Polynomial Time Algorithms for Tracking Path Problems

Abstract

Given a graph G, and terminal vertices s and t, the Tracking Paths problem asks to compute a set of minimum number of vertices to be marked as trackers, such that the sequence of trackers encountered in each $$s$$ s -$$t$$ t path is unique. Tracking Paths is NP-hard in both directed and undirected graphs in general. In this paper we give a collection of polynomial time algorithms for some restricted versions of Tracking Paths. We prove that Tracking Paths is polynomial time solvable for undirected chordal graphs and tournament graphs. We also show that Tracking Paths is NP-hard in graphs with bounded maximum degree $$\Delta \ge 6$$ Δ ≥ 6 , and give a $$2(\Delta +1)$$ 2 ( Δ + 1 ) -approximate algorithm for this case. Further, we give a polynomial time algorithm which, given an undirected graph G, a tracking set $$T\subseteq V(G)$$ T ⊆ V ( G ) , and a sequence of trackers $$\pi $$ π , returns the unique $$s$$ s -$$t$$ t path in G that corresponds to $$\pi $$ π , if one exists. Finally we analyze the version of tracking $$s$$ s -$$t$$ t paths where paths are tracked using edges instead of vertices, and we give a polynomial time algorithm for the same.

Structural Parameterizations of Tracking Paths Problem

“…For a subgraph G , we can check if a pair of vertices a, b ∈ V (G ) forms a local source-destination pair if there exists disjoint paths from s to a and b to t in the graph G \ G ∪ {a, b}, in quadratic time using the disjoint path algorithm from [14]. The concept of local source-destination pair has been used to obtain efficient algorithms for Tracking Paths (see [3,6,10]). If u, v form a local source-destination pair for a subgraph G , we refer to them as a local s-t pair.…”

“…They also gave a linear time algorithm for bounded clique-width graphs. Recently we gave polynomial time algorithms for some restricted cases of Tracking Paths [6].…”

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

“…Related Work. Tracking has been shown to be NP-Complete [4], even when the input graph is planar [23] or has bounded degree [13]. It is fixed-parameter tractable (FPT): when parameterized by the solution size (a.k.a., the natural parameter), it admits a quadratic kernel in general and a linear kernel when the graph is planar [14] (other parameterizations have been studied in [15]).…”

“…It is fixed-parameter tractable (FPT): when parameterized by the solution size (a.k.a., the natural parameter), it admits a quadratic kernel in general and a linear kernel when the graph is planar [14] (other parameterizations have been studied in [15]). Further, it admits approximation ratios of 4 [23] for planar graphs and of 2∆ + 1 [13] for degree-∆ graphs. Exact polynomial time algorithms exist for bounded clique-width graphs [23], as well as chordal and tournament graphs [13].…”

“…Further, it admits approximation ratios of 4 [23] for planar graphs and of 2∆ + 1 [13] for degree-∆ graphs. Exact polynomial time algorithms exist for bounded clique-width graphs [23], as well as chordal and tournament graphs [13]. For the NP-hard variant of tracking only shortest paths between multiple start-finish pairs, there exists a O( √ n lg n)-approximation [8].…”