coordinated search for an object hidden on the line we consider the coordinated search problem faced by two searchers who start together at zero and can move at speed one to find an object symmetrically distributed on the line in particular we fully analyze the case of the negative exponential distribution given by the density f x e x 2 0 the searchers wish to minimize the expected time to find the object and meet back together with the object at zero we give necessary and sufficient conditions for the existence of an optimal search strategy when the target density is continuous and decreasing we show that for the negative exponential distribution the optimal time is between 4 728 and 4 729 a strategy with expected time in this interval begins with the searchers going in opposite directions and returning to the origin after searching up to successive distances 0 745 2 11 3 9 6 8 4 these results extend the theory of coordinated search to unbounded regions it has previously been studied for objects hidden on a circle by thomas and on an interval by the author

types of arcs in a fuzzy graph the concept of connectivity plays an important role in both theory and applications of fuzzy graphs depending on the strength of an arc this paper classifies arcs of a fuzzy graph into three types namely a strong b strong and d arcs the advantage of this type of classification is that it helps in understanding the basic structure of a fuzzy graph completely we analyze the relation between strong paths and strongest paths in a fuzzy graph and obtain characterizations for fuzzy bridges fuzzy trees and fuzzy cycles using the concept of a strong b strong and d arcs an arc of a fuzzy tree is a strong if and only if it is an arc of its unique maximum spanning tree also we identify different types of arcs in complete fuzzy graphs

searching a variable speed network a point lies on a network according to some unknown probability distribution starting at a specified root of the network a searcher moves to find this point at speeds that depend on his location and direction he seeks the randomized search algorithm that minimizes the expected search time this is equivalent to modeling the problem as a zero sum hide and seek game whose value is called the search value of the network we make a new and direct derivation of an explicit formula for the search value of a tree proving that it is equal to half the sum of the minimum tour time of the tree and a quantity called its incline the incline of a tree is an average over the leaf nodes of the difference between the time taken to travel from the root to a leaf node and the time taken to travel from a leaf node to the root this difference can be interpreted as height of a leaf node 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 that differ in a single arc some simple networks have very complicated optimal strategies that 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

rendezvous search on the labeled line the rendezvous search problem is the problem of finding optimal search strategies for two people who are placed randomly on a known search region and want to meet each other in minimal expected time we focus on initial location distributions that are centrally symmetric and nonincreasing as one moves away from the center including the discretized and or truncated gaussian densities when the search region is a discrete or a continuous interval and the interval is labeled so that the searchers know their own location at all times we prove that the optimal strategy for both searchers is to go directly to the center and wait there the same result also holds for rendezvous search on the infinite line

rendezvous search on the line with indistinguishable players alpern introduced a problem in which two players are placed on the real line at a distance drawn from a bounded distribution f known to both they can move at maximum velocity one and wish to meet as soon as possible neither knows the direction of the other nor do they have a common notion of a positive direction on the line it is required to find the symmetric rendezvous value r s f which is the minimum expected meeting time achievable by players using the same mixed strategy this corresponds to the case where the players are indistinguishable they both take directions from a controller who does not know their names in this paper we give a mixed strategy which has an expected meeting time of 1 78d mu 2 where d is the maximum of f and mu its mean this leads to an upper bound r s f le 1 78d mu 2 on the symmetric rendezvous value which is better than the upper bound r s f le 2d mu 2 obtained by alpern

the complexity of searching a graph t parsons originally proposed and studied the following pursuit evasion problem on graphs members of a team of searchers traverse the edges of a graph g in pursuit of a fugitive who moves along the edges of the graph with complete knowledge of the locations of the pursuers what is the smallest number s g of searchers that will suffice for guaranteeing capture of the fugitive it is shown that determining whether s g k for a given integer k is np complete for general graphs but can be solved in linear time for trees we also provide a structural characterization of those graphs g with s g k for k 1 2 3

