Bounds on the max and min bisection of random cubic and random 4-regular graphs

In this paper, we present a randomized algorithm to compute the bisection width of cubic and 4-regular graphs. The analysis of the proposed algorithms on random graphs provides asymptotic upper bounds for the bisection width of random cubic and random 4-regular graphs with n vertices, giving upper bounds of 0.174039n for random cubic, and of 0.333333n for random 4-regular. We also obtain asymptotic lower bounds for the size of the maximum bisection, for random cubic and random 4-regular graphs with n vertices, of 1.32697n and 1.66667n, respectively. The randomized algorithms are derived from initial greedy algorithm and their analysis is based on the differential equation method.

[1]  N. Wormald.  The differential equation method for random graph processes and greedy algorithms , 1999 .

[2]  Robert Elsässer,et al.  New Spectral Lower Bounds on the Bisection Width of Graphs , 2000, WG.

[3]  Marek Karpinski,et al.  Approximation Algorithms for MAX-BISECTION on Low Degree Reg ular Graphs and Planar Graphs , 2000, Electron. Colloquium Comput. Complex..

[4]  Béla Bollobás,et al.  Random Graphs , 1985 .

[5]  Frank Thomson Leighton,et al.  Graph Bisection Algorithms with Good Average Case Behavior , 1984, FOCS.

[6]  Marek Karpinski,et al.  Polynomial time approximation schemes for dense instances of NP-hard problems , 1995, STOC '95.

[7]  Robert Krauthgamer,et al.  A Polylogarithmic Approximation of the Minimum Bisection , 2002, SIAM J. Comput..

[8]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[9]  L. S. Mel'nikov,et al.  On Bounds of the Bisection Width of Cubic Graphs , 1992 .

[10]  Michael Langberg,et al.  The RPR2 rounding technique for semidefinite programs , 2001, J. Algorithms.

[11]  Robert Krauthgamer,et al.  A polylogarithmic approximation of the minimum bisection , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[12]  Robert Elsässer,et al.  New spectral lower bounds on the bisection width of graphs , 2000, Theor. Comput. Sci..

[13]  M. Fiedler.  A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory , 1975 .

[14]  Nicholas C. Wormald,et al.  Analysis of greedy algorithms on graphs with bounded degrees , 2003, Discret. Math..

[15]  JOSEP DÍAZ,et al.  A survey of graph layout problems , 2002, CSUR.

[16]  Marek Karpinski,et al.  A note on approximating Max-Bisection on regular graphs , 2001, Inf. Process. Lett..

[17]  N. Wormald,et al.  Models of the , 2010 .

[18]  Klaus Jansen,et al.  Polynomial Time Approximation Schemes for MAX-BISECTION on Planar and Geometric Graphs , 2001, SIAM J. Comput..

[19]  Maria J. Serna,et al.  Bisection of Random Cubic Graphs , 2002, RANDOM.

[20]  Béla Bollobás,et al.  The Isoperimetric Number of Random Regular Graphs , 1988, Eur. J. Comb..