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The Complexity of Tree Multicolorings [chapter]

Dániel Marx
2002 Lecture Notes in Computer Science  
The list of the central node v contains every color. List multicoloring is NP-complete in trees.  ...  binary trees, even if every demand is polynomially bounded (in the size of the tree) As a first step of the proof, we introduce another problem where trees are difficult to color: List multicoloring  ...  Returning to minimum sum multicoloring. (There are no lists, the goal is to The NP-hardness of minimum sum coloring in trees is proved by a similar reduction.  ... 
doi:10.1007/3-540-45687-2_44 fatcat:yasjoin64fdrphzk5wxr6son2u

On the minimum monochromatic or multicolored subgraph partition problems

Xueliang Li, Xiaoyan Zhang
2007 Theoretical Computer Science  
We investigate the complexity of the problems for finding the minimum number of monochromatic or multicolored subgraphs, such as cliques, cycles, trees and paths, partitioning V (G), depending on the number  ...  We show that unless NP ⊆ DTIME(N O(log log N) ), for any ≥ 0 there is no approximation algorithm for finding the minimum number of multicolored trees partitioning V (G) with performance 50/521(1 − ) ln  ...  Acknowledgements The authors are indebted to Prof. Baogang Xu and Prof. Shenggui Zhang for helpful discussions and the referees for their helpful suggestions.  ... 
doi:10.1016/j.tcs.2007.04.033 fatcat:6ny45ast65c4bmtbal3nxxbkk4

Multicoloring trees

Magnús M. Halldórsson, Guy Kortsarz, Andrzej Proskurowski, Ravit Salman, Hadas Shachnai, Jan Arne Telle
2003 Information and Computation  
We study these versions of the multicoloring problem on trees, under the sum-of-completion-times objective.  ...  Scheduling jobs with pairwise conflicts is modeled by the graph multicoloring problem.  ...  Notice that for both models, the algorithms of this paper have considerably better complexity bounds for the case of trees (k ¼ 1).  ... 
doi:10.1016/s0890-5401(02)00032-9 fatcat:4jfprdb7nfc7pe6svjt3wyg2sa

Efficient list cost coloring of vertices and/or edges of bounded cyclicity graphs

Krzysztof Giaro, Marek Kubale
2009 Discussiones Mathematicae Graph Theory  
We consider a list cost coloring of vertices and edges in the model of vertex, edge, total and pseudototal coloring of graphs.  ...  We use a dynamic programming approach to derive polynomial-time algorithms for solving the above problems for trees.  ...  The extension of coloring to multicoloring causes a dramatic increase in computational complexity. Theorem 3.1 ([7] ).  ... 
doi:10.7151/dmgt.1452 fatcat:az4plrwi3nfb3g6nlbwo3bbgv4

Reconstruction of Ancestral Genomes in Presence of Gene Gain and Loss

Pavel Avdeyev, Shuai Jiang, Sergey Aganezov, Fei Hu, Max A. Alekseyev
2016 Journal of Computational Biology  
At the same time, a larger number of input genomes may actually simplify the problem in practice as it was earlier illustrated with MGRA, a state-of-the-art software tool for reconstruction of ancestral  ...  One of the key obstacles for MGRA and other similar tools is presence of breakpoint reuses when the same breakpoint region is broken by several different genome rearrangements in the course of evolution  ...  Acknowledgement The work was supported by the National Science Foundation under the grant No. IIS-1462107.  ... 
doi:10.1089/cmb.2015.0160 pmid:26885568 fatcat:hip5sakfhzfudankl2iazhxesy

The Square Root Phenomenon in Planar Graphs [chapter]

Dániel Marx
2013 Lecture Notes in Computer Science  
Example: Transforming an Independent Set instance (G , k) into a Clique instance (G , k) is a parameterized reduction. 38 Multicolored Clique A useful variant of Clique: Multicolored Clique: The vertices  ...  Complexity: NP-complete NP-complete Parameterized complexity Problem: Vertex Cover Independent Set Input: Graph G , integer k Graph G , integer k Question: Is it possible to cover the edges with k vertices  ...  Input: A k × k matrix and a set of pairs S i,j ⊆ [D] × [D] for each cell. Find: A pair s i,j ∈ S i,j for each cell such that Vertical neighbors agree in the 1st coordinate.  ... 
doi:10.1007/978-3-642-39212-2_4 fatcat:aoetw6pginbj3gkfymdj3f54fq

The Square Root Phenomenon in Planar Graphs [chapter]

Dániel Marx
2013 Lecture Notes in Computer Science  
Example: Transforming an Independent Set instance (G , k) into a Clique instance (G , k) is a parameterized reduction. 38 Multicolored Clique A useful variant of Clique: Multicolored Clique: The vertices  ...  Complexity: NP-complete NP-complete Parameterized complexity Problem: Vertex Cover Independent Set Input: Graph G , integer k Graph G , integer k Question: Is it possible to cover the edges with k vertices  ...  Input: A k × k matrix and a set of pairs S i,j ⊆ [D] × [D] for each cell. Find: A pair s i,j ∈ S i,j for each cell such that Vertical neighbors agree in the 1st coordinate.  ... 
doi:10.1007/978-3-642-38756-2_1 fatcat:ldsbn2zrdje4tlferdy2z7cg5u

Reconstruction of ancestral genomes in presence of gene gain and loss [article]

