The solution of some random NP-hard problems in polynomial expected time
Journal of Algorithms
Finding good approximate vertex and edge partitions is NP-hard
Information Processing Letters
Randomized algorithms
Complexity of graph partition problems
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Hiding cliques for cryptographic security
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Computers and Intractability; A Guide to the Theory of NP-Completeness
Computers and Intractability; A Guide to the Theory of NP-Completeness
Algorithms for Graph Partitioning on the Planted Partition Model
RANDOM-APPROX '99 Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization Problems: Randomization, Approximation, and Combinatorial Algorithms and Techniques
A Generalized Encryption Scheme Based on Random Graphs
WG '91 Proceedings of the 17th International Workshop
Heuristics for Finding Large Independent Sets, with Applications to Coloring Semi-random Graphs
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Graph partitioning via adaptive spectral techniques
Combinatorics, Probability and Computing
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Consider the general partitioning (GP) problem defined as follows: Partition the vertices of a graph into k parts W1, ..., Wk 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 Wi forms a sparse or dense subgraph and which pairs Wi, Wj 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(Wi) = 0; each d(Wi, Wj) is arbitrary), bisection (k = 2; |W1| = |W2|; d(W1, W2) ≤ 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).