A Parameterized Approximation Algorithm for the Chromatic k-Median Problem

Chromatic <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>-median is a frequently encountered problem in the determination of the topological structures of chromosomes. This problem considers a set <inline-formula> <tex-math notati...

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Bibliographic Details
Published in:IEEE access Vol. 9; pp. 31678 - 31683
Main Authors: Zhang, Zhen, Zhang, Jinchuan, Zhu, Lingzhi
Format: Journal Article
Language:English
Published: IEEE 2021
Subjects:
Approximation
Approximation algorithms
Cells (biology)
clustering
Clustering algorithms
Color
Extraterrestrial measurements
fixed-parameter tractability
ISSN:2169-3536, 2169-3536
Online Access:Get full text
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Summary:Chromatic <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>-median is a frequently encountered problem in the determination of the topological structures of chromosomes. This problem considers a set <inline-formula> <tex-math notation="LaTeX">\mathcal {C} </tex-math></inline-formula> of colored clients and a set <inline-formula> <tex-math notation="LaTeX">\mathcal {F} </tex-math></inline-formula> of facilities located in a metric space, where <inline-formula> <tex-math notation="LaTeX">|\mathcal {C}\cup \mathcal {F}|=n </tex-math></inline-formula>. The goal is to open <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> facilities and assign each client to an opened facility, such that clients with the same color are assigned to different facilities and the sum of the distance from each client to the corresponding facility is minimized. It was known that the chromatic <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>-median problem is W[2]-hard if parameterized by <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>. This rules out the probability of obtaining an exact FPT(<inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>)-time algorithm for the problem. In this paper, we give an FPT(<inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>)-time approximation algorithm for chromatic <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>-median. The algorithm achieves a <inline-formula> <tex-math notation="LaTeX">(3+\epsilon) </tex-math></inline-formula>-approximation and runs in <inline-formula> <tex-math notation="LaTeX">(k\epsilon ^{-1})^{O(k)}n^{O(1)} </tex-math></inline-formula> time. We propose a different random sampling algorithm for opening facilities, which is the crucial step in getting the constant factor parameterized approximation.
ISSN:2169-3536
2169-3536
DOI:10.1109/ACCESS.2021.3060422