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On the max coloring problem

Leah Epstein, Asaf Levin
2012 Theoretical Computer Science  
We consider 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.  ... 
doi:10.1016/j.tcs.2012.07.037 fatcat:eof7puuvf5ac3ohk6xp5nmnspi

On the max-weight edge coloring problem

Giorgio Lucarelli, Ioannis Milis, Vangelis T. Paschos
2009 Journal of combinatorial optimization  
We study 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.  ... 
doi:10.1007/s10878-009-9223-z fatcat:fuvlvykj5vbzzk243cys3w4g54

Colored de Bruijn Graphs and the Genome Halving Problem

Max Alekseyev, Pavel Pevzner
2007 IEEE/ACM Transactions on Computational Biology & Bioinformatics  
We further use 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.  ... 
doi:10.1109/tcbb.2007.1002 pmid:17277417 fatcat:r3p3dokojjanpfeza3ur6q375e

On the Max Coloring Problem [chapter]

Leah Epstein, Asaf Levin
Approximation and Online Algorithms  
We consider 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.  ... 
doi:10.1007/978-3-540-77918-6_12 dblp:conf/waoa/EpsteinL07 fatcat:dspl4hoxbrezzb65v2dcc223oi

On the Complexity of the Max-Edge-Coloring Problem with Its Variants [chapter]

Chang Wu Yu
Lecture Notes in Computer Science  
In 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.  ... 
doi:10.1007/978-3-540-74450-4_32 fatcat:jidsfd37izdqtkx4qrdhhpkvtq

Max-coloring paths: tight bounds and extensions

Telikepalli Kavitha, Julián Mestre
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.  ... 
doi:10.1007/s10878-010-9290-1 fatcat:7x5tp2qymbcldjh4ikz7jhpnsq

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

Telikepalli Kavitha, Julián Mestre
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.  ... 
doi:10.1007/978-3-642-10631-6_11 fatcat:xl5zxt7dn5detau3pt2fhir4wm

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

Timothy Highley, Hoang Le
2018 arXiv   pre-print
TROPICAL-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] .  ... 
arXiv:1610.05115v4 fatcat:76gbiemtfjc6ngl3ezn5iltusy

Scheduling Multiprocessor Tasks with Equal Processing Times as a Mixed Graph Coloring Problem

Yuri N. Sotskov, Еvangelina I. Mihova
2021 Algorithms  
Contrary to a classical shop-scheduling 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.  ... 
doi:10.3390/a14080246 fatcat:v2qyrvrxfnec7giy7gqnp3oy24

Approximation Algorithms for the Max-coloring Problem [chapter]

Sriram V. Pemmaraju, Rajiv Raman
2005 Lecture Notes in Computer Science  
Given a graph G = (V, E) and positive integral vertex weights w : V → N, 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.  ... 
doi:10.1007/11523468_86 fatcat:2rbsrydsbzgtplrj3f7s4a27de

Max-coloring and online coloring with bandwidths on interval graphs

Sriram V. Pemmaraju, Rajiv Raman, Kasturi Varadarajan
2011 ACM Transactions on Algorithms  
We make a connection between 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.  ... 
doi:10.1145/1978782.1978790 fatcat:2vuasr6trjcdjft3tjcf7ttnuy

The Parameterized Complexity of Happy Colorings [article]

Neeldhara Misra, I. Vinod Reddy
2017 arXiv   pre-print
We show that 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.  ... 
arXiv:1708.03853v1 fatcat:mi3tnmhtknetlhvvdtt7mgxwme

The Parameterized Complexity of Happy Colorings [chapter]

Neeldhara Misra, I. Vinod Reddy
2018 Lecture Notes in Computer Science  
We show that 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.  ... 
doi:10.1007/978-3-319-78825-8_12 fatcat:37n7irliirgbrasbgdtsflv6oq

Approximate Constrained Bipartite Edge Coloring [chapter]

Ioannis Caragiannis, Afonso Ferreira, Christos Kaklamanis, Stéphane Pérennes, Pino Persiano, Hervé Rivano
2001 Lecture Notes in Computer Science  
In previous work Caragiannis et al. [2] consider two special cases of 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.  ... 
doi:10.1007/3-540-45477-2_4 fatcat:qjwmv2iywzathj36moshxkwfee

Vectorial solutions to list multicoloring problems on graphs [article]

Yves Aubry, Olivier Togni
2012 arXiv   pre-print
Furthermore, we describe 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.  ... 
arXiv:1202.4842v1 fatcat:raprjfpkfrdmnhxefexevnatgq
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