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Greedy, Prohibition, and Reactive Heuristics for Graph Partitioning
April 1999 (vol. 48 no. 4)
pp. 361-385

Abstract—New heuristic algorithms are proposed for the Graph Partitioning problem. A greedy construction scheme with an appropriate tie-breaking rule (Min-Max-Greedy) produces initial assignments in a very fast time. For some classes of graphs, independent repetitions of Min-Max-Greedy are sufficient to reproduce solutions found by more complex techniques. When the method is not competitive, the initial assignments are used as starting points for a prohibition-based scheme, where the prohibition is chosen in a randomized and reactive way, with a bias towards more successful choices in the previous part of the run. The relationship between prohibition-based diversification (Tabu Search) and the variable-depth Kernighan-Lin algorithm is discussed. Detailed experimental results are presented on benchmark suites used in the previous literature, consisting of graphs derived from parametric models (random graphs, geometric graphs, etc.) and of “real-world” graphs of large size. On the first series of graphs, a better performance for equivalent or smaller computing times is obtained, while, on the large “real-world” instances, significantly better results than those of multilevel algorithms are obtained, but for a much larger computational effort.

[1] E.H.L. Aarts, J.H.M. Korst, and P.J. Zwietering, “Deterministic and Randomized Local Search,” Mathematical Perspectives on Neural Networks, M. Mozer, P. Smolensky, and D. Rumelhart, eds. Hillsdale, N.J.: Lawrence Erlbaum, to appear.
[2] C.J. Alpert and A.B. Kahng, “Recent Directions in Netlist Partitioning: A Survey,” VLSI J., vol. 19, nos. 1-2, pp. 1–81, 1995.
[3] F. Barahona, M. Grötschel, M. Jünger, and G. Reinelt, “An Application of Combinatorial Optimization to Statistical Physics and Circuit Layout Design,” Operations Research, vol. 36, pp. 493-513 1988.
[4] S.T. Barnard and H.D. Simon, “Fast Multilevel Implementation of Recursive Spectral Bisection for Partitioning Unstructured Problems,” Concurrency: Practice and Experience vol. 6, no. 2, pp. 101-117 1994.
[5] R. Battiti and A.A. Bertossi, “Differential Greedy for the 0-1 Equicut Problem,” Network Design: Connectivity and Facilities Location, D.Z. Du and P.M. Pardalos, eds., pp. 3-22, 1997.
[6] R. Battiti and M. Protasi, “Reactive Local Search for the Maximum Clique Problem,” Algorithmica, to appear.
[7] R. Battiti and G. Tecchiolli, “The Reactive Tabu Search,” ORSA J. Computing, vol. 6, no. 2, pp. 126-140, 1994.
[8] R. Battiti, “Reactive Search: Toward Self-tuning Heuristics,” Modern Heuristic Search Methods, V.J. Rayward-Smith, ed., chapter 4, pp. 61-83. John Wiley and Sons Ltd., 1996.
[9] R.B. Boppana, “Eigenvalues and Graph Bisection: An Average-Case Analysis,” Proc. 28th Symp. Foundations of Computer Science, pp. 280-285, 1987.
[10] T.N. Bui, S. Chaudhuri, F.T. Leighton, and M. Sipser, "Graph Bisection Algorithms with Good Average Case Behavior," Combinatorica, vol. 7, no. 2, pp. 171-191, 1987.
[11] T.N. Bui and C. Jones, "Finding Good Approximate Vertex and Edge Partitions is NP-Hard," Information Processing Letters, vol. 42, pp. 153-159, 1992.
[12] T.N. Bui and C. Jones, “A Heuristic for Reducing Fill in Sparse Matrix Factorization,” Proc. Sixth SIAM Conf. Parallel Processing for Scientific Computing, pp. 445-452, 1993.
[13] T.N. Bui and A. Peck, "Partitioning Planar Graphs," SIAM J. Computing, vol. 21, no. 2, pp. 203-215, 1992.
[14] T.N. Bui and B.R. Moon, Genetic Algorithm and Graph Partitioning IEEE Trans. Computers, vol. 45, no. 7, pp. 841-855, July 1996.
[15] T.-S. Chiang and Y. Chow, “On the Convergence Rate of Annealing Processes,” SIAM J. Control and Optimization, vol. 26, no. 6, pp. 1,455-1,470, 1988.
[16] M. Dell'Amico and F. Maffioli, “A New Tabu Search Approach to the 0-1 Equicut Problem.” Meta-Heuristics 1995: The State of the Art, pp. 361-377. Kluwer Academic, 1996.
