Polynomial Kernels for Tracking Shortest Paths

TL;DR

This paper introduces the first polynomial kernel for the Tracking Shortest Paths problem, providing efficient preprocessing bounds and an exponential algorithm for planar graphs.

Contribution

It establishes polynomial kernels for Tracking Shortest Paths and related problems, improving computational efficiency and offering new algorithmic insights.

Findings

Polynomial kernel with O(k^4) vertices for general graphs

Kernel with O(k^2) vertices for planar graphs

Single exponential algorithm for planar graphs

Abstract

Given an undirected graph $G=(V,E)$, vertices $s,t\in V$, and an integer $k$, Tracking Shortest Paths requires deciding whether there exists a set of $k$ vertices $T\subseteq V$ such that for any two distinct shortest paths between $s$ and $t$, say $P_1$ and $P_2$, we have $T\cap V(P_1)\neq T\cap V(P_2)$. In this paper, we give the first polynomial size kernel for the problem. Specifically we show the existence of a kernel with $\mathcal{O}(k^2)$ vertices and edges in general graphs and a kernel with $\mathcal{O}(k)$ vertices and edges in planar graphs for the Tracking Paths in DAG problem. This problem admits a polynomial parameter transformation to Tracking Shortest Paths, and this implies a kernel with $\mathcal{O}(k^4)$ vertices and edges for Tracking Shortest Paths in general graphs and a kernel with $\mathcal{O}(k^2)$ vertices and edges in planar graphs. Based on the above we also give a single exponential algorithm for Tracking Shortest Paths in planar graphs.