search for an immobile hider in a known subset of a network abstract a unit speed searcher constrained to start in a given closed set s wishes to quickly find a point x known to be located in a given closed subset h of a metric network q this defines a game g g q h s where the payoff to the maximizing hider is the time for the searcher path to reach x lengths on q are defined by a measure which then defines distance as least length of connecting path for trees q we find that the minimax search time value v of g is given by v h d h s 2 where d h s is what we call the h diameter of s and equals the usual diameter d s of s in the case h q for the classical case of gal where the s is a singleton and h q our formula reduces to his result v q if s h q our formula gives dagan and gal s result v q d q 2 in all other cases our result is new optimal searches consist of minimum length paths covering h which start and end at points of s traversed equiprobably in either direction

search for an immobile entity on a network we consider the problem of searching for a single uniformly distributed immobile entity on an undirected network this problem differs from edge covering problems e g the chinese postman problem cpp since the objective here is not to find the minimum length tour that covers all the edges at least once but instead to minimize the expected time to find the entity we introduce a heuristic algorithm to deal with the search process given that the entity is equally likely to be at any point on the network computational results are presented

an evasion game on a finite tree the paper considers the following two person zero sum multistage game an evader starts at a given vertex of a tree and at discrete intervals of time chooses either to move to one of the vertices adjacent to him or stay where he is a gunner with a single bullet may at each of the same discrete intervals of time either fire the bullet at any of the vertices of the tree or hold his fire the gunner always hits the vertex at which he aims and the bullet takes one unit of time to reach its target the payoff to the gunner is 1 if he hits the evader mu where mu 1 if he fires and misses and 0 if he never fires it is shown that whatever vertex the evader starts at the value of the game is 1 mu 2

on the approximation ratio of the random chinese postman tour for network search abstract we consider a classic search problem first proposed by 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 alpern in which the searcher must return to his starting point after finding the hider

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 combinatorial 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 simple motion 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 th

rendezvous evasion search in two boxes an agent who may or may not want to be found is located in one of two boxes at time 0 suppose that he is in box b with probability p he wishes to be found in which case he has been asked to stay in box b with probability 1 p he tries to evade the searcher in which case he may move between boxes a and b the searcher looks into one of the boxes at times 1 2 3 between each search the agent may change boxes if he wants the searcher is trying to minimise the expected time to discovery the agent is trying to minimise this time if he wants to be found but trying to maximise it otherwise this paper finds a solution to this game in a sense defined in the paper associated strategies for the searcher and each type of agent and a continuous value function v p giving the expected time for the agent to be discovered the solution method which is to solve an associated zero sum game would apply generally to this type of game of incomplete information

the evolutionary game of pressure or interference resistance and collaboration in this paper we extend the framework of the evolutionary inspection game put forward recently by the author and coworkers to a large class of conflict interactions to address the pressure executed by the major player or principal on the large group of small players who can resist this pressure or collaborate with the major player we prove rigorous results on the convergence of various markov decision models of interacting small agents including evolutionary growth i e pairwise in groups and by coalition formation to a deterministic evolution on the distributions of the state spaces of small players paying main attention to situations with an infinite state space of small players we supply precise rates of convergence the theoretical results of the paper are applied to the analysis of the processes of inspection corruption cyber security counter terrorism banks and firms merging strategically enhanced preferential attachment and many other

mining coal or finding terrorists the expanding search paradigm 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 v 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 a1 a2 of the arcs of a spanning subtree such that the root node is in a1 and every arc ai is adjacent to a previous arc aj j i if ak contains h the search time t is a1 ak5 where is length measure on q for more general distributions v an expanding search s is described by the nested family of connected sets s4t5 that specify the area of q that has been covered by time t s 0 is the root s t t and t min t h 2 s t for a known hider distribution v on a tree q the expected time minimizing strategy sn begins with the rooted subtree q maximizing the density v q q for arbitrary networks we use this criterion on all spanning subtrees the search sn 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 payoff 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

rendezvous search on the line we present two new results for the asymmetric rendezvous problem on the line we first show that it is never optimal for one player to be stationary during the entire search period in the two player rendezvous then we consider the meeting time of n players in the worst case and show that it has an asymptotic behavior of n 2 o log n