[17] R. Diekmann, B. Monien, and R. Preis, “Using Helpful Sets to Improve Graph Bisections,” Interconnection Networks and Mapping and Scheduling Parallel Computations, D.F. Hsu, A.L. Rosenberg, and D. Sotteau, eds., pp. 57-73, 1995.
[18] I.S. Duff, R. Grimes, and J. Lewis, “Sparse Matrix Test Problems,” ACM Trans. Mathematical Software, vol. 15, pp. 1–14, Mar. 1989.
[19] A. Dunlop and B. Kernighan, “A Procedure for Placement of Standard-Cell VLSI Circuits,” IEEE Trans. Computer-Aided Design of Integrated Circuits and Systems, vol. 1, pp. 92-98, 1985.
[20] C. Farhat, “A Simple and Efficient Automatic FEM Domain Decomposer,” Computers and Structures, vol. 28, no. 5, pp. 579-602, 1988.
[21] C. Farhat, S. Lanteri, and H.D. Simon, “TOP/DOMDEC—A Software Tool for Mesh Partitioning and Parallel Processing,” J. Computing Systems in Eng., vol. 6, no. 1, pp. 13-26, 1995.
[22] A.G. Ferreira and J. Zerovnik, “Bounding the Probability of Success of Stochastic Methods for Global Optimization,” Computer Math. Applications, vol. 25, pp. 1-8, 1993.
[23] C.M. Fiduccia and R.M. Mattheyses, "A Linear Time Heuristic for Improving Network Partitions," Proc. 19th Design Automation Conf., pp. 175-181, 1982.
[24] G.C. Fox, “A Review of Automatic Load Balancing an Decomposition Methods for the Hypercube,” Numerical Algorithms for Modern Parallel Computer Architectures, M. Schultz, ed., pp. 63-76. New York: Springer-Verlag, 1988.
[25] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness.New York: W.H. Freeman, 1979.
[26] J.R. Gilbert, G.L. Miller, and S.H. Teng, “Geometric Mesh Partitioning: Implementation and Experiments,” Proc. Ninth Int'l Parallel Processing Symp., pp. 418-427, Apr. 1995.
[27] F. Glover, “Tabu Search—Part I,” ORSA J. Computing, vol. 1, no. 3, pp. 190-260, 1989.
[28] S.W. Hammond, “Mapping Unstructured Grid Computations to Massively Parallel Computers,” Technical Report 92.14, RIACS, Nasa Ames, 1992.
[29] P. Hansen and B. Jaumard, "Algorithms for the maximum satisfiability problem," Computing, vol. 44, pp. 279-303, 1990.
[30] B. Hendrickson and R. Leland, “The Chaco User's guide,” Technical Report SAND94-2692, SANDIA Nat'l Laboratories, Albuquerque, N.M., 1994.
[31] B. Hendrickson and R. Leland, “A Multilevel Algorithm for Partitioning Graphs,” Technical Report SAND93-1301, SANDIA Nat'l Laboratory, 1993.
[32] B. Hendrickson and R. Leland, "An Improved Spectral Graph Partitioning Algorithm for Mapping Parallel Computations," SIAM J. Scientific Computation, vol. 16, no. 2, Mar. 1995, pp. 452-469.
[33] D.S. Johnson, C. Aragon, L. McGeoch, and C. Schevon, "Optimization by Simulated Annealing: An Experimental Evaluation, Part 1, Graph Partitioning," Operations Research, vol. 37, pp. 865-892, 1989.
[34] G. Karypis and V. Kumar, “A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs,” Technical Report 95-035, Univ. of Minnesota, Dept. of Computer Science, 1995.
[35] B. Kernighan and S. Lin, “An Efficient Heuristic Procedure for Partitioning Graphs,” Bell Systems Technical J., vol. 49, pp. 291-307, Feb. 1970.
[36] S. Kirkpatrick, “Optimization by Simulated Annealing: Quantitative Studies,” J. Statistical Physics, vol. 34, pp. 975-986, 1984.
[37] S. Kirkpatrick, C. D. Gelatt Jr., and M.P. Vecchi, “Optimization by Simulated Annealing,” Science, vol. 220, no. 4,598, pp. 671-680, May 1983.
[38] M. Laguna, T.A. Feo, and H.C. Elrod, “A Greedy Randomized Adaptive Search Procedure for the Two-Partition Problem,” Operations Research, vol. 42, pp. 677-687, 1994.
[39] F.T. Leighton and S. Rao, "An Approximate Max-Flow Min-Cut Theorem for Uniform Multicommodity Flow Problems with Applications to Approximation Algorithms," Proc. 29th Symp. Foundations of Computer Science, pp. 422-431, 1988.
