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The Complexity of Tree Multicolorings
[chapter]

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. ...

##
###
On the minimum monochromatic or multicolored subgraph partition problems

2007
*
Theoretical Computer Science
*

We investigate

doi:10.1016/j.tcs.2007.04.033
fatcat:6ny45ast65c4bmtbal3nxxbkk4
*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. ...##
###
Multicoloring trees

2003
*
Information and Computation
*

We study these versions

doi:10.1016/s0890-5401(02)00032-9
fatcat:4jfprdb7nfc7pe6svjt3wyg2sa
*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). ...##
###
Efficient list cost coloring of vertices and/or edges of bounded cyclicity graphs

2009
*
Discussiones Mathematicae Graph Theory
*

We consider a list cost coloring

doi:10.7151/dmgt.1452
fatcat:az4plrwi3nfb3g6nlbwo3bbgv4
*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] ). ...##
###
Reconstruction of Ancestral Genomes in Presence of Gene Gain and Loss

2016
*
Journal of Computational Biology
*

At

doi:10.1089/cmb.2015.0160
pmid:26885568
fatcat:hip5sakfhzfudankl2iazhxesy
*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. ...##
###
The Square Root Phenomenon in Planar Graphs
[chapter]

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

doi:10.1007/978-3-642-39212-2_4
fatcat:aoetw6pginbj3gkfymdj3f54fq
*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. ...##
###
The Square Root Phenomenon in Planar Graphs
[chapter]

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

doi:10.1007/978-3-642-38756-2_1
fatcat:ldsbn2zrdje4tlferdy2z7cg5u
*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. ...##
###
Reconstruction of ancestral genomes in presence of gene gain and loss
[article]

2016
*
bioRxiv
*
pre-print

At

doi:10.1101/040196
fatcat:7mgw55xeefexbc2dhlsfkx3vde
*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. ...##
###
On the Complexity of Problems on Tree-structured Graphs
[article]

2022
*
arXiv
*
pre-print

Moreover, we introduce '

arXiv:2206.11828v1
fatcat:rq2mmd5hgbeadiydafwomuyjuy
*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. ...##
###
Extreme Distances in Multicolored Point Sets
[chapter]

2002
*
Lecture Notes in Computer Science
*

We also give some combinatorial bounds on

doi:10.1007/3-540-47789-6_2
fatcat:3t6wfw7jpjdwhgotxo3lxajtye
*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. ...##
###
Extreme Distances in Multicolored Point Sets

2004
*
Journal of Graph Algorithms and Applications
*

We also give some combinatorial bounds on

doi:10.7155/jgaa.00080
fatcat:ifx4tkwmc5f63eafchegfuw2ba
*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. ...##
###
Tools for Multicoloring with Applications to Planar Graphs and Partial k-Trees

2002
*
Journal of Algorithms
*

Finally, we give a PTAS for

doi:10.1006/jagm.2001.1210
fatcat:eyi2w2ayy5ebjh62sehoxr7h6y
*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. ...##
###
Extreme Distances in Multicolored Point Sets
[chapter]

2006
*
Graph Algorithms and Applications 5
*

We also give some combinatorial bounds on

doi:10.1142/9789812773289_0003
fatcat:snrojxoy7rhqzp76praebp5rje
*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. ...##
###
Resource Allocation Problems in Multifiber WDM Tree Networks
[chapter]

2003
*
Lecture Notes in Computer Science
*

In this paper, we consider

doi:10.1007/978-3-540-39890-5_19
fatcat:bluc7kautvgzxasm7gzlbj3xry
*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. ...##
###
Incidence algebra antipodes and lagrange inversion in one and several variables

1989
*
Journal of combinatorial theory. Series A
*

By

doi:10.1016/0097-3165(89)90013-7
fatcat:smfk5hfkqvb73eupuzs63iw3em
*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 ...
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