The average-case complexity of counting cliques in Erdős-Rényi Hypergraphs
Name
1227521043-MIT.pdf
Size
676.76 KB
Format
Adobe PDF
Checksum (MD5)
88e29c74071888757c9fbf4a1797abcf
Author(s)
Boix, Enric.
Advisor(s)
Guy Bresler.
Date Issued
2020
Publisher
Massachusetts Institute of Technology
Abstract
The complexity of clique problems on Erdős-Rényi random graphs has become a central topic in average-case complexity. Algorithmic phase transitions in these problems have been shown to have broad connections ranging from mixing of Markov chains and statistical physics to information-computation gaps in high-dimensional statistics. We consider the problem of counting k-cliques in s-uniform Erdős-Rényi hypergraphs G(n, c, s) with edge density c and show that its fine-grained average-case complexity can be based on its worst-case complexity. We give a worst-case to average-case reduction for counting k-cliques on worst-case hypergraphs given a blackbox solving the problem on G(n, c, s) with low error probability. Our approach is closely related to [Goldreich and Rothblum, FOCS18], which showed a worst-case to average-case reduction for counting cliques for an efficiently-sampleable distribution on graphs. Our reduction has the following implications: --
Dense Erdős-Rényi graphs and hypergraphs: Counting k-cliques on G(n, c, s) with k and c constant matches its worst-case complexity up to a polylog(n) factor. Assuming ETH, it takes n(superscript [omega](k) time to count k-cliques in G(n, c, s) if k and c are constant. --
Sparse Erdős-Rényi graphs and hypergraphs: When c = [theta](n⁻[superscript alpha]), for each fixed [alpha] our reduction yields different average-case phase diagrams depicting a tradeoff between runtime and k. Assuming the best known worst-case algorithms are optimal, in the graph case of s = 2, we establish that the exponent in n of the optimal running time for k-clique counting in G(n, c, s) is wk/3 - C[alpha](k 2) + [omicron][subscript k,[alpha]](1), where w/9 /= 3, we show a lower bound at the exponent of k - [alpha](k s) + [omicron][subscript k,[alpha]](1) which surprisingly is tight against algorithmic achievability exactly for the set of c above the Erdős-Rényi k-clique percolation threshold. Our reduction yields the first average-case hardness result for a problem over Erdős-Rényi hypergraphs based on a corresponding worst-case hardness assumption.
Moreover, because we consider sparse Erdős-Rényi hypergraphs, for each n, k, and s we actually have an entire family of problems parametrized by the edge probability c and the behavior changes as a function of c; this is the first worst-to-average-case hardness result we are aware of for which the complexity of the same problem over worst-case versus average-case inputs is completely different. We also analyze several natural algorithms for counting k-cliques in G(n, c, s) that establish our upper bounds in the sparse case c = [theta](n⁻[superscript alpha]). The ingredients in our worst-case to average-case reduction include: (1) several new techniques for the self-reducibility of counting k-cliques as a low-degree polynomial; and (2) a finite Fourier analytic method to bound the total variation convergence of random biased binary expansions to the uniform distribution over residues in F[subscript p].
Description
Thesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, September, 2020
Cataloged from student-submitted PDF version of thesis.
Includes bibliographical references (pages 77-82).
Subjects
Electrical Engineering and Computer Science.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Terms of Use
MIT theses may be protected by copyright. Please reuse MIT thesis content according to the MIT Libraries Permissions Policy, which is available through the URL provided.
Persistent DSpace Link
