How to Catch Marathon Cheaters: New Approximation Algorithms for Tracking Paths

TL;DR

This paper introduces new approximation algorithms for the NP-complete tracking paths problem in graphs, with applications in animal migration and marathon course-cutting detection, providing improved ratios and kernels for various graph classes.

Contribution

The paper presents novel approximation algorithms with specific ratios for different graph classes and improves kernelization techniques for the tracking paths problem.

Findings

Approximation ratios of (1+ε), O(log OPT), and O(log n) for different graph classes.

A linear kernel for H-minor-free graphs.

Improvements to the quadratic kernel for general graphs.

Abstract

Given an undirected graph, $G$, and vertices, $s$ and $t$ in $G$, the tracking paths problem is that of finding the smallest subset of vertices in $G$ whose intersection with any $s$-$t$ path results in a unique sequence. This problem is known to be NP-complete and has applications to animal migration tracking and detecting marathon course-cutting, but its approximability is largely unknown. In this paper, we address this latter issue, giving novel algorithms having approximation ratios of $(1+\epsilon)$, $O(\lg OPT)$ and $O(\lg n)$, for $H$-minor-free, general, and weighted graphs, respectively. We also give a linear kernel for $H$-minor-free graphs and make improvements to the quadratic kernel for general graphs.