Algorithms for graph partitioning on the planted partition model

The NP-hard graph bisection problem is to partition the nodes of an undirected graph into two equal-sized groups so as to minimize the number of edges that cross the partition. The more general graph l-partition problem is to partition the nodes of an undirected graph into l equal-sized groups so as to minimize the total number of edges that cross between groups. We present a simple, linear-time algorithm for the graph l-partition problem and analyze it on a random \planted l-partition" model. In this model, the n nodes of a graph are partitioned into l groups, each of size n=l; two nodes in the same group are connected by an edge with some probability p, and two nodes in diierent groups are connected by an edge with some probability r < p. We show that if p ? r n ?1=2+ for some constant , then the algorithm nds the optimal partition with probability 1 ? exp(?n ()).

[1]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[2]  Brian W. Kernighan,et al.  An efficient heuristic procedure for partitioning graphs , 1970, Bell Syst. Tech. J..

[3]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[4]  R. M. Mattheyses,et al.  A Linear-Time Heuristic for Improving Network Partitions , 1982, 19th Design Automation Conference.

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

[6]  T. N. Bui.  ON BISECTING RANDOM GRAPHS , 1983 .

[7]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

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

[9]  Martin E. Dyer,et al.  Fast solution of some random NP-hard problems , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[10]  Ravi B. Boppana,et al.  Eigenvalues and graph bisection: An average-case analysis , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[11]  Cecilia R. Aragon,et al.  Optimization by Simulated Annealing: An Experimental Evaluation; Part I, Graph Partitioning , 1989, Oper. Res..

[12]  Martin E. Dyer,et al.  The Solution of Some Random NP-Hard Problems in Polynomial Expected Time , 1989, J. Algorithms.

[13]  David Williams,et al.  Probability with Martingales , 1991, Cambridge mathematical textbooks.

[14]  A. Shiryayev.  On Sums of Independent Random Variables , 1992 .

[15]  Mark Jerrum,et al.  Simulated annealing for graph bisection , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[16]  G. Weiss.  Aspects and Applications of the Random Walk , 1994 .

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

[18]  J. Douglas.  Aspects and applications of the random walk , 1995 .

[19]  Ludek Kucera,et al.  Expected Complexity of Graph Partitioning Problems , 1995, Discret. Appl. Math..

[20]  Byung Ro Moon,et al.  Genetic Algorithm and Graph Partitioning , 1996, IEEE Trans. Computers.

[21]  Alan M. Frieze,et al.  The regularity lemma and approximation schemes for dense problems , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[22]  Mark Jerrum,et al.  The Metropolis Algorithm for Graph Bisection , 1998, Discret. Appl. Math..