Average-case complexity of detecting cliques
Name
711003585-MIT.pdf
Description
Full printable version
Size
4.58 MB
Format
Adobe PDF
Checksum (MD5)
4b746d3cafced532b65e95decafd4477
Author(s)
Rossman, Benjamin (Benjamin E.)
Advisor(s)
Madhu Sudan.
Date Issued
2010
Publisher
Massachusetts Institute of Technology
Abstract
The computational problem of testing whether a graph contains a complete subgraph of size k is among the most fundamental problems studied in theoretical computer science. This thesis is concerned with proving lower bounds for k-CLIQUE, as this problem is known. Our results show that, in certain models of computation, solving k-CLIQUE in the average case requires Q(nk/4) resources (moreover, k/4 is tight). Here the models of computation are bounded-depth Boolean circuits and unbounded-depth monotone circuits, the complexity measure is the number of gates, and the input distributions are random graphs with an appropriate density of edges. Such random graphs (the well-studied Erdos-Renyi random graphs) are widely believed to be a source of computationally hard instances for clique problems (as Karp suggested in 1976). Our results are the first unconditional lower bounds supporting this hypothesis. For bounded-depth Boolean circuits, our average-case hardness result significantly improves the previous worst-case lower bounds of Q(nk/Poly(d)) for depth-d circuits. In particular, our lower bound of Q(nk/ 4 ) has no noticeable dependence on d for circuits of depth d ; k- log n/log log n, thus bypassing the previous "size-depth tradeoffs". As a consequence, we obtain a novel Size Hierarchy Theorem for uniform AC0 . A related application answers a longstanding open question in finite model theory (raised by Immerman in 1982): we show that the hierarchy of bounded-variable fragments of first-order logic is strict on finite ordered graphs. Additional results of this thesis characterize the average-case descriptive complexity of k-CLIQUE through the lens of first-order logic.
Description
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 79-83).
Subjects
Electrical Engineering and Computer Science.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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