Exact and approximation algorithms for the expanding search problem

Abstract

Suppose a target is hidden in one of the vertices of an edge-weighted graph according to a known probability distribution. The expanding search problem asks for a search sequence of the vertices so as to minimize the expected time for finding the target, where the time for reaching the next vertex is determined by its distance to the region that was already searched. This problem has numerous applications, such as searching for hidden explosives, mining coal, and disaster relief. In this paper, we develop exact algorithms and heuristics, including a branch-and-cut procedure, a greedy algorithm with a constant-factor approximation guarantee, and a novel local search procedure based on a spanning tree neighborhood. Computational experiments show that our branch-and-cut procedure outperforms all existing methods for general instances and both heuristics compute near-optimal solutions with little computational effort.

Minimizing the total weighted pairwise connection time in network construction problems

“…However, FNCP is polynomially solvable on trees [3]; in contrast, as we show below, Problem A is strongly NP-hard even on so special case of trees as stars. There is an interesting connection between FNCP and the search theory: FNCP is equivalent to the expanding search problem introduced in Alpern and Lidbetter [2] and further studied in Hermans et al [10]. Hence, Problem A is a generalization of the expanding search problem from [2,10] as well.…”

“…There is an interesting connection between FNCP and the search theory: FNCP is equivalent to the expanding search problem introduced in Alpern and Lidbetter [2] and further studied in Hermans et al [10]. Hence, Problem A is a generalization of the expanding search problem from [2,10] as well.…”

“…An important special case is where all r-pairs have a common vertex; as mentioned in Section 2, for the total weighted connection time objective this special case is equivalent to the FNCP from [3] or the expanding search problem from [2,10]. In this case, there are no more than r −1 connection vertices in an r-forest (which in fact will be a tree); hence, the algorithm outlined above will have complexity O(n r−1 ).…”

“…The same setting but with a different objective (minimizing the maximum lateness of connection times with respect to a given set of due dates) was considered in [4]. As we discuss in Section 2, a problem which mathematically is equivalent to a special case of our problem was studied in [3,2,10].…”

“…Third, for the problem on a general network with a fixed number r of relevant pairs (vertex pairs with non-zero weights whose connection times are relevant for the objective function), we develop a polynomial O(n 2r−2 ) exact algorithm with an O(n 3 ) pre-processing. For the important special case of all relevant pairs having a common vertex (which, as we show in Section 2, is equivalent to the problem studied in [3], [2] and [10]), the complexity is reduced to O(n r−1 ) with O(n 3 ) pre-processing. The results for the problem with a fixed number of relevant pairs are applicable to other objectives that are monotonically non-decreasing functions of the connection times of the relevant pairs, e.g.…”