Computational Symposium on Graph Coloring and its
Generalizations
Review by Michael Trick
Carnegie Mellon
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.
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
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
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
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)
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
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– …
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
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
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
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
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)
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
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
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
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
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
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.)
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
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
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
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
Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002
Need from You
• Instances– Particularly for generalizations
• Papers– Particularly for generalizations
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
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