Abstract
We introduce new lower bounds for the minimum graph bisection problem. Within a branch-and-bound framework, they enable the solution of a wide variety of instances with tens of thousands of vertices to optimality. Our algorithm compares favorably with the best previous approaches, solving long-standing open instances in minutes.
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References
Armbruster, M.: Branch-and-Cut for a Semidefinite Relaxation of Large-Scale Minimum Bisection Problems. PhD thesis, Technische Universität Chemnitz (2007)
Armbruster, M.: Graph Bisection and Equipartition (2007), http://www.tu-chemnitz.de/mathematik/discrete/armbruster/diss/
Armbruster, M., Fügenschuh, M., Helmberg, C., Martin, A.: A Comparative Study of Linear and Semidefinite Branch-and-Cut Methods for Solving the Minimum Graph Bisection Problem. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 112–124. Springer, Heidelberg (2008)
Bader, D.A., Meyerhenke, H., Sanders, P., Wagner, D.: 10th DIMACS Implementation Challenge: Graph Partitioning and Graph Clustering (2011), http://www.cc.gatech.edu/dimacs10/index.shtml
Brunetta, L., Conforti, M., Rinaldi, G.: A branch-and-cut algorithm for the equicut problem. Mathematical Programming 78, 243–263 (1997)
Budiu, M., Delling, D., Werneck, R.F.: DryadOpt: Branch-and-bound on distributed data-parallel execution engines. In: IPDPS, pp. 1278–1289 (2011)
Bui, T.N., Chaudhuri, S., Leighton, F., Sipser, M.: Graph bisection algorithms with good average case behavior. Combinatorica 7(2), 171–191 (1987)
Delling, D., Goldberg, A.V., Razenshteyn, I., Werneck, R.F.: Exact combinatorial branch-and-bound for graph bisection. In: ALENEX, pp. 30–44 (2012)
Ferreira, C.E., Martin, A., de Souza, C.C., Weismantel, R., Wolsey, L.A.: The node capacitated graph partitioning problem: A computational study. Mathematical Programming 81, 229–256 (1998)
Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of \(\mathcal{NP}\)-Completeness. W.H. Freeman and Company (1979)
Hager, W.W., Phan, D.T., Zhang, H.: An exact algorithm for graph partitioning. Mathematical Programming, 1–26 (2011)
Johnson, D.S., Aragon, C.R., McGeoch, L.A., Schevon, C.: Optimization by simulated annealing: an experimental evaluation; part I, graph partitioning. Operations Research 37(6), 865–892 (1989)
Karisch, S.E., Rendl, F., Clausen, J.: Solving graph bisection problems with semidefinite programming. INFORMS Journal on Computing 12, 177–191 (2000)
Karypis, G., Kumar, G.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Scientific Computing 20(1), 359–392 (1999)
Koch, T., Martin, A., Voß, S.: SteinLib: An updated library on Steiner tree problems in graphs. Technical Report 00-37, Konrad-Zuse-Zentrum Berlin (2000)
Land, A.H., Doig, A.G.: An automatic method of solving discrete programming problems. Econometrica 28(3), 497–520 (1960)
Rendl, F., Rinaldi, G., Wiegele, A.: Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Math. Programming 121, 307–335 (2010)
Sander, P.V., Nehab, D., Chlamtac, E., Hoppe, H.: Efficient traversal of mesh edges using adjacency primitives. ACM Trans. on Graphics 27, 144:1–144:9 (2008)
Sanders, P., Schulz, C.: Distributed evolutionary graph partitioning. In: ALENEX, pp. 16–29. SIAM (2012)
Sellmann, M., Sensen, N., Timajev, L.: Multicommodity Flow Approximation Used for Exact Graph Partitioning. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 752–764. Springer, Heidelberg (2003)
Sensen, N.: Lower Bounds and Exact Algorithms for the Graph Partitioning Problem Using Multicommodity Flows. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 391–403. Springer, Heidelberg (2001)
Soper, A.J., Walshaw, C., Cross, M.: The Graph Partitioning Archive (2004), http://staffweb.cms.gre.ac.uk/~c.walshaw/partition/
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Delling, D., Werneck, R.F. (2012). Better Bounds for Graph Bisection. In: Epstein, L., Ferragina, P. (eds) Algorithms – ESA 2012. ESA 2012. Lecture Notes in Computer Science, vol 7501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33090-2_36
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DOI: https://doi.org/10.1007/978-3-642-33090-2_36
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