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dc.contributor.advisorGuy Bresler.en_US
dc.contributor.authorBoix, Enric.en_US
dc.contributor.otherMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Science.en_US
dc.date.accessioned2021-01-06T19:36:13Z
dc.date.available2021-01-06T19:36:13Z
dc.date.copyright2020en_US
dc.date.issued2020en_US
dc.identifier.urihttps://hdl.handle.net/1721.1/129260
dc.descriptionThesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, September, 2020en_US
dc.descriptionCataloged from student-submitted PDF version of thesis.en_US
dc.descriptionIncludes bibliographical references (pages 77-82).en_US
dc.description.abstractThe 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: --en_US
dc.description.abstractDense 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. --en_US
dc.description.abstractSparse 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 </= C </= 1 and w is the matrix multiplication constant. In the hypergraph case of s >/= 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.en_US
dc.description.abstractMoreover, 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].en_US
dc.description.statementofresponsibilityby Enric Boix.en_US
dc.format.extent82 pagesen_US
dc.language.isoengen_US
dc.publisherMassachusetts Institute of Technologyen_US
dc.rightsMIT 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.en_US
dc.rights.urihttp://dspace.mit.edu/handle/1721.1/7582en_US
dc.subjectElectrical Engineering and Computer Science.en_US
dc.titleThe average-case complexity of counting cliques in Erdős-Rényi Hypergraphsen_US
dc.typeThesisen_US
dc.description.degreeS.M.en_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Scienceen_US
dc.identifier.oclc1227521043en_US
dc.description.collectionS.M. Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Scienceen_US
dspace.imported2021-01-06T19:36:12Zen_US
mit.thesis.degreeMasteren_US
mit.thesis.departmentEECSen_US


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