Constant Factor Approximate Solutions for Expanding Search on General Networks

TL;DR

This paper develops near-optimal expanding search strategies for locating a hidden point on general networks, extending previous work to more network types and providing strategies with expected search times close to the theoretical minimum.

Contribution

It introduces strategy classes applicable to any network that achieve search times within a small factor of the optimal, expanding the scope of previous results.

Findings

Strategies with expected search times close to the minimax value.

Extension of the search problem solution to additional network families.

Identification of cases where strategies are proven optimal.

Abstract

We study the classical problem introduced by R. Isaacs and S. Gal of minimizing the time to find a hidden point $H$ on a network $Q$ moving from a known starting point. Rather than adopting the traditional continuous unit speed path paradigm, we use the ``expanding search'' paradigm recently introduced by the authors. Here the regions $S\left( t\right) $ that have been searched by time $t$ are increasing from the starting point and have total length $t$. Roughly speaking the search follows a sequence of arcs $a_{i}$ such that each one starts at some point of an earlier one. This type of search is often carried out by real life search teams in the hunt for missing persons, escaped convicts, terrorists or lost airplanes. The paper which introduced this type of search solved the adversarial problem (where $H$ is hidden to take a long time to find) for the cases where $Q$ is a tree or is 2-arc-connected. This paper solves the game on some additional families of networks. However the main contribution is to give strategy classes which can be used on any network and have expected search times which are within a factor close to 1 of the value of the game (minimax search time). We identify cases where our strategies are in fact optimal.