1. Computational Symposium on Graph Coloring and its Generalizations Review by Michael Trick Carnegie Mellon

2. 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.

3. 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

4. Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002 Graph Coloring and its Generalizations Graph Coloring Graph: Assign colors to nodes Different colors at end of each edge Minimize number of colors used

5. 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

6. Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002 Generalization: Bandwidth 2 1 3 2 4 1 2 2 1 3 5 1 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)

7. Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002 2 2 2 2 Generalization: Bandwidth plus Multicoloring 2 1 3 2 4 1 2 2 1 3 5 1 2 7 6 3 6 8 Values on both edges and nodes: bandwidth and multicoloring Minimize maximum color value

8. 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 –…

9. 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

10. 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

11. 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

12. Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002 Presentations Instance 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

13. 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)

14. 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

15. 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, 386 1996: 86, 189, 734, 2993 Can standardize times to get rough comparisons Time in seconds

16. 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

17. 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

18. 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

19. 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.)

20. 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

21. 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

22. 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

23. 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

24. Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002 Need from You Instances –Particularly for generalizations Papers –Particularly for generalizations

25. Computational Symposium on Graph Coloring and Generalizations, CP2002, Sept. 2002 Keeping in Touch http://mat.gsia.cmu.edu/COLOR02 trick@cmu.edu Thanks to co-organizers Anuj Mehrotra and David Johnson and program members Ed Sewell and Joe Culberson