ILP-Based Local Search for Graph Partitioning

Computing high-quality graph partitions is a challenging problem with numerous applications. In this paper, we present a novel meta-heuristic for the balanced graph partitioning problem. Our approach is based on integer linear programs that solve the partitioning problem to optimality. However, since those programs typically do not scale to large inputs, we adapt them to heuristically improve a given partition. We do so by defining a much smaller model that allows us to use symmetry breaking and other techniques that make the approach scalable. For example, in Walshaw's well-known benchmark tables we are able to improve roughly half of all entries when the number of blocks is high.

[1]  Peter Sanders,et al.  Optimized Hybrid Parallel Lattice Boltzmann Fluid Flow Simulations on Complex Geometries , 2012, Euro-Par.

[2]  Andrew B. Kahng,et al.  Spectral Partitioning with Multiple Eigenvectors , 1999, Discret. Appl. Math..

[3]  Michael Armbruster,et al.  Branch-and-Cut for a Semidefinite Relaxation of Large-scale Minimum Bisection Problems , 2007 .

[4]  Vipin Kumar,et al.  Graph partitioning for high-performance scientific simulations , 2003 .

[5]  Franz Rendl,et al.  Graph partitioning using linear and semidefinite programming , 2003, Math. Program..

[6]  Peter Widmayer,et al.  An $\mathcal{O}(n^4)$ Time Algorithm to Compute the Bisection Width of Solid Grid Graphs , 2011, ESA.

[7]  Daniel Delling,et al.  Better Bounds for Graph Bisection , 2012, ESA.

[8]  A. Land,et al.  An Automatic Method for Solving Discrete Programming Problems , 1960, 50 Years of Integer Programming.

[9]  Peter Widmayer,et al.  An $$O(n^4)$$O(n4) Time Algorithm to Compute the Bisection Width of Solid Grid Graphs , 2014, Algorithmica.

[10]  Andrew B. Kahng,et al.  Recent directions in netlist partitioning: a survey , 1995, Integr..

[11]  Christian Schulz,et al.  ILP-Based Local Search for Graph Partitioning , 2020 .

[12]  Peter Sanders,et al.  Recent Advances in Graph Partitioning , 2013, Algorithm Engineering.

[13]  Peter Sanders,et al.  Distributed Time-Dependent Contraction Hierarchies , 2010, SEA.

[14]  Franz Rendl,et al.  Solving Graph Bisection Problems with Semidefinite Programming , 2000, INFORMS J. Comput..

[15]  Laurence A. Wolsey,et al.  The node capacitated graph partitioning problem: A computational study , 1998, Math. Program..

[16]  A. George.  Nested Dissection of a Regular Finite Element Mesh , 1973 .

[17]  Charles M. Fiduccia,et al.  A linear-time heuristic for improving network partitions , 1988, 25 years of DAC.

[18]  Curt Jones,et al.  Finding Good Approximate Vertex and Edge Partitions is NP-Hard , 1992, Inf. Process. Lett..

[19]  Michael Armbruster,et al.  A Comparative Study of Linear and Semidefinite Branch-and-Cut Methods for Solving the Minimum Graph Bisection Problem , 2008, IPCO.

[20]  Christian Schulz,et al.  Graph Partitioning: Formulations and Applications to Big Data , 2019, Encyclopedia of Big Data Technologies.

[21]  Peter Sanders,et al.  Distributed Evolutionary Graph Partitioning , 2011, ALENEX.

[22]  William W. Hager,et al.  An exact algorithm for graph partitioning , 2013, Math. Program..

[23]  Andrew V. Goldberg,et al.  Exact Combinatorial Branch-and-Bound for Graph Bisection , 2012, ALENEX.

[24]  Dorothea Wagner,et al.  Partitioning graphs to speedup Dijkstra's algorithm , 2007, ACM J. Exp. Algorithmics.

[25]  S.,et al.  An Efficient Heuristic Procedure for Partitioning Graphs , 2022 .

[26]  Dennis Luxen,et al.  Candidate Sets for Alternative Routes in Road Networks , 2012, SOCS.

[27]  Meinolf Sellmann,et al.  Multicommodity Flow Approximation Used for Exact Graph Partitioning , 2003, ESA.

[28]  Chris Walshaw,et al.  A Combined Evolutionary Search and Multilevel Optimisation Approach to Graph-Partitioning , 2004, J. Glob. Optim..

[29]  Ariel Felner,et al.  Finding optimal solutions to the graph partitioning problem with heuristic search , 2005, Annals of Mathematics and Artificial Intelligence.

[30]  Andrew V. Goldberg,et al.  Customizable Route Planning , 2011, SEA.

[31]  Ailsa H. Land,et al.  An Automatic Method of Solving Discrete Programming Problems , 1960 .

[32]  Peter Sanders,et al.  Think Locally, Act Globally: Highly Balanced Graph Partitioning , 2013, SEA.

[33]  David A. Bader,et al.  Benchmarking for Graph Clustering and Partitioning , 2014, Encyclopedia of Social Network Analysis and Mining.

[34]  Matthias Hein,et al.  Beyond Spectral Clustering - Tight Relaxations of Balanced Graph Cuts , 2011, NIPS.

[35]  C. Walshaw.  JOSTLE : parallel multilevel graph-partitioning software – an overview , 2008 .

[36]  Dorothea Wagner,et al.  A Multi-Level Framework for Bisection Heuristics , 2009 .

[37]  Norbert Sensen,et al.  Lower Bounds and Exact Algorithms for the Graph Partitioning Problem Using Multicommodity Flows , 2001, ESA.

[38]  Vipin Kumar,et al.  A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs , 1998, SIAM J. Sci. Comput..

[39]  Peter Sanders,et al.  Engineering Multilevel Graph Partitioning Algorithms , 2010, ESA.

[40]  Philippe Galinier,et al.  An efficient memetic algorithm for the graph partitioning problem , 2011, Ann. Oper. Res..