{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/62441"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/62441","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Average-case complexity of detecting cliques","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Rossman, Benjamin (Benjamin E.)"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science","school":null,"contributors":[],"advisors":["Madhu Sudan."],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010","date_published":"2010","updated_at":"2026-07-22T22:20:52Z","subjects":["Electrical Engineering and Computer Science."],"languages":["eng"],"rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. 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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. 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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. 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