[40] S. Lin, “Computer Solutions of the Traveling Salesman Problems,” BSTJ, vol. 44, no. 10, pp. 2,245-2,269, 1965.
[41] O.C. Martin and S.W. Otto, “Partitioning of Unstructured Meshes for Load Balancing,” Technical Report CSE-94-017, Oregon Graduate Inst. of Science and Tech nology, 1994. Concurrency: Practice and Experience,to appear.
[42] C.C. McGeoch, “Toward an Experimental Method for Algorithm Simulation,” INFORMS J. Computing, vol. 8, no. 1, pp. 1-28, 1996.
[43] B. Monien and R. Diekmann, “A Local Graph Partitioning Heuristic Meeting Bisection Bounds,” Proc. Eighth SIAM Conf. Parallel Processing for Scientific Computing, 1997.
[44] F. Pellegrini and J. Roman, “SCOTCH: A Software Package for Static Mapping by Dual Recursive Bipartitioning of Process and Architecture Graphs,” Lecture Notes in Computer Science, vol. 1067, pp. 493-499, 1996.
[45] H. Pirkul and E. Rolland, "New heuristic solution procedures for the uniform graph partitioning problem: Extensions and evaluation." Computers and Operations Research, Oct. 1994.
[46] A. Pothen, H. Simon, and K. Liou, "Partitioning Sparse Matrices with Eigenvectors of Graphs," SIAM J. Matrix Analysis and Application, vol. 11, pp. 430-352, July 1990.
[47] R. Preis and R. Diekmann, “The Party Partitioning Library, User Guide,” Technical Report TR-RSFB-96-024, Univ. of Paderborn, Germany, 1996.
[48] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling”, Numerical Recipes in C. Cambridge Univ. Press, 1988.
[49] E. Rolland and H. Pirkul, “Heuristic Solution Procedures for the Graph Partitioning Problem,” Computer Science and Operations Research: New Developments in Their Interfaces, O. Balci, ed. Oxford: Pergamon Press, 1992.
[50] E. Rolland, H. Pirkul, and F. Glover, “A Tabu Search for Graph Partitioning,” Annals of Operations Research, Metaheuristics in Combinatorial Optimization, vol. 63, 1996.
[51] R. Rutenbar, "Simulated Annealing Algorithms: An Overview," IEEE Circuit and Devices Magazine, pp. 19-26, 1989.
[52] Y.G. Saab, ”A Fast and Robust Network Bisection Algorithm,” IEEE Trans. Computers, vol. 44, no. 7, July 1995.
[53] L.A. Sanchis, “Multiple Way Network Partitioning,” IEEE Trans. Computers, Vol. 38, No. 1, Jan. 1989, pp. 62‐81.
[54] C. Sechen and A. Sangiovanni-Vincentelli, "Timberwolf3.2: A New Standard Cell Placement and Global Routing Package," Proc. 23rd ACM/IEEE Design Automation Conf., pp. 432-439, 1986.
[55] H.D. Simon, “Partitioning of Unstructured Problems for Parallel Processing,” Computing Systems in Eng., vol. 2, pp. 135-148, 1991.
[56] K. Steiglitz and P. Weiner, “Some Improved Algorithms for Computer Solution of the Traveling Salesman Problem,” Proc. Sixth Allerton Conf. Circuit and System Theory, Urbana, Illi nois, pp. 814-821, 1968.
[57] W. Sun and C. Sechen, "Efficient and Effective Placement for Very Large Circuits," IEEE Trans. Computer-Aided Design, vol. 14, no. 3, pp. 349-359, 1995.
[58] E. Taillard, “Robust Taboo Search for the Quadratic Assignment Problem,” Parallel Computing, vol. 17, pp. 443-455, 1991.
[59] C. Walshaw and M. Berzins, “Dynamic Load-Balancing for PDE Solvers on Adaptive Unstructured Meshes,” Concurrency: Practice and Experience, vol. 7, no. 1, pp. 17-28, 1995.
[60] R. Williams, “Unification of Spectral and Inertial Bisection,” technical report, California Inst. of Tech nology, 1994. Available at:http://www.cacr.caltech. edu/~roypapers/.

Index Terms:
Graph bisection, graph partitioning, heuristic algorithms, iterative improvement, local search, reactive search.
Citation:
Roberto Battiti, Alan Albert Bertossi, "Greedy, Prohibition, and Reactive Heuristics for Graph Partitioning," IEEE Transactions on Computers, vol. 48, no. 4, pp. 361-385, April 1999, doi:10.1109/12.762522
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