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On the Sum Coloring Problem on Interval Graphs

S. Nicoloso, M. Sarrafzadeh, X. Song
1999 Algorithmica  
Acknowledgments We would like to thank the anonymous referees for their thorough reading the manuscript and valuable comments which considerably improved the contents of current paper, specially the proof  ...  Hence the coloring problem of interval graphs is in P, see [6] for more details.  ...  It is known that the coloring and the clique problems of interval graphs both belong to P. In this section these problems for signed interval graphs are studied.  ... 
doi:10.1007/pl00009252 fatcat:oc5sjdsqhfg73bdsgf5znzjde4

On sum coloring and sum multi-coloring for restricted families of graphs

Allan Borodin, Ioana Ivan, Yuli Ye, Bryce Zimny
2012 Theoretical Computer Science  
We consider the sum coloring (chromatic sum) problem and the sum multi-coloring problem for restricted families of graphs.  ...  Finally, we study priority algorithms as a model for greedy algorithms for the sum coloring problem and the sum multi-coloring problem.  ...  The priority algorithm schema For completeness, we provide the schema for an adaptive priority algorithm introduced in Borodin, Nielsen, and Rackoff [15] as a model for greedy algorithms.  ... 
doi:10.1016/j.tcs.2011.11.010 fatcat:jrr7kt7zsremhac7ikqtad54gy

Sum coloring and interval graphs: a tight upper bound for the minimum number of colors

S Nicoloso
2004 Discrete Mathematics  
In this note we prove that the number of colors required to attain a minimum valued sum on arbitrary interval graphs does not exceed min{n; 2 (G) − 1}.  ...  The SUM COLORING problem consists of assigning a color c(vi) ∈ Z + to each vertex vi ∈ V of a graph G = (V; E) so that adjacent nodes have di erent colors and the sum of the c(vi)'s over all vertices vi  ...  Acknowledgements The author wish to thank the two anonymous referees for the helpful and accurate comments.  ... 
doi:10.1016/j.disc.2003.06.015 fatcat:tjuxhw37a5adzn7dlea7xc3tlm

A short proof of the NP-completeness of minimum sum interval coloring

Dániel Marx
2005 Operations Research Letters  
In the minimum sum coloring problem we have to assign positive integers to the vertices of a graph in such a way that neighbors receive different numbers and the sum of the numbers is minimized.  ...  Szkalicki [9] has shown that minimum sum coloring is NP-hard for interval graphs. Here we present a simpler proof of this result.  ...  The chromatic sum of a graph is the smallest sum that a proper coloring can have. In the minimum sum coloring problem we have to find a coloring that minimizes the sum.  ... 
doi:10.1016/j.orl.2004.07.006 fatcat:74wbtz2uynapldhzfelg5fykm4

Improved Bounds for Sum Multicoloring and Scheduling Dependent Jobs with Minsum Criteria [chapter]

Rajiv Gandhi, Magnús M. Halldórsson, Guy Kortsarz, Hadas Shachnai
2005 Lecture Notes in Computer Science  
In particular, we obtain the first constant factor approximation ratio for npSMC on interval graphs, on which our problems have numerous applications.  ...  An input to our problems can be modeled as an instance of the sum multicoloring (SMC) problem: Given a graph and the number of colors required by each vertex, find a proper multicoloring which minimizes  ...  The wire-minimization problem then corresponds to sum coloring an interval graph.  ... 
doi:10.1007/978-3-540-31833-0_8 fatcat:w4rvcm4nprcjhnthpcnppi3jxq

Batch Coloring of Graphs [article]

Joan Boyar, Leah Epstein, Lene M. Favrholdt, Kim S. Larsen, Asaf Levin
2016 arXiv   pre-print
For classic graph coloring, the goal is to minimize the maximum color used, and for the sum coloring problem, the goal is to minimize the sum of colors assigned to all input vertices.  ...  We provide several results, including a general result for sum coloring and results for the classic graph coloring problem on restricted graph classes: We show tight bounds for any graph class containing  ...  The sum coloring problem (also called chromatic sum) was introduced in [16] (see [15] for a survey of results on this problem).  ... 
arXiv:1610.02997v1 fatcat:5mpykp4xpzcklbdliqdzoddtg4

Minimum sum set coloring of trees and line graphs of trees

Flavia Bonomo, Guillermo Durán, Javier Marenco, Mario Valencia-Pabon
2011 Discrete Applied Mathematics  
In this paper, we study the minimum sum set coloring (MSSC) problem which consists in assigning a set of x(v) positive integers to each vertex v of a graph so that the intersection of sets assigned to  ...  We show that the MSSC problem is strongly NP-hard both in the preemptive case on trees and in the non-preemptive case in line graphs of trees.  ...  Acknowledgements We thank the anonymous referees for their helpful comments that improved the presentation of this paper. Third author's work was partially supported by UBACyT Grant X069 (Argentina).  ... 
doi:10.1016/j.dam.2010.11.018 fatcat:hfu5ekcs5zglxliak2yd6jcnqa

Sum Multi-coloring of Graphs [chapter]

