computational symposium on graph coloring and its generalizations review by michael trick carnegie...

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.


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


Graph Coloring and its Generalizations

• Graph ColoringGraph:

Assign colors to nodesDifferent colors at end of each edge

Minimize number of colors used


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


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)


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


Why Graph Coloring?

• Useful in a number of applications– Register Allocation– Frequency assignment– Timetabling– Combinatorial designs– …


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


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


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


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


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)


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 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


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


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


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


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.)


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


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


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


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


Need from You

• Instances– Particularly for generalizations

• Papers– Particularly for generalizations


Keeping in Touch

• http://mat.gsia.cmu.edu/COLOR02

• [email protected]

• Thanks to co-organizers Anuj Mehrotra and David Johnson and program members Ed Sewell and Joe Culberson

computational symposium on graph coloring and its generalizations review by michael trick carnegie...

Documents

computational symposium

colors colors

computational issues

computational comparisons

regular coloring

computational work instances

symposium participants

generalizations review