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Computational Symposium on Graph Coloring and its Generalizations

Computational Symposium on Graph Coloring and its Generalizations. Review by Michael Trick Carnegie Mellon. What is a Computational Symposium?. Invitation to present work on computational issues for a particular problem domain Not limited to any particular computational approach

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Computational Symposium on Graph Coloring and its Generalizations

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

  2. 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. 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. 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. Generalization: Multicoloring Value on node Number of colors to assign 2 1 2 2 All colors must differ around edge 2 Objective: minimize number of colors used 1 Easy to convert to regular coloring: more effective ways?

  6. 5 1 3 6 3 1 Generalization: Bandwidth Values on edges: required difference in colors 4 2 Colors in range 1..k 1 2 3 Absolute value of difference in colors at least edge value 1 2 2 Objective: minimize k (sometimes number of different colors)

  7. 5 6 1 3 8 3 2 7 6 1 Generalization: Bandwidth plus Multicoloring Values on both edges and nodes: bandwidth and multicoloring 2 4 2 1 2 2 Minimize maximum color value 2 3 2 1 2 2

  8. Why Graph Coloring? • Useful in a number of applications • Register Allocation • Frequency assignment • Timetabling • Combinatorial designs • …

  9. 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. 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. 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. 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. 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. 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 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. 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. 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. 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. 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. 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. 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. 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. 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. Need from You • Instances • Particularly for generalizations • Papers • Particularly for generalizations

  25. 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

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