General Partitioning on Random Graphs (original) (raw)
2002, Journal of Algorithms
Consider the general partitioning (GP) problem defined as follows: Partition the vertices of a graph into k parts W 1 W k satisfying a polynomial time verifiable property. In particular, consider properties (introduced by T. Feder, P. Hell, S. Klein, and R. Motwani, in "Proceedings of the Annual ACM Symposium on Theory of Computing (STOC '99), 1999" and) specified by a pattern of requirements as to which W i forms a sparse or dense subgraph and which pairs W i , W j form a sparse or dense or an arbitrary (no restriction) bipartite subgraph. The sparsity or density is specified by upper or lower bounds on the edge density d ∈ 0 1 , which is the fraction of actual edges present to the maximum number of edges allowed. This problem is NP-hard even for some fixed patterns and includes as special cases well-known NP-hard problems like k-coloring (each d W i = 0; each d W i W j is arbitrary), bisection (k = 2; W 1 = W 2 ; d W 1 W 2 ≤ b), and also other problems like finding a clique/independent set of specified size. We show that GP is solvable in polynomial time almost surely over random instances with a planted partition of desired type, for several types of pattern requirement. The algorithm is based on the approach of growing BFS trees outlined by C. R. Subramanian (in "Proceedings of the 8th Annual European Symposium on Algorithms (ESA '00), 2000," pp. 415-426).
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