Approximate Hypergraph Partitioning and Applications

Szemeredi's regularity lemma is a cornerstone result in extremal combinatorics. It (roughly) asserts that any dense graph is composed of a finite number of pseudorandom graphs. The regularity lemma has found many applications in theoretical computer science, and thus a lot of attention was given to designing algorithmic versions of this lemma. Our main results in this paper are the following: (i) We introduce a new approach to the problem of constructing regular partitions of graphs, which results in a surprisingly simple $O(n)$ time algorithmic version of the regularity lemma, thus improving over the previous $O(n^2)$ time algorithms. Furthermore, unlike all the previous approaches for this problem (see [N. Alon and A. Naor, SIAM J. Comput., 35 (2006), pp. 787-803], [R. A. Duke, H. Lefmann, and V. Rodl, SIAM J. Comput., 24 (1995), pp. 598-620], [A. Frieze and R. Kannan, Electron. J. Combin., 6 (1999), article 17], [A. Frieze and R. Kannan, “The regularity lemma and approximation schemes for dense problems,” in Proceedings of the 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996), IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 12-20], and [Y. Kohayakawa, V. Rodl, and L. Thoma, SIAM J. Comput., 32 (2003), pp. 1210-1235]), which only guaranteed to find tower-size partitions, our algorithm will find a small regular partition, if one exists in the graph. (ii) For any constant $r\geq3$ we give an $O(n)$ time randomized algorithm for constructing regular partitions of $r$-uniform hypergraphs, thus improving the previous $O(n^{2r-1})$ time (deterministic) algorithms [A. Czygrinow and V. Rodl, SIAM J. Comput., 30 (2000), pp. 1041-1066], [A. Frieze and R. Kannan, “The regularity lemma and approximation schemes for dense problems,” in Proceedings of the 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996), IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 12-20]. These two results are obtained as an application of an efficient algorithm for approximating partition problems of hypergraphs which we obtain here: Given a (directed) hypergraph with bounded edge arities, a set of constraints on the set sizes and densities of a possible partition of its vertex set, and an approximation parameter, we provide in $O(n)$ time a partition approximating the constraints if a partition satisfying them exists. We can also test in $O(1)$ time for the existence of such a partition given the approximation parameter. This algorithm extends the result of Goldreich, Goldwasser, and Ron for graph partition problems [O. Goldreich, S. Goldwasser, and D. Ron, J. ACM, 45 (1998), pp. 653-750] and encompasses more recent hypergraph-related results such as the maximal constraint satisfaction approximation of [G. Andersson and L. Engebretsen, Random Structures Algorithms, 21 (2002), pp. 14-32].