Year: 2010
Type: dissertation
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 AC 0 .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.
Authors: Madhu Sudan, Benjamin Rossman
Institution: Massachusetts Institute of Technology
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2169644995