Pavel Avdeyev, Shuai Jiang, Sergey Aganezov, Fei Hu, Max A. Alekseyev
2016 bioRxiv   pre-print
At the same time, a larger number of input genomes may actually simplify the problem in practice as it was earlier illustrated with MGRA, a state-of-the-art software tool for reconstruction of ancestral  ...  One of the key obstacles for MGRA and other similar tools is presence of breakpoint reuses when the same breakpoint region is broken by several different genome rearrangements in the course of evolution  ...  Acknowledgement The work was supported by the National Science Foundation under the grant No. IIS-1462107.  ... 
doi:10.1101/040196 fatcat:7mgw55xeefexbc2dhlsfkx3vde

On the Complexity of Problems on Tree-structured Graphs [article]

Hans L. Bodlaender and Carla Groenland and Hugo Jacob and Marcin Pilipczuk and Michal Pilipczuk
2022 arXiv   pre-print
Moreover, we introduce 'tree-shaped' variants of Weighted CNF-Satisfiability and Multicolor Clique that are XALP-complete.  ...  We give a number of equivalent characterisations of the class XALP, e.g., XALP is the class of problems solvable by an Alternating Turing Machine whose runs have tree size at most f(k)n^O(1) and use f(  ...  XALP-completeness for a tree-chained variant of Multicolor Clique Our first XALP-complete problem is a 'tree' variant of the well-known Multicolor Clique problem.  ... 
arXiv:2206.11828v1 fatcat:rq2mmd5hgbeadiydafwomuyjuy

Extreme Distances in Multicolored Point Sets [chapter]

Adrian Dumitrescu, Sumanta Guha
2002 Lecture Notes in Computer Science  
We also give some combinatorial bounds on the multiplicities of extreme distances in this setting.  ...  of different colors.  ...  Since for arbitrary graphs, the time complexity of a minimum spanning tree computation is the same as that of a maximum spanning tree computation, we have: Open Problem.  ... 
doi:10.1007/3-540-47789-6_2 fatcat:3t6wfw7jpjdwhgotxo3lxajtye

Extreme Distances in Multicolored Point Sets

Adrian Dumitrescu, Sumanta Guha
2004 Journal of Graph Algorithms and Applications  
We also give some combinatorial bounds on the maximum multiplicity of extreme bichromatic distances in the plane.  ...  We do this by solving the more general problem of maintaining a bichromatic edge of minimum (resp. maximum) weight in an undirected weighted graph with colored vertices, when vertices dynamically change  ...  We wish to thank the anonymous referee for a thorough reading and useful suggestions that have lead to improved results in Section 2.2.  ... 
doi:10.7155/jgaa.00080 fatcat:ifx4tkwmc5f63eafchegfuw2ba

Tools for Multicoloring with Applications to Planar Graphs and Partial k-Trees

Magnús M. Halldórsson, Guy Kortsarz
2002 Journal of Algorithms  
Finally, we give a PTAS for the makespan of a preemptive multicoloring of partial k-trees that uses only O(log n) preemptions.  ...  For non-preemptive multicoloring sum of partial k-trees, we obtain a fully polynomial time approximation scheme.  ...  Acknowledgements This paper owes its existence to Amotz Bar-Noy, for bringing the authors together. We thank him for his support and many early discussions.  ... 
doi:10.1006/jagm.2001.1210 fatcat:eyi2w2ayy5ebjh62sehoxr7h6y

Extreme Distances in Multicolored Point Sets [chapter]

Adrian Dumitrescu, Sumanta Guha
2006 Graph Algorithms and Applications 5  
We also give some combinatorial bounds on the maximum multiplicity of extreme bichromatic distances in the plane.  ...  We do this by solving the more general problem of maintaining a bichromatic edge of minimum (resp. maximum) weight in an undirected weighted graph with colored vertices, when vertices dynamically change  ...  We wish to thank the anonymous referee for a thorough reading and useful suggestions that have lead to improved results in Section 2.2.  ... 
doi:10.1142/9789812773289_0003 fatcat:snrojxoy7rhqzp76praebp5rje

Resource Allocation Problems in Multifiber WDM Tree Networks [chapter]

Thomas Erlebach, Aris Pagourtzis, Katerina Potika, Stamatis Stefanakos
2003 Lecture Notes in Computer Science  
In this paper, we consider the problem of minimizing the total number of fibers necessary to establish a given set of requests with a bounded number w of wavelengths, and the problem of maximizing the  ...  We study both problems in undirected tree networks T = (V, E) and present approximation algorithms with ratio 1 + 4|E| log |V |/OPT and 4 for the former and with ratio 2.542 for the latter.  ...  We are indebted to Chandra Chekuri for suggesting the current proof of Corollary 2.  ... 
doi:10.1007/978-3-540-39890-5_19 fatcat:bluc7kautvgzxasm7gzlbj3xry

Incidence algebra antipodes and lagrange inversion in one and several variables

Mark Haiman, William Schmitt
1989 Journal of combinatorial theory. Series A  
By the definition of < in P", blocks common to more than one of the xi have the same color in all of them; thus T(C) is a multicolored tree, the multicolored tree associated with C. LEMMA 6.  ...  For the multicolored Faa di Bruno Hopf algebra, the antipode is given by (14) where the sum ranges over multicolored trees (i.e., trees with a multicolored set of elements) having Jnl labelled leaves  ... 
doi:10.1016/0097-3165(89)90013-7 fatcat:smfk5hfkqvb73eupuzs63iw3em
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