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. Starting from a fixed root node, an expanding search visits the vertices sequentially until it finds the target, where the next vertex can be reached from any of the previously visited vertices. That is, the time to reach the next vertex equals the shortest-path distance from the set of all previously visited vertices. The expanding search problem then asks for a sequence of the nodes, so as to minimize the expected time to finding the target. 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 local search procedure based on a spanning-tree neighborhood. Computational experiments show that our branch-and-cut procedure outperforms existing methods for instances with nonuniform probability distributions and that both our heuristics compute near-optimal solutions with little computational effort. Summary of Contribution: This paper studies new algorithms for the expanding search problem, which asks to search a graph for a target hidden in one of the nodes according to a known probability distribution. This problem has applications such as searching for hidden explosives, mining coal, and disaster relief. We propose several new algorithms, including a branch-and-cut procedure, a greedy algorithm, and a local search procedure; and we analyze their performance both experimentally and theoretically. Our analysis shows that the algorithms improve on the performance of existing methods and establishes the first constant-factor approximation guarantee for this problem.

Competitive search in a network

“…It captures settings in which the cost of re-exploration is negligible (e.g., in applications such as minesweeping and coal mining). Expanding search, in trees and general networks, has been studied in several works, e.g., (Angelopoulos et al, 2019, Angelopoulos and Lidbetter, 2020, Hermans et al, 2022, Alpern and Lidbetter, 2019, Griesbach et al, 2023; in this paper, we enhance the search with a prediction that describes, roughly speaking, a connected substree in which the Searcher is expected to hide.…”

Hiring Secretaries over Time: The Benefit of Concurrent Employment

“…Proof. Hermans et al [27] argue that maximum density subtrees can be computed in polynomial time by the dynamic program given in [4] when the underlying graph is a tree. Hence, the density-greedy algorithm can be implemented in polynomial time on trees.…”

“…As the computation of a subtree of maximum density is NP-hard (Lau et al [33]), there is no polynomial implementation of this step, unless P = NP. However, Hermans et al [27] give a polynomial 2-approximation for the computation of a subtree with maximal density. In order to obtain a polynomial algorithm for the incremental prize-collecting Steiner-tree problem on general graphs, we need to do the following adaptations to Algorithm 2:…”

“…1. In Step 4, we use the approximation algorithm of Hermans et al [27] to obtain a tree T * of G/T with density d G/T (T * ) ≥ d * /2 where d * is the maximum density of a rooted subtree in G/T .…”