Improving graph partitions using submodular functions

We investigate into the role of submodular functions in designing new heuristics and approximate algorithms to some NP-hard problems arising in the field of VLSI Design Automation. In particular, we design and implement efficient heuristic for improving a bipartition of a graph in the sense of ratioCut (Discrete Appl. Math. 90 (1999) 3; 29th Annual Symposium on Foundations of Computer Science, 1988, p. 422). We also design an approximate algorithm for another NP-hard problem which is a dual of the well-known NP-hard problem of finding a densest k-subgraph of a graph (see J. Algorithms 34 (2000) 203; Proceedings of the 34th Annual Symposium on Foundations of Computer Science, 1993, p. 692). Our algorithms are based on submodular function and are implementable in polynomial time using efficient network flow based subroutines. To the best of our knowledge our algorithms are the first ones to use submodular functions based approach for the problems considered here. We also describe the experimental results which provide the evidence of our heuristic for improving the ratioCut.

[1]  Dan Gusfield,et al.  Computing the Strength of a Graph , 1991, SIAM J. Comput..

[2]  H. Narayanan.  Submodular functions and electrical networks , 1997 .

[3]  J. Picard,et al.  Selected Applications of Minimum Cuts in Networks , 1982 .

[4]  Sachin B. Patkar,et al.  Approximation Algorithms for Min-k-Overlap Problems Using the Principal Lattice of Partitions Approach , 1994, J. Algorithms.

[5]  Frank Thomson Leighton,et al.  An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[6]  Sachin B. Patkar,et al.  Fast On-Line/Off-Line Algorithms for Optimal Reinforcement of a Network and Its Connections with Principal Partition , 2000, FSTTCS.

[7]  Charles J. Alpert,et al.  Spectral Partitioning: The More Eigenvectors, The Better , 1995, 32nd Design Automation Conference.

[8]  Robert E. Tarjan,et al.  A Fast Parametric Maximum Flow Algorithm and Applications , 1989, SIAM J. Comput..

[9]  Thomas Lengauer,et al.  Combinatorial algorithms for integrated circuit layout , 1990, Applicable theory in computer science.

[10]  Werner Dinkelbach.  On Nonlinear Fractional Programming , 1967 .

[11]  Jack Edmonds,et al.  Submodular Functions, Matroids, and Certain Polyhedra , 2001, Combinatorial Optimization.

[12]  Hisao Tamaki,et al.  Greedily Finding a Dense Subgraph , 2000, J. Algorithms.

[13]  Dorit S. Hochbaum,et al.  Polynomial algorithm for the k-cut problem , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[14]  Harold N. Gabow,et al.  Algorithms for graphic polymatroids and parametric s-Sets , 1995, SODA '95.

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

[16]  Eddie Cheng,et al.  A Faster Algorithm for Computing the Strength of a Network , 1994, Inf. Process. Lett..

[17]  Satoru Fujishige,et al.  Submodular functions and optimization , 1991 .