research-article

A Colored Path Problem and Its Applications

Authors:

Iyad KanjAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 16, Issue 4

Article No.: 47, Pages 1 - 48

https://doi.org/10.1145/3396573

Published: 21 June 2020 Publication History

Abstract

Given a set of obstacles and two points in the plane, is there a path between the two points that does not cross more than k different obstacles? Equivalently, can we remove k obstacles so that there is an obstacle-free path between the two designated points? This is a fundamental NP-hard problem that has undergone a tremendous amount of research work. The problem can be formulated and generalized into the following graph problem: Given a planar graph G whose vertices are colored by color sets, two designated vertices s, t ∈ V(G), and k ∈ N, is there an s-t path in G that uses at most k colors? If each obstacle is connected, then the resulting graph satisfies the color-connectivity property, namely that each color induces a connected subgraph.

We study the complexity and design algorithms for the above graph problem with an eye on its geometric applications. We prove a set of hardness results, including a result showing that the color-connectivity property is crucial for any hope for fixed-parameter tractable (FPT) algorithms. We also show that our hardness results translate to the geometric instances of the problem.

We then focus on graphs satisfying the color-connectivity property. We design an FPT algorithm for this problem parameterized by both k and the treewidth of the graph and extend this result further to obtain an FPT algorithm for the parameterization by both k and the length of the path. The latter result implies and explains previous FPT results for various obstacle shapes.

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Cited By

de Andrade RCastelo ESaraiva R(2023)Valid inequalities for the k-Color Shortest Path ProblemEuropean Journal of Operational Research10.1016/j.ejor.2023.12.014Online publication date: Dec-2023
https://doi.org/10.1016/j.ejor.2023.12.014
Kumar NLokshtanov DSaurabh SSuri SMarx D(2021)A constant factor approximation for navigating through connected obstacles in the planeProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458116(822-839)Online publication date: 10-Jan-2021
https://dl.acm.org/doi/10.5555/3458064.3458116

Index Terms

A Colored Path Problem and Its Applications
1. Theory of computation
  1. Design and analysis of algorithms
    1. Graph algorithms analysis
    2. Parameterized complexity and exact algorithms
      1. Fixed parameter tractability
      2. W hierarchy

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 16, Issue 4

October 2020

404 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3407674

Editor:
Aravind Srinivasan
University of Maryland, USA

Issue’s Table of Contents

Copyright © 2020 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 21 June 2020

Online AM: 07 May 2020

Accepted: 01 April 2020

Revised: 01 October 2019

Received: 01 January 2018

Published in TALG Volume 16, Issue 4

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Cited By

de Andrade RCastelo ESaraiva R(2023)Valid inequalities for the k-Color Shortest Path ProblemEuropean Journal of Operational Research10.1016/j.ejor.2023.12.014Online publication date: Dec-2023
https://doi.org/10.1016/j.ejor.2023.12.014
Kumar NLokshtanov DSaurabh SSuri SMarx D(2021)A constant factor approximation for navigating through connected obstacles in the planeProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458116(822-839)Online publication date: 10-Jan-2021
https://dl.acm.org/doi/10.5555/3458064.3458116

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