Pursuit on a Graph under Partial Information from Sensors

TL;DR

This paper studies pursuit-evasion on directed acyclic graphs with sensor-based partial information, proving NP-hardness of key problems and proposing a linear-time algorithm for a specific class of policies on trees.

Contribution

It establishes NP-hardness results for pursuit under partial information and introduces a linear-time algorithm for maximum delay computation in certain policies on trees.

Findings

NP-hardness of pursuit delay problems even on trees

Approximation of maximum pursuer delay is NP-hard

Linear-time algorithm for node-sweeping policies on bounded-degree trees

Abstract

We consider a class of pursuit-evasion problems where an evader enters a directed acyclic graph and attempts to reach one of the terminal nodes. A pursuer enters the graph at a later time and attempts to capture the evader before it reaches a terminal node. The pursuer can only obtain information about the evader's path via sensors located at each node in the graph; the sensor measurements are either green or red (indicating whether or not the evader has passed through that node). We first show that it is NP-hard to determine whether the pursuer can enter with some nonzero delay and still be guaranteed to capture the evader, even for the simplest case when the underlying graph is a tree. This also implies that it is NP-hard to determine the largest delay at which the pursuer can enter and still have a guaranteed capture policy. We further show that it is NP-hard to approximate (within any constant factor) the largest delay at which the pursuer can enter. Finally, we provide an algorithm to compute the maximum pursuer delay for a class of node-sweeping policies on tree networks and show that this algorithm runs in linear-time for bounded-degree trees.