Markovian Network Interdiction and the Four Color Theorem

TL;DR

This paper proves that the problem of optimally placing sensors to detect Markovian evaders on a network is NP-hard even with only two evaders, linking it to graph coloring and highlighting the need for approximation algorithms.

Contribution

It establishes NP-hardness for the sensor placement problem with two evaders using a connection to the Four Color Theorem, extending previous results.

Findings

NP-hardness with 2 evaders proven

Connection to planar graph coloring established

Open problem for 1-evader case remains

Abstract

The Unreactive Markovian Evader Interdiction Problem (UME) asks to optimally place sensors on a network to detect Markovian motion by one or more "evaders". It was previously proved that finding the optimal sensor placement is NP-hard if the number of evaders is unbounded. Here we show that the problem is NP-hard with just 2 evaders using a connection to coloring of planar graphs. The results suggest that approximation algorithms are needed even in applications where the number of evaders is small. It remains an open problem to determine the complexity of the 1-evader case or to devise efficient algorithms.