Amotz Bar-Noy, Magnús M. Halldórsson, Guy Kortsarz, Ravit Salman, Hadas Shanhnai
1999 Lecture Notes in Computer Science  
Scheduling dependent jobs on multiple machines is modeled by the graph multi-coloring problem. In this paper, we consider the problem of minimizing the average completion time of all jobs.  ...  We concentrate on nding approximation algorithms since the special case of the sum coloring problem is already hard to solve and hard to approximate.  ...  In fact, it can be shown, that any on-line deterministic algorithm is doomed to a poor performance, even for the sum coloring problem on interval graphs.  ... 
doi:10.1007/3-540-48481-7_34 fatcat:grw6hvyiujaz7c3vjxa4bqym74

The chromatic sum of a graph: history and recent developments

Ewa Kubicka
2004 International Journal of Mathematics and Mathematical Sciences  
The chromatic sum of a graph is the smallest sum of colors among all proper colorings with natural numbers.  ...  The strength of a graph is the minimum number of colors necessary to obtain its chromatic sum.  ...  The chromatic sum problem restricted to interval graphs is NPcomplete. We will finish the survey by presenting a new result concerning split graphs.  ... 
doi:10.1155/s0161171204306216 fatcat:sszoqjoienh5tnzxucompvcoqy

Routing with Minimum Wire Length in the Dogleg-Free Manhattan Model is $\cal NP$-Complete

Tibor Szkaliczki
1999 SIAM journal on computing (Print)  
We show that this problem is N P-complete. This implies the N P-completeness of other problems including the minimum wire length routing and the sum coloring on interval graphs.  ...  The results on dogleg-free Manhattan routing can be connected with other application areas related to interval graphs. In this paper we define the minimum value interval placement problem.  ...  Similarly to the weighted case, the N P-completeness of the chromatic sum problem and the sum coloring problem on interval graphs follows from the N P-completeness of the unweighted version of the minimum  ... 
doi:10.1137/s0097539796303123 fatcat:6zeslg6k5fholgql3dkpyku65y

"Rent-or-Buy" Scheduling and Cost Coloring Problems [chapter]

Takuro Fukunaga, Magnús M. Halldórsson, Hiroshi Nagamochi
2007 Lecture Notes in Computer Science  
We give exact and approximation algorithms for RBC and three other cost coloring problems (including the previously studied Probabilistic coloring problem), both on interval and on perfect graphs.  ...  We study several cost coloring problems, where we are given a graph and a cost function on the independent sets and are to find a coloring that minimizes the function costs of the color classes.  ...  They give a polynomial time algorithm for all such problems on co-interval graphs (complements of interval graphs).  ... 
doi:10.1007/978-3-540-77050-3_7 fatcat:lz7s3zadwretdmitnw46varigq

Complexity results on restricted instances of a paint shop problem for words

P. Bonsma, Th. Epping, W. Hochstättler
2006 Discrete Applied Mathematics  
The goal is to minimize the number of color changes between adjacent letters. This is a special case of the paint shop problem for words, which was previously shown to be NP-complete.  ...  A solution is a 2-coloring of its letters such that the two occurrences of every letter are colored with different colors.  ...  As the row sums are pairwise disjoint it suffices to show that there is at least one color change in any odd row sum I 1 I 2 . . . I 2k+1 . Let c denote the number of color changes in I .  ... 
doi:10.1016/j.dam.2005.05.033 fatcat:fd3mhjgypvaftkqhi4hdtb52mq

SUB-COLORING AND HYPO-COLORING INTERVAL GRAPHS

RAJIV GANDHI, BRADFORD GREENING, SRIRAM PEMMARAJU, RAJIV RAMAN
2010 Discrete Mathematics, Algorithms and Applications (DMAA)  
In this paper, we study the sub-coloring and hypo-coloring problems on interval graphs.  ...  In the hypo-coloring problem, given a graph G, and integral weights on vertices, we want to find a partition of the vertices of G into sub-color classes such that the sum of the weights of the heaviest  ...  Our proof is heavily influenced by the NP-completeness proof of minimum sum coloring on interval graphs by D.  ... 
doi:10.1142/s1793830910000693 fatcat:v7qgias2w5dorf37vmdo4cu77e

Sub-coloring and Hypo-coloring Interval Graphs [chapter]

Rajiv Gandhi, Bradford Greening, Sriram Pemmaraju, Rajiv Raman
2010 Lecture Notes in Computer Science  
In this paper, we study the sub-coloring and hypo-coloring problems on interval graphs.  ...  In the hypo-coloring problem, given a graph G, and integral weights on vertices, we want to find a partition of the vertices of G into sub-color classes such that the sum of the weights of the heaviest  ...  Our proof is heavily influenced by the NP-completeness proof of minimum sum coloring on interval graphs by D.  ... 
doi:10.1007/978-3-642-11409-0_11 fatcat:amagyd2x6bb5jejehhy2nuvsvq

Matchings, coverings, and Castelnuovo-Mumford regularity

Russ Woodroofe
2014 Journal of Commutative Algebra  
We show that the co-chordal cover number of a graph G gives an upper bound for the Castelnuovo-Mumford regularity of the associated edge ideal.  ...  Several known combinatorial upper bounds of regularity for edge ideals are then easy consequences of covering results from graph theory, and we derive new upper bounds by looking at additional covering  ...  Hence, boxicity of G is the co-interval cover # of G . Remark: any covering problem on G has a dual intersection problem on G , as we've seen with colorings and boxicity.  ... 
doi:10.1216/jca-2014-6-2-287 fatcat:p5m3v7pppvbyxcmopzjqvfltdi
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