A Colored Path Problem and Its Applications

Author:

Eiben Eduard1,Kanj Iyad2

Affiliation:

1. Royal Holloway, University of London, Egham, United Kingdom

2. School of Computing, DePaul University, Chicago, IL, USA

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 , tV ( 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.

Publisher

Association for Computing Machinery (ACM)

Subject

Mathematics (miscellaneous)

Cited by 2 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Fixed-Parameter Tractability of Maximum Colored Path and Beyond;ACM Transactions on Algorithms;2024-06-28

2. Valid inequalities for the k-Color Shortest Path Problem;European Journal of Operational Research;2023-12

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