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Hardness of the Generalized Coloring Numbers [article]

Michael Breen-McKay, Brian Lavallee, Blair D. Sullivan
2022 arXiv   pre-print
The generalized coloring numbers of Kierstead and Yang offer an algorithmically useful characterization of graph classes with bounded expansion.  ...  In this work, we consider the hardness and approximability of these parameters. First, we complete the work of Grohe et al. by showing that computing the weak 2-coloring number is NP-hard.  ...  The generalized coloring numbers (consisting of the r-coloring numbers and the weak r-coloring numbers) extend the notion of degeneracy to neighborhoods of radius r.  ... 
arXiv:2112.10562v2 fatcat:epid7antvzgi5laci6ha6ybezi

Constructive generation of very hard 3-colorability instances

Kazunori Mizuno, Seiichi Nishihara
2008 Discrete Applied Mathematics  
We demonstrated experimentally that the computational cost to solve our generated instances is of an exponential order of the number of vertices by using a few actual coloring algorithms and constraint  ...  We propose a rather systematic or constructive algorithm that repeats the embedding of 4-critical graphs to arbitrarily generate large extraordinarily hard 3-colorability instances.  ...  The number of assignments of colors to vertices is counted as search cost until the colorability is determined.  ... 
doi:10.1016/j.dam.2006.07.015 fatcat:wnmphqnqcvhofpezjscrad442q

Learning to Sample Hard Instances for Graph Algorithms [article]

Ryoma Sato, Makoto Yamada, Hisashi Kashima
2019 arXiv   pre-print
However, they generate only finite number of hard instances. The merit of such methods is limited because it is difficult to extract meaningful patterns from small number of instances.  ...  We seek for a probabilistic generator of hard instances.  ...  When the algorithm is coloring V \(C ∪ P ), the number of color candidates of vertices of P are at most two, and the degrees of them are only two.  ... 
arXiv:1902.09700v2 fatcat:uvacyzjdtndubo53ozqwipsz7u

Complexity of the General Chromatic Art Gallery Problem [article]

Sándor P. Fekete and Stephan Friedrichs and Michael Hemmer
2014 arXiv   pre-print
Let χ_G(P), the chromatic number of P, denote the minimum number of colors required to color any guard cover of P. We show that determining, whether χ_G(P) ≤ k is -hard for all k ≥ 2.  ...  In the original Art Gallery Problem (AGP), one seeks the minimum number of guards required to cover a polygon P. We consider the Chromatic AGP (CAGP), where the guards are colored.  ...  Acknowledgement Michael Hemmer received support by the DFG, contract KR 3133/1-1 (Kunst!).  ... 
arXiv:1403.2972v1 fatcat:vyeqmyi5sngxrdzmm2uv3mqdq4

On the Hardness of 4-Coloring a 3-Colorable Graph

Venkatesan Guruswami, Sanjeev Khanna
2004 SIAM Journal on Discrete Mathematics  
This highlights a qualitative difference between the known hardness result for coloring £ -colorable graphs and the factor ¤ ¦ ¥ hardness for approximating the chromatic number of general graphs, as the  ...  Another aspect in which our proof is different is that using the PCP theorem we can show that § -coloring of £ -colorable graphs remains NP-hard even on bounded-degree graphs (this hardness result does  ...  the hardness of general graph coloring and coloring -colorable graphs.  ... 
doi:10.1137/s0895480100376794 fatcat:mbhydwqlhbe6flj35b77dqr7oe

Planar graph coloring is not self-reducible, assuming P ≠ NP

Samir Khuller, Vijay V. Vazirani
1991 Theoretical Computer Science  
One consequence of our result is that the schema of Jerrum et al. (1986) cannot be used for approximately counting the number of four colorings of a planar graph.  ...  We show that obtaining the lexicographically first four coloring of a planar graph is NP-hard. This shows that planar graph four-coloring is not self-reducible, assuming Pf NP.  ...  85-52938 and PYI matching funds from AT&T Bell Labs and Sun Microsystems, Inc. at Cornell University. problem of approximately counting the number of solutions of an NP problem to generating a random  ... 
doi:10.1016/0304-3975(91)90081-c fatcat:hwcsjnc2gzew3mf5csloiizqpi

Balance and Filtering in Structured Satisfiable Problems (Preliminary Report)

Henry Kautz, Yongshao Ruan, Dimitris Achlioptas, Carla Gomes Bart Selman, Mark Stickel
2001 Electronic Notes in Discrete Mathematics  
More generally, we show that the generator is one member of a family of related models that generate distributions ranging from ones that are everywhere tractable to ones that exhibit a sharp hardness  ...  We also discuss the critical role of the problem encoding in the performance of both systematic and local search solvers.  ...  QCP is an NP-complete problem, and random instances exhibit a peak in problem hardness in the area of the phase transition in the percentage of satisfiable instances generated as the ratio of the number  ... 
doi:10.1016/s1571-0653(04)00310-5 fatcat:p3ovbo6yhjaavgbm7t47hpozcy

A Survey on the Complexity of Flood-Filling Games [chapter]

