Parameterized and Approximation Algorithms for the Load Coloring Problem

Let c ,  k be two positive integers. Given a graph G = ( V , E ) , the c - Load Coloring problem asks whether there is a c -coloring φ : V → [ c ] such that for every i ∈ [ c ] , there are at least k edges with both endvertices colored i . Gutin and Jones (Inf Process Lett 114:446–449, 2014 ) studie...

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Bibliographic Details
Published in:Algorithmica Vol. 79; no. 1; pp. 211 - 229
Main Authors: Barbero, F., Gutin, G., Jones, M., Sheng, B.
Format: Journal Article
Language:English
Published: New York Springer US 01.09.2017
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Approximation
Coloring
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Decision trees
Graph theory
Integers
Mathematical analysis
Mathematics of Computing
Theory of Computation
Kernels
Approximation algorithms
Parameterized algorithms
FPT
Load coloring
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:Let c ,  k be two positive integers. Given a graph G = ( V , E ) , the c - Load Coloring problem asks whether there is a c -coloring φ : V → [ c ] such that for every i ∈ [ c ] , there are at least k edges with both endvertices colored i . Gutin and Jones (Inf Process Lett 114:446–449, 2014 ) studied this problem with c = 2 . They showed 2- Load Coloring to be fixed-parameter tractable (FPT) with parameter k by obtaining a kernel with at most 7 k vertices. In this paper, we extend the study to any fixed c by giving both a linear-vertex and a linear-edge kernel. In the particular case of c = 2 , we obtain a kernel with less than 4 k vertices and less than 6 k + ( 3 + 2 ) k + 4  edges. These results imply that for any fixed c ≥ 2 , c - Load Coloring is FPT and the optimization version of c - Load Coloring (where k is to be maximized) has an approximation algorithm with a constant ratio.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-016-0259-z