Branch and bound methods for a search problem

Abstract

The problem of searching for randomly moving targets such as children and submarines is known to be fundamentally difficult, but finding efficient methods for generating optimal or near optimal solutions is nonetheless an important practical problem. This paper investigates the efficiency of Branch and Bound methods, with emphasis on the tradeoff between the accuracy of the bound employed and the time required to compute it. A variety of bounds are investigated, some of which are new. In most cases the best bounds turn out to be imprecise, but very easy to compute.

Discounted MEAN bound for the optimal searcher path problem with non-uniform travel times

“…Washburn (1995Washburn ( , 1998) details a number of known bounding techniques and examines the trade off between bound tightness and calculation speed. The methods reviewed will be referred to by the same names in this paper for consistency.…”

“…Eschewing sharpness for calculation speed, Martins (1993)'s MEAN method made linear relaxations through transforming the OSP into a longest path problem maximising the expected number of detections that preserves both searcher indivisibility and path constraints. When the path constraints are relaxed to form a reward collection problem, the more easily evaluated bound PROP (Washburn, 1998) is obtained. Lastly, ERGO2 (Washburn, 1998) estimates bounds with even less computation by directly using a stationary target distribution rather than the actual distribution at each time.…”

“…When the path constraints are relaxed to form a reward collection problem, the more easily evaluated bound PROP (Washburn, 1998) is obtained. Lastly, ERGO2 (Washburn, 1998) estimates bounds with even less computation by directly using a stationary target distribution rather than the actual distribution at each time.…”

“…A search plan respecting searcher travel constraints is sought to maximise the probability of detection within a specified time frame. Much work has been done in the past on this path constrained or optimal searcher path problem (OSP) (Stewart, 1979;Eagle and Yee, 1990;Dell et al 1996;Hohzaki, 1997;Washburn 1998), for the specific case where the environment is discretised into a grid of identically-sized cells and a searcher can immediately search one cell after another in successive time periods. In contrast, a platform searching a built environment cannot move from one region to another without expending some travel time for each transition.…”

Search and pursuit-evasion in mobile robotics

“…Relation to the vehicle routing problem (e.g., see Lenstra and Kan (1981); Toth and Vigo (2002)) in operations research offers numerous approximation algorithms, often relying on customized branch-and-bound approaches (Washburn, 1998). These works formulate optimization objective functions with relevance to both types of search performance measures, namely maximal detection probability and minimal (expected) detection time.…”