The expanding search ratio of a graph

Abstract

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.

Competitive search in a network

“…For normalized objectives (Angelopoulos et al, 2019) recently studied expanding search in a fixed (finite) graph in which the Hider can only hide on vertices. In terms of finding a strategy of optimal deterministic competitive ratio (Angelopoulos et al, 2019) showed that the problem is NP-hard, and gave a 4 ln 4 approximation. Concerning the randomized competitive ratio, the same work presented a strategy that is a 5/4-approximation in the special case of tree graphs.…”

“…In this section we give a search strategy that is a 5/4-approximation of the optimal randomized search. This is inspired by the strategy of (Angelopoulos et al, 2019) for the discrete case, namely for searching in a given graph when the Hider can only hide at a vertex.…”

“…This class of strategies is based on iterative applications of Randomized Depth-First-Search, in randomly chosen and increasingly large subsets of the network. This strategy is inspired by a randomized strategy used for tree graphs in the discrete setting, namely when the Hider can only hide over vertices of a given, finite tree (Angelopoulos et al, 2019). We emphasize that, unlike (Angelopoulos et al, 2019), in this work, the search domain may be substantially more complex than a tree, and it may also be unbounded.…”

“…This theorem appeared without proof as Theorem 6 of(Angelopoulos, Dürr, & Lidbetter, 2016). 3 This proposition appeared without proof as Proposition 7 of(Angelopoulos, Dürr, & Lidbetter, 2016).…”

Searching for multiple objects in multiple locations

“…This type of payoff function was introduced in search theory independently by Beck (1964) and Bellman (1963) in the context of the linear search problem, later extended by Gal (1974). The linear search problem is framed in a continuous setting, but multiplicative regret for search theory has also been studied in a discrete setting in Koutsoupias et al (1996) and Angelopoulos et al (2016).…”