search games on trees with asymmetric travel times a point h is hidden in a rooted tree q which is endowed with asymmetric distances travel times between nodes we determine the randomized search strategy starting from the root which minimizes the expected time to reach h in the worst case this is equivalent to a zero sum search game gamma left q right with minimizing searcher maximizing hider and payoff equal to the capture time the worst hiding distribution over the leaves from the searcher s viewpoint is one where at every node i the probability of each branch is proportional to the minimum time required to tour it from i the optimal randomized search is a mixture over depth first searches we also consider briefly some other networks and the possibility of a mobile hider our formulation with asymmetric travel times generalizes that of gal siam j control optim 17 1979 pp 99 122 for symmetric travel times and also the search games of kikuta j oper res 38 1995 pp 70 88 and kikuta and ruckle naval res logist 41 1994 pp 821 831 who posited search costs c i at each node i which were added to the travel time to obtain the payoff we also briefly consider what happens if we allow the searcher hider to start hide at any leaf node we determine when properties found by dagan and gal networks 52 2008 pp 156 161 for the symmetric version of such games hold in our asymmetric context

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 bimodal search game 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 i 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 i 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

a mixed strategy minimax theorem without compactness minimax theorems for infinite games generally require that both players choose their pure strategies from compact sets and have semicontinuity requirements in both variables this paper proves the following theorem let x be a compact hausdorfl space and let y a be a measurable space let f x times y be a measurable function which is bounded below and lower semicontinuous on x for all fixed y in y let m be any convex set of probability measures mixed strategies on y a then mathop min limits mu in b x mathop sup limits rho in m iint f x y d mu d rho mathop sup limits rho in m mathop min limits mu in b x iint f x y d mu d rho where b x denotes the regular borel probability measures on x

generalized best first search strategies and the optimality of a this paper reports several properties of heuristic best first search strategies whose scoring functions depend on all the information available from each candidate path not merely on the current cost g and the estimated completion cost h it is shown that several known properties of a retain their form with the minmax of f playing the role of the optimal cost which helps establish general tests of admissibility and general conditions for node expansion for these strategies on the basis of this framework the computational optimality of a in the sense of never expanding a node that can be skipped by some other algorithm having access to the same heuristic information that a uses is examined a hierarchy of four optimality types is defined and three classes of algorithms and four domains of problem instances are considered computational performances relative to these algorithms and domains are appraised for each class domain combination we then identify the strongest type of optimality that exists and the algorithm for achieving it the main results of this paper relate to the class of algorithms that like a return optimal solutions i e admissible when all cost estimates are optimistic i e h h on this class a is shown to be not optimal and it is also shown that no optimal algorithm exists but if the performance tests are confirmed to cases in which the estimates are also consistent then a is indeed optimal additionally a is also shown to be optimal over a subset of the latter class containing all best first algorithms that are guided by path dependent evaluation functions

technical note trading off quick versus slow actions in optimal search we consider the search for a target whose precise location is uncertain the search region is divided into grid cells and the searcher decides which cell to visit next and whether to search it quickly or slowly a quick search of a cell containing the target may damage it resulting in a failed search or it may locate the target safely if the target is not in the cell the search continues over the remaining cells if a slow search is performed on a cell then the search ends in failure with a fixed probability regardless of whether or not the target is in that cell e g because of enemy fire while performing the slow search if the slow search survives this failure possibility then the search ends in success if the target is in that cell otherwise the search continues over the remaining cells we seek to minimize the probability of the search ending in failure and consider two types of rules for visiting cells the unconstrained search in which the searcher may visit cells in any order and the constrained search in which the searcher may only visit adjacent cells e g up down left or right of cells already visited we prove that the optimal policy for the unconstrained search is to search quickly some initial set of cells with the lowest probabilities of containing the target before slowly searching the remaining cells in decreasing order of probabilities for the special case in which a quick search on a cell containing the target damages it with certainty the optimal policy is to search all cells slowly in decreasing order of probabilities we use the optimal solution of the unconstrained search in a branch and bound optimal solution algorithm for the constrained search for larger instances we evaluate heuristics and approximate dynamic programming approaches for finding good solutions