Fixed-parameter tractable algorithms for Tracking Shortest Paths

Banik, Aritra; Choudhary, Pratibha; Raman, Venkatesh; Saurabh, Saket

doi:10.1016/j.tcs.2020.09.006

Cited by 10 publications

(9 citation statements)

References 19 publications

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“…A lot of problems including Feedback Vertex Set and Feedback Arc Set are known to be NP-hard in tournaments [9,26]. It is known that Tracking Paths is NP-hard for directed acyclic graphs [6]. This implies that Tracking Paths is NP-hard for directed graphs as well.…”

Section: Tracking Paths In Tournamentsmentioning

confidence: 99%

Polynomial Time Algorithms for Tracking Path Problems

Choudhary

2022

Algorithmica

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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.

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Section: Tracking Paths In Tournamentsmentioning

confidence: 99%

Polynomial Time Algorithms for Tracking Path Problems

Choudhary

2022

Algorithmica

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“…There has been a lot of heuristic based work on the problem of tracking moving objects in a network [29,35,38]. Parameterized complexity of Tracking Shortest Paths and Tracking Paths was studied in [4,5,8,12,13,17]. Feedback Vertex Set is known to admit a 2-approximation algorithm which is tight under UGC [3,14].…”

Section: Theoremmentioning

confidence: 99%

Constant Factor Approximation for Tracking Paths and Fault Tolerant Feedback Vertex Set

Blažej

Choudhary

Knop

et al. 2021

Lecture Notes in Computer Science

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Consider a vertex-weighted graph G with a source s and a target t. Tracking Paths requires finding a minimum weight set of vertices (trackers) such that the sequence of trackers in each path from s to t is unique. In this work, we derive a factor 66-approximation algorithm for Tracking Paths in weighted graphs and a factor 4-approximation algorithm if the input is unweighted. This is the first constant factor approximation for this problem. While doing so, we also study approximation of the closely related r-Fault Tolerant Feedback Vertex Set problem. There, for a fixed integer r and a given vertex-weighted graph G, the task is to find a minimum weight set of vertices intersecting every cycle of G in at least $$r+1$$ r + 1 vertices. We give a factor $$\mathcal {O}(r^2)$$ O ( r 2 ) approximation algorithm for r-Fault Tolerant Feedback Vertex Set if r is a constant.

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“…There has been a lot of heuristic based work on the problem of tracking moving objects in a network [30,36,39]. Parameterized complexity of Tracking Shortest Paths and Tracking Paths was studied in [4,5,8,12,13,17]. Feedback Vertex Set is known to admit a 2-approximation algorithm which is tight under UGC [3,14].…”

Section: Tracking Pathsmentioning

confidence: 99%

Constant Factor Approximation for Tracking Paths and Fault Tolerant Feedback Vertex Set

Blažej¹,

Choudhary²,

Knop³

et al. 2021

Preprint

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Consider a vertex-weighted graph G with a source s and a target t. Tracking Paths requires finding a minimum weight set of vertices (trackers) such that the sequence of trackers in each path from s to t is unique. In this work, we derive a factor 66-approximation algorithm for Tracking Paths in weighted graphs and a factor 4-approximation algorithm if the input is unweighted. This is the first constant factor approximation for this problem. While doing so, we also study approximation of the closely related r-Fault Tolerant Feedback Vertex Set problem. There, for a fixed integer r and a given vertex-weighted graph G, the task is to find a minimum weight set of vertices intersecting every cycle of G in at least r + 1 vertices. We give a factor O(r 2 ) approximation algorithm for r-Fault Tolerant Feedback Vertex Set if r is a constant.⋆ The authors acknowledge the support of the OP VVV MEYS funded project CZ.

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Fixed-parameter tractable algorithms for Tracking Shortest Paths

Cited by 10 publications

References 19 publications

Polynomial Time Algorithms for Tracking Path Problems

Polynomial Time Algorithms for Tracking Path Problems

Constant Factor Approximation for Tracking Paths and Fault Tolerant Feedback Vertex Set

Constant Factor Approximation for Tracking Paths and Fault Tolerant Feedback Vertex Set

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