Michael R. Fellows, Frances A. Rosamond, Maise Dantas da Silva, Uéverton S. Souza
2018 Lecture Notes in Computer Science  
The Flood-It problem on trees and the Restricted Shortest Common Supersequence (RSCS) problem are analogous. Flood-It remains NP-hard when played on 3-colored trees.  ...  In this survey, we review recent results on one-player floodfilling games on graphs, Flood-It and Free-Flood-It, in which the player aims to make the board monochromatic with a minimum number of flooding  ...  The parameter neighborhood diversity is a natural generalization of the vertex cover number.  ... 
doi:10.1007/978-3-319-98355-4_20 fatcat:2fxb6b5f7jdgxkiwbeei3ytsxe

Reliability of Connections in Multilayer Networks under Shared Risk Groups and Costs Constraints

D. Coudert, F. Huc, F. Peix, M.-E. Voge
2008 2008 IEEE International Conference on Communications  
This problem has been proved NP-complete and hard to approximate in general, even when routing a single request.  ...  The reliability of a connection therefore depends on the number of SRRGs through which it is routed. Consequently, this number has to be minimized.  ...  The second criterion, related to the safety of the connections, is the average number of colors -SRGs -to which belongs a path routing one unit of flow, that is the sum of the numbers of colors on which  ... 
doi:10.1109/icc.2008.971 dblp:conf/icc/CoudertHPV08 fatcat:ewvzqzufirhavlwtqgwrfeenpu

On the Complexity of System Throughput Derivation for Static 802.11 Networks

Gene Cheung, Jeongkeun Lee, Sung-Ju Lee, Puneet Sharma
2010 IEEE Communications Letters  
assignment, association control and transmission scheduling, is NP-hard and hard to approximate in polynomial time.  ...  We show that for a particular network setting, the derivation of the system throughput (where system throughput is aggregate throughput of all clients or maxmin throughput), for any combination of channel  ...  Consider now the general case where the number of nodes in is and the chromatic number (minimum number of colors to color a graph) of is , where ≥ ≥ +1.  ... 
doi:10.1109/lcomm.2010.081910.100835 fatcat:xbimcezcavcjrd2ua4pio3pyfm

On Coloring Resilient Graphs [article]

Jeremy Kun, Lev Reyzin
2014 arXiv   pre-print
We prove lower bounds on the NP-hardness of coloring resiliently colorable graphs, and provide an algorithm that colors sufficiently resilient graphs.  ...  We introduce a new notion of resilience for constraint satisfaction problems, with the goal of more precisely determining the boundary between NP-hardness and the existence of efficient algorithms for  ...  In approximating the chromatic number of a general graph, the first results were of Garey and Johnson, giving a performance guarantee of O(n/ log n) colors [18] and proving that it is NP-hard to approximate  ... 
arXiv:1402.4376v2 fatcat:pfyvaddu7bfdjnrjnoosntvq4e

Solving Over-Constrained Problems with SAT [chapter]

Josep Argelich, Felip Manyà
2005 Lecture Notes in Computer Science  
We first define a clausal form formalism that deals with blocks of clauses and distinguish between hard and soft blocks.  ...  We present a new generic problem solving approach for overconstrained problems based on Max-SAT.  ...  The parameters of the generator are: number of vertices (n), optimum number of colors to get a valid coloring (k), and number of colors we use to color the graph (c).  ... 
doi:10.1007/11564751_76 fatcat:unin35gvyvgfzhjnu6mseckzwe

Refining the phase transition in combinatorial search

Tad Hogg
1996 Artificial Intelligence  
These are shown to reduce the variation in some cases. This also provides further insight into the nature of intrinsically hard search problems.  ...  These studies show that a simple parameter describing the problem structure predicts the difficulty of solving the problem, on average.  ...  Williams contributed to many helpful discussions and the development of the search programs used to produce these results.  ... 
doi:10.1016/0004-3702(95)00050-x fatcat:s4x5fisnlzao5c2wfvjkiqts3y

Lightpath routing for maximum reliability in optical mesh networks

Shengli Yuan, Saket Varma, Jason P. Jue
2008 Journal of Optical Networking  
We prove the problems to be NP-hard. We then extend the results to the general problems where the failure probabilities of the SRLGs may differ.  ...  We formulate the problems as minimum-color lightpath problems. By minimizing the number of colors on the lightpaths, the failure probability of the lightpaths can be minimized.  ...  Once the NP-hardness of the problems has been proved for this special case, we discuss the general case in which the SRLGs have different failure probabilities.  ... 
doi:10.1364/jon.7.000449 fatcat:vxil632rxffp7c4swit5ygk2mi

Generating hard and diverse test sets for NP-hard graph problems

Laura A. Sanchis
1995 Discrete Applied Mathematics  
We present algorithms for efficiently generating test instances for some NP-hard graph problems in such a way that the sets of instances generated can be shown to be both diverse and computationally hard  ...  The techniques used involve combining extremal graph theory results with NP-hardness reductions. ' Part of this work was done while the author was an AT&T Bell Labs Scholar at the University of Rochester  ...  Acknowledgements The author is grateful to Mark Fulk and Mandayam Srinivas for many useful discussions.  ... 
doi:10.1016/0166-218x(93)e0140-t fatcat:y55esbkdfnckrpuyhld5sa3aii
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