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On the max coloring problem

2012
*
Theoretical Computer Science
*

We consider

doi:10.1016/j.tcs.2012.07.037
fatcat:eof7puuvf5ac3ohk6xp5nmnspi
*max**coloring**on*hereditary graph classes.*The**problem*is defined as follows. ... Given a graph G = (V , E) and positive node weights w : V → (0, ∞),*the*goal is to find a proper node*coloring*of G whose*color*classes C 1 , C 2 , . . . , C k minimize k i=1*max*v∈C i w(v). ... There are further results*on**the**max**coloring**problem*[8, 6, 9, 15, 37, 17] and even studies of*the**max*edge*coloring**problem*(which is*the**max**coloring*of line graphs) (see e.g. ...##
###
On the max-weight edge coloring problem

2009
*
Journal of combinatorial optimization
*

We study

doi:10.1007/s10878-009-9223-z
fatcat:fuvlvykj5vbzzk243cys3w4g54
*the*following generalization of*the*classical edge*coloring**problem*: Given a weighted graph, find a partition of its edges into matchings (*colors*), each*one*of weight equal to*the*maximum weight ... We explore*the*frontier between polynomial and NP-hard variants of*the**problem*as well as*the*approximability of*the*NP-hard variants. ... Due to*the*fact that*the*weight w i of each*color*is defined as*the*maximum weight of*the*edges*colored*i, in*the*following we shall refer to this*problem*as*Max*-weight Edge*Coloring*(MEC)*problem*. ...##
###
Colored de Bruijn Graphs and the Genome Halving Problem

2007
*
IEEE/ACM Transactions on Computational Biology & Bioinformatics
*

We further use

doi:10.1109/tcbb.2007.1002
pmid:17277417
fatcat:r3p3dokojjanpfeza3ur6q375e
*the*generalized breakpoint graphs to study*the*Genome Halving*Problem*(first introduced and solved by Nadia El-Mabrouk and David Sankoff). ...*The*El-Mabrouk-Sankoff algorithm is rather complex, and, in this paper, we present an alternative approach that is based*on*generalized breakpoint graphs. ... In this paper, we focus*on**the*former and harder*problem*: Weak Genome Halving*Problem*. ...##
###
On the Max Coloring Problem
[chapter]

*
Approximation and Online Algorithms
*

We consider

doi:10.1007/978-3-540-77918-6_12
dblp:conf/waoa/EpsteinL07
fatcat:dspl4hoxbrezzb65v2dcc223oi
*max**coloring**on*hereditary graph classes.*The**problem*is defined as follows. ... Given a graph G = (V , E) and positive node weights w : V → (0, ∞),*the*goal is to find a proper node*coloring*of G whose*color*classes C 1 , C 2 , . . . , C k minimize k i=1*max*v∈C i w(v). ... There are further results*on**the**max**coloring**problem*[8, 6, 9, 15, 37, 17] and even studies of*the**max*edge*coloring**problem*(which is*the**max**coloring*of line graphs) (see e.g. ...##
###
On the Complexity of the Max-Edge-Coloring Problem with Its Variants
[chapter]

*
Lecture Notes in Computer Science
*

In

doi:10.1007/978-3-540-74450-4_32
fatcat:jidsfd37izdqtkx4qrdhhpkvtq
*the*work, we discuss*the*complexity issues*on**the*new graph*problem*and its variants. Specifically, we design a 2-approximmation algorithm for*the**max*-edge-*coloring**problem**on*biplanar graphs. ...*The**max*-edge-*coloring**problem*(MECP) is finding an edge*colorings*{E 1 , E 2 , E 3 , ..., E z } of a weighted graph G=(V, E) to minimize { } ∑ = ∈ z i i k k E e e w 1 ) (*max*, where w(e k ) is*the*weight ...*max*-*coloring**problem**on*interval graphs. ...##
###
Max-coloring paths: tight bounds and extensions

2010
*
Journal of combinatorial optimization
*

*The*

*max*-

*coloring*

*problem*is to compute a legal

*coloring*of

*the*vertices of a graph G = (V , E) with vertex weights w such that k i=1

*max*v∈C i w(v i ) is minimized, where C 1 , . . . , C k are

*the*various ... For general graphs,

*max*-

*coloring*is as hard as

*the*classical vertex

*coloring*

*problem*, a special case of

*the*former where vertices have unit weight. ...

*the*original author(s) and source are credited. ...

##
###
Max-Coloring Paths: Tight Bounds and Extensions
[chapter]

2009
*
Lecture Notes in Computer Science
*

*The*

*max*-

*coloring*

*problem*is to compute a legal

*coloring*of

*the*vertices of a graph G = (V , E) with vertex weights w such that k i=1

*max*v∈C i w(v i ) is minimized, where C 1 , . . . , C k are

*the*various ... For general graphs,

*max*-

*coloring*is as hard as

*the*classical vertex

*coloring*

*problem*, a special case of

*the*former where vertices have unit weight. ...

*the*original author(s) and source are credited. ...

