Computational Symposium on Graph Coloring and its Generalizations
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Transcript of Computational Symposium on Graph Coloring and its Generalizations
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Computational Symposium on Graph Coloring and its
Generalizations
Review by Michael Trick
Carnegie Mellon
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
What is a Computational Symposium?
• Invitation to present work on computational issues for a particular problem domain
• Not limited to any particular computational approach
• Papers can be a mix of instance generators, codes, heuristics, computational comparisons, etc.
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Goals for Symposium
• Participants– Provide resources to ease computational work
• Instances, bibliographies, comparison codes
– Provide outlet for computational work– Let results be greater than sum of parts
• Field– Give a snapshot of “state of the art”– Provide insights generalizable to other domains
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Graph Coloring and its Generalizations
• Graph ColoringGraph:
Assign colors to nodesDifferent colors at end of each edge
Minimize number of colors used
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Generalization: Multicoloring
1 2
2
1
2
2
Value on node Number of colors to assign
All colors must differ around edge
Easy to convert to regular coloring: more effective ways?
Objective: minimize number of colors used
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Generalization: Bandwidth
21
32
4
12
2
1
3
51
6
3
Values on edges: required difference in colors
Colors in range 1..k
Absolute value of difference in colors at least edge value
Objective: minimize k (sometimes number of different colors)
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
2
2
2
2
Generalization: Bandwidth plus Multicoloring
21
32
4
12
2
1
3
51
2
76
3
6
8
Values on both edges and nodes: bandwidth and multicoloring
Minimize maximum color value
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Why Graph Coloring?
• Useful in a number of applications– Register Allocation– Frequency assignment– Timetabling– Combinatorial designs– …
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Why Graph Coloring?
• Lots of algorithmic choices– IP, CP, hybrid, combinatorial bounds, heuristics,
etc. etc.
• No current clear winner• Accessible small instances (compare viz.
TSP)• Part of DIMACS Challenge (1993) with
published results in 1996– Can repeat instances, and determine advances
in state-of-the art
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Participation
• Open for any work in this area– Instance generators– Exact algorithms
• Constraint, integer, semidefinite, nonlinear approaches
– Heuristic Methods• Metaheuristics (tabu, simulated annealing, genetic
algorithms, ant systems), incomplete methods
– Applications and Instances– Evaluation of Methods
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Process
• Initial announcement January, 2002 to all standard electronic outlets
• Mailing list set up for communication: 60 subscribers
• Instances collected (approx 80 for coloring)• Papers/extended abstracts due mid July• Presentations September 8, just before CP
2002
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
PresentationsInstance Generators• Toward Ordered Generation of Exceptionally Hard Instances for Graph 3-Colorability, Mizuno and
Nishihara• Graph Coloring in the Estimation of Mathematical Derivatives, Hossain and Steihaug• 2+p-COL, Walsh• Completing Quasigroups or Latin Squares: A Structured Graph Coloring Problem, Gomes and Shmoys
Exact Methods• Vertex Coloring by Multistage Branch and Bound, Caramia and Dell'Olmo• Another Look at Graph Coloring via Propositional Satisfiability, Van Gelder• A Branch-and-Cut Algorithm for Graph Coloring, Mendez Diaz and Zabala
Genetic Algorithms and Ant Systems• A New Genetic Graph Coloring Heuristic, Croitoru, Luchian, Gheorghies, and Apetrei• Adaptive Memory Algorithms for Graph Coloring, Galinier, Hertz, and Zufferey • An Ant System for Coloring Graphs, Bui and Patel Local Search and Simulated Annealing• Coloring Graphs with a General Heuristic Search Engine, Phan and Skiena• A Combined Algorithm for Graph Coloring in Register Allocation, Allen, Kumaran, and Liu• An Application of Iterated Local Search to Graph Coloring Problem, Chiarandini and Stuetzle• Constrained Bandwidth Multicoloration Neighborhoods, Prestwich
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Some General Conclusions
• New applications for graph coloring continue to be found– Matrix decomposition in estimating
mathematical derivatives uses graph coloring to determine a good partition of rows and columns to exploit sparcity (Hossain and Steihaug)
– Completing latin squares can create very difficult coloring instances (Gomes and Shmoys)
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Instance Generation
• 3-colorability can generate hard instances– Non 3-color without
(Mizono and Nishihara)
– Instances that mix 2-coloring (easy) with 3-coloring (Walsh): interesting phase transition issues
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Computational Results
• Easy to compare to 1996 papers.• All solved a standard instance with a
standard code.• Computers are faster but not
excessively so:2002: 16, 24, 3861996: 86, 189, 734, 2993
• Can standardize times to get rough comparisons
Time in seconds
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Exact methods
• 3 different approaches:– Caramia and Dell’Olmo: Combinatorial
branch and bound– Van Gelder: translation to SAT– Mendez Diaz and Zabala: Branch and
Cut
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Exact Methods
• Great improvements since 1996– 125 node, .5 density random graphs now
solvable (before only 80)– Specific test instances solved for first
time: myciel6, leighton5x
• No one method best: all show promise
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Heuristic Methods
• Lots of Variety– Croitoru et al.: Genetic Algorithms– Galanier et al.: Adaptive Memory – Bui and Patel: Ant Systems– Phan and Skiena: Simulated Annealing– Allen et al.: Randomized Greedy and
restarts– Chiarandini and Stuetzle: Iterated Local
Search
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Winner?
• No clear winner– Difficult to compare (different time limits,
instances solved)– Approach of Bui and Patel generally
successful
• Aggregate advance over 1996– More variety, interesting methods for
combining solutions– Better solutions more consistently for a
number of graphs (leighton, etc.)
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Multicoloring and Bandwidth
• Surprisingly not well studied– Prestwich formulated as ILP and
experimented with incomplete search methods
– Phan and Skiena adapted their general search methods (simulated annealing, multiple start methods)
• Instance class may be limited
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Surprises
• Heuristics generally continue to do poorly on relatively small random graphs (gap of 18 versus 12 on 125 node instance)
• Lack of interest in mulicoloring and bandwidth problems
• No pure CP approaches (global constraints, propagation, etc.) and little IP methods
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Future Plans
• We aren’t done yet!
• Need to– Add instances (particularly hard 3-
coloring instances) to suite, remove “easy” instances
– Determine suitable testing procedure(s) for heuristics
– Get word out wider
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Future Plans
• Refereed volume: Call for Papers next year– Not to late to work on this– Current papers updated based on Symposium
results
• Possible mini-Symposium at next year’s Mathematical Programming Symposium– August, Copenhagen– Goal is to attract wide variety of papers in area
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Need from You
• Instances– Particularly for generalizations
• Papers– Particularly for generalizations
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Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Keeping in Touch
• http://mat.gsia.cmu.edu/COLOR02
• Thanks to co-organizers Anuj Mehrotra and David Johnson and program members Ed Sewell and Joe Culberson