthquake survivors. In such problems, knowing only the list of possible locations, the Searcher needs to find all the hidden objects by visiting these locations one by one. To study this problem, we formulate new game-theoretic models of discrete search between a Hider and a Searcher. The Hider hides $k$ balls in $n$ boxes, and the Searcher opens the boxes one by one with the aim of finding all the balls. Every time the Searcher opens a box she must pay its search cost, and she either finds one of the balls it contains or learns that it is empty. If the Hider is an adversary, an appropriate payoff function may be the expected total search cost paid to find all the balls, while if the Hider is Nature, a more appropriate payoff function may be the difference between the total amount paid and the amount the Searcher would have to pay if she knew the locations of the balls a priori (the regret). We give a full solution to the regret version of this game, and a partial solution to the search cost version. We also consider variations on these games for which the Hider can hide at most one ball in each box. The search cost version of this game has already been solved in previous work, and we give a partial solution in the regret version.

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arXiv2017

The expanding search ratio of a graph

We study the problem of searching for a hidden target in an environment that is modeled by an edge-weighted graph. A sequence of edges is chosen starting from a given root vertex such that each edge is adjacent to a previously chosen edge. This search paradigm, known as expanding search was recently introduced by Apern and Lidbetter [2013] for modeling problems such as searching for coal or minesweeping in which the cost of re-exploration is negligible. It can also be used to model a team of searchers successively splitting up in the search for a hidden adversary or explosive device, for example. We define the search ratio of an expanding search as the maximum over all vertices of the ratio of the time taken to reach the vertex and the shortest-path cost to it from the root. This can be interpreted as a measure of the multiplicative regret incurred in searching, and similar objectives have previously been studied in the context of conventional (pathwise) search. In this paper we address algorithmic and computational issues of minimizing the search ratio over all expanding searches, for a variety of search environments, including general graphs, trees and star-like graphs. Our main results focus on the problem of finding the randomized expanding search with minimum expected search ratio, which is equivalent to solving a zero-sum game between a Searcher and a Hider. We solve these problems for certain classes of graphs, and obtain constant-factor approximations for others.

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arXiv2017

Bounds on the burning numbers of spiders and path-forests

Graph burning is one model for the spread of memes and contagion in social networks. The corresponding graph parameter is the burning number of a graph $G$, written $b(G)$, which measures the speed of the social contagion. While it is conjectured that the burning number of a connected graph of order $n$ is at most $\lceil \sqrt{n} \rceil$, this remains open in general and in many graph families. We prove the conjectured bound for spider graphs, which are trees with exactly one vertex of degree at least 3. To prove our result for spiders, we develop new bounds on the burning number for path-forests, which in turn leads to a $\frac 3 2$-approximation algorithm for computing the burning number of path-forests.

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arXiv2017

Periodic Patrols on the Line and Other Networks

We consider a patrolling game on a graph recently introduced by Alpern et al. (2011) where the Patroller wins if he is at the attacked node while the attack is taking place. This paper studies the periodic patrolling game in the case that the attack duration is two periods. We show that if the Patroller's period is even, the game can be solved on any graph by finding the fractional covering number and fractional independence number of the graph. We also give a complete solution to the periodic patrolling game on line graphs of arbitrary size, extending the work of Papadaki et al. (2016) to the periodic domain. This models the patrolling problem on a border or channel, which is related to a classical problem of operational research going back to Morse and Kimball (1951). A periodic patrol is required to start and end at the same location, for example the place where the Patroller leaves his car to begin a foot patrol.

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arXiv2017

On the Approximation Ratio of the Random Chinese Postman Tour for Network Search

We consider a classic search problem first proposed by S. 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 S. Alpern in which the Searcher must return to his starting point after finding the Hider.

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arXiv2016

Constant Factor Approximate Solutions for Expanding Search on General Networks

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\left( t\right) $ 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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arXiv2015

The solution to an open problem for a caching game

In a caching game introduced by Alpern et al., a Hider who can dig to a total fixed depth normalized to $1$ buries a fixed number of objects among $n$ discrete locations. A Searcher who can dig to a total depth of $h$ searches the locations with the aim of finding all of the hidden objects. If he does so, he wins, otherwise the Hider wins. This zero-sum game is complicated to analyze even for small values of its parameters, and for the case of $2$ hidden objects has been completely solved only when the game is played in up to $3$ locations. For some values of $h$ the solution of the game with $2$ objects hidden in $4$ locations is known, but the solution in the remaining cases was an open question recently highlighted by Fokkink et al. Here we solve the remaining cases of the game with $2$ objects hidden in $4$ locations. We also give some more general results for the game, in particular using a geometrical argument to show that when there are $2$ objects hidden in $n$ locations and $n \rightarrow \infty$, the value of the game is asymptotically equal to $h/n$ for $h \ge n/2$.

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