The expanding search ratio of a graph

TL;DR

This paper investigates the problem of optimizing expanding search strategies in graphs to minimize the maximum ratio of search time to shortest-path distance, addressing algorithmic challenges and providing solutions for various graph classes.

Contribution

It introduces the problem of minimizing the search ratio in expanding searches, analyzes its computational complexity, and offers algorithms and approximations for different graph types.

Findings

Optimal randomized search strategies are identified for certain graph classes.

Constant-factor approximation algorithms are developed for general graphs.

The problem is formulated as a zero-sum game between the Searcher and Hider.

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 Apern 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 shortest-path 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.