##
###
Tropical Vertex-Disjoint Cycles of a Vertex-Colored Digraph: Barter Exchange with Multiple Items Per Agent
[article]

2018
*
arXiv
*
pre-print

TROPICAL-

arXiv:1610.05115v4
fatcat:76gbiemtfjc6ngl3ezn5iltusy
*MAX*-SIZE-EXCHANGE is a similar*problem*, where*the*goal is to find a set of vertex-disjoint cycles that contains*the*maximum number of vertices and also contains all of*the**colors*in*the*graph. ... It is known that*the**problem*of finding a set of vertex-disjoint cycles with*the*maximum total number of vertices (*MAX*-SIZE-EXCHANGE) can be solved in polynomial time. ... We explore*problems*that are similar to*the*Assignment*Problem*and kidney exchange*problem*but based*on*vertex-*colored*graphs. Vertex-*colored*graphs have also been called tropical graphs [19] . ...##
###
Scheduling Multiprocessor Tasks with Equal Processing Times as a Mixed Graph Coloring Problem

2021
*
Algorithms
*

Contrary to a classical shop-scheduling

doi:10.3390/a14080246
fatcat:v2qyrvrxfnec7giy7gqnp3oy24
*problem*, several processors must fulfill a multiprocessor task. Furthermore, two types of*the*precedence constraints may be given*on**the*task set . ... We prove that*the*extended scheduling*problem*with integer release times of*the*jobs to minimize schedule length may be solved as an optimal mixed graph*coloring**problem*that consists of*the*assignment ... Conflicts of Interest:*The*authors declare no conflict of interest. ...##
###
Approximation Algorithms for the Max-coloring Problem
[chapter]

2005
*
Lecture Notes in Computer Science
*

Given a graph G = (V, E) and positive integral vertex weights w : V → N,

doi:10.1007/11523468_86
fatcat:2rbsrydsbzgtplrj3f7s4a27de
*the**max*-*coloring**problem*seeks to find a proper vertex*coloring*of G whose*color*classes C1, C2, . . . , C k , minimize L. ... We would like to thank*the*anonymous referees for suggestions that led to a simplified proof of Theorem 1, and also for pointing out*the*work done in [3] . ... Our PTAS for*the**max*-*coloring**problem**on*trees relied*on**the*fact that*the*FEASIBLE k-*COLORING**problem**on*trees can be solved in polynomial time for any k. ...##
###
Max-coloring and online coloring with bandwidths on interval graphs

2011
*
ACM Transactions on Algorithms
*

We make a connection between

doi:10.1145/1978782.1978790
fatcat:2vuasr6trjcdjft3tjcf7ttnuy
*max*-*coloring*and*on*-line graph*coloring*and use this to devise a simple 2-approximation algorithm for*max*-*coloring**on*interval graphs. ... We also show that*the**max*-*coloring**problem*is NP-hard.*The**problem*of online*coloring*of intervals with bandwidths is a simultaneous generalization of online interval*coloring*and online bin packing. ... Finally, we thank*the*anonymous referees whose suggestions have improved*the*paper. ...##
###
The Parameterized Complexity of Happy Colorings
[article]

2017
*
arXiv
*
pre-print

We show that

arXiv:1708.03853v1
fatcat:mi3tnmhtknetlhvvdtt7mgxwme
*the*maximum happy vertex (edge)*problem*is*on*split graphs and bipartite graphs and polynomially solvable*on*cographs. ... Given a partial*coloring*c of V,*the*Maximum Happy Vertex (Edge)*problem*asks for a total*coloring*of V extending c to all vertices of V that maximises*the*number of happy vertices (edges). ... Note that*coloring*v with*the*same*color*as*the**one*used*on**max*(d j ) makes*max*(d j ) edges happy. ...##
###
The Parameterized Complexity of Happy Colorings
[chapter]

2018
*
Lecture Notes in Computer Science
*

We show that

doi:10.1007/978-3-319-78825-8_12
fatcat:37n7irliirgbrasbgdtsflv6oq
*the*maximum happy vertex (edge)*problem*is NP-hard*on*split graphs and bipartite graphs and polynomially solvable*on*cographs. ... Given a partial*coloring*c of V,*the*Maximum Happy Vertex (Edge)*problem*asks for a total*coloring*of V extending c to all vertices of V that maximises*the*number of happy vertices (edges). ... Note that*coloring*v with*the*same*color*as*the**one*used*on**max*(d j ) makes*max*(d j ) edges happy. ...##
###
Approximate Constrained Bipartite Edge Coloring
[chapter]

2001
*
Lecture Notes in Computer Science
*

In previous work Caragiannis et al. [2] consider two special cases of

doi:10.1007/3-540-45477-2_4
fatcat:qjwmv2iywzathj36moshxkwfee
*the**problem*and proved tight bounds*on**the*optimal number of*colors*by decomposing*the*bipartite graph into matchings which are*colored*... In this paper we present a randomized (1.37 + o(1))-approximation algorithm for*the*general*problem*in*the*case where*max*{l, c} = ω(ln n). ... This*problem*can be modelled as an edge*coloring**problem**on*a bipartite graph. ...##
###
Vectorial solutions to list multicoloring problems on graphs
[article]

2012
*
arXiv
*
pre-print

Furthermore, we describe

arXiv:1202.4842v1
fatcat:raprjfpkfrdmnhxefexevnatgq
*the*set of solutions to*the**on*call*problem*: when w is not a permissible weight, we find all*the*nearest permissible weights w'. ... For a graph G with a given list assignment L*on**the*vertices, we give an algebraical description of*the*set of all weights w such that G is (L,w)-*colorable*, called permissible weights. ...*The**on*call*problem**The**on*call*problem*can be modelized as follows : for w / ∈ − → W (G, L), find w * ∈ − → W (G, L) such that w * ≤ w and w − w * is minimal. ...
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