We show how to optimize the search for a hidden object, terrorist, or simply Hider, located at a point H according to a known or unknown distribution on a rooted network Q. We modify the traditional 'pathwise search' approach to a more general notion of 'expanding search'. When the Hider is restricted to the nodes of Q; an expanding search S consists of an ordering (a 1 ; a 2 ; : : : ) of the arcs of a spanning subtree such that the root node is in a 1 and every arc a i is adjacent to a previous arc a j ; j < i: If a k contains H; the search time T is (a 1 ) + + (a k ) ; where is length measure on Q: For more general distributions ; an expanding search S is described by the nested family of connected sets S (t) which specify the area of Q that has been covered by time t: S (0) is the root, (S (t)) = t; and T = min ft : H 2 S (t)g : For a known Hider distribution on a tree Q; the expected time minimizing strategy S begins with the rooted subtree Q 0 maximizing the 'density' (Q 0 ) = (Q 0 ) : (For arbitrary networks, we use this criterion on all spanning subtrees.) The search S can be interpreted as the optimal method of mining known coal seams, when the time to move miners or machines is negligible compared to digging time. When the Hider distribution is unknown, we consider the zero-sum search game where the Hider picks H; the Searcher S and the payo¤ is T: For trees Q; the value is V = ( (Q) + D) =2, where D is a mean distance from root to leaf nodes. If Q is 2 arc connected, V = (Q) =2. Applications and interpretations of the expanding search paradigm are given, particularly to multiple agent search.

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.

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 ⌈ √ n⌉, 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 3 2 -approximation algorithm for computing the burning number of path-forests.1991 Mathematics Subject Classification. 05C05,05C85.

Scatter hoarders are animals (e.g. squirrels) who cache food (nuts) over a number of sites for later collection. A certain minimum amount of food must be recovered, possibly after pilfering by another animal, in order to survive the winter. An optimal caching strategy is one that maximizes the survival probability, given worst case behaviour of the pilferer. We modify certain 'accumulation games' studied by Kikuta & Ruckle (2000 J. Optim. Theory Appl.) and Kikuta & Ruckle (2001 Naval Res. Logist.), which modelled the problem of optimal diversification of resources against catastrophic loss, to include the depth at which the food is hidden at each caching site. Optimal caching strategies can then be determined as equilibria in a new 'caching game'. We show how the distribution of food over sites and the site-depths of the optimal caching varies with the animal's survival requirements and the amount of pilfering. We show that in some cases, 'decoy nuts' are required to be placed above other nuts that are buried further down at the same site. Methods from the field of search games are used. Some empirically observed behaviour can be shown to be optimal in our model.

Abstract. We consider a class of zero-sum search games in which a Searcher seeks to minimize the expected time to find several objects hidden by a Hider. We begin by analyzing a game in which the Searcher wishes to find k balls hidden among n > k boxes. There is a known cost of searching each box, and the Searcher seeks to minimize the total expected cost of finding all the objects in the worst case. We show that it is optimal for the Searcher to begin by searching a k-subset H of boxes with probability ν(H), which is proportional to the product of the search costs of the boxes in H. The Searcher should then search the n − k remaining boxes in a random order. A worst-case Hider distribution is the distribution ν. We distinguish between the case of a smart Searcher who can change his search plan as he goes along and a normal Searcher who has to set out his plan from the beginning. We show that a smart Searcher has no advantage. We then show how the game can be formulated in terms of a more general network search game, and we give upper and lower bounds for the value of the game on an arbitrary network. For 2-arc connected networks (networks that cannot be disconnected by the removal of fewer than two arcs), we solve the game for a smart Searcher and give an upper bound on the value for a normal Searcher. This bound is tight if the network is a circle. 1. Introduction. The first problem we consider is that of a Searcher who wishes to find a number of objects (or balls) hidden among a set of discrete locations (or boxes), each of which has a designated search cost. The Searcher looks in the boxes one by one, paying the search costs associated with the boxes he looks in, until he has found all the balls. He wishes to minimize the total search cost of finding the balls in the worst case, so we view the problem as a zero-sum game between the Searcher and a malevolent Hider who wishes to maximize the total search cost. This is a natural problem to consider and one which we face on an everyday basis. For example, before leaving the house in the morning we may wish to locate certain essential items such as wallet, phone, and keys. There is a set of discrete locations around the house where these objects may be hidden, each of which takes a particular amount of time to search, and we wish to minimize the total time it takes to find all the items. The problem provides a simple model for other practical search scenarios, such as a search for a number of corrupted files which may be distributed among several folders, or a search for bombs hidden in several locations. The problem is also relevant to studies such as [23], which have examined how scatter hoarders like squirrels search for food they have previously cached over a number of sites.In section 2.1, after we have formally defined the problem, we will see that if both parties are allowed to use randomized strategies it is optimal for the Searcher to begin his search with a subset of k boxes chosen with probability proportional to the product of their search costs and t...

Patrolling games were recently introduced by Alpern, Morton and Papadaki to model the problem of protecting the nodes of a network from an attack. Time is discrete and in each time unit the Patroller can stay at the same node or move to an adjacent node. The Attacker chooses when to attack and which node to attack, and needs m consecutive time units to carry it out. The Attacker wins if the Patroller does not visit the chosen node while it is being attacked; otherwise the Patroller wins. This paper studies the patrolling game where the network is a line graph of n nodes, which models the problem of guarding a channel or protecting a border from infiltration. We solve the patrolling game for any values of m and n, providing an optimal Patroller strategy, an optimal Attacker strategy and the value of the game (optimal probability that the attack is intercepted).

A point H lies on a network Q according to some unknown distribution. A Searcher starts at a given point O of Q and moves to …nd H at speeds which depend on his location and direction. He seeks the randomized search algorithm which minimizes the expected search time. This is equivalent to modeling the problem as a zero-sum hide-and-seek game whose value V is called the search value of (Q; O).We make a new and direct derivation of an explicit formula V = (1=2) ( + ) for the search value of a tree, where is the minimum tour time of Q and (called the incline of Q) is an average over the leaf nodes i of the di¤er-; where d (x; y) is the time to go from x to y: The function can be interpreted as height, assuming uphill is slower than downhill. We then apply this formula to obtain numerous results for general networks. We also introduce a new general method of comparing the search value of networks which di¤er in a single arc. Some simple networks have very complicated optimal strategies which require mixing of a continuum of pure strategies. Many of our results generalize analogous ones obtained for constant velocity (in both directions) by S. Gal, but not all of those results can be extended.

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 Alpern 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 shortestpath 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.