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Oct D 5 2010 Libraries Average-Case Complexity of Detecting Cliques by Benjamin Rossman Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy MASSACHUSETTS INSTITUTE OF TECHOLOGY at the OCT D5 2010 MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2010 LIBRARIES @ Benjamin Rossman, 2010. All rights reserved. ARCHNES The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author........... .......................... Department of Electrical Enginf ring and Computer Science September 3, 2010 I I A 1, It t\ Certified by.............. Madhu Sudan Professor of Electrical Engineering and Computer Science Thesis Supervisor A ccepted by ................................. Ter.. .......... Terry P. Orlando Chairman, Department of Electrical Engineering and Computer Science Average-Case Complexity of Detecting Cliques by Benjamin Rossman Submitted to the Department of Electrical Engineering and Computer Science on September 3, 2010, in partial fulfillment of the requirements for the degree of Doctor of Philosophy 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 im- proves the previous worst-case lower bounds of Q(nk/Poly(d)) for depth-d circuits. In partic- ular, 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 conse- quence, 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 descrip- tive complexity of k-CLIQUE through the lens of first-order logic. Thesis Supervisor: Madhu Sudan Title: Professor of Electrical Engineering and Computer Science Acknowledgments It was my extreme good fortunate to have Madhu Sudan as an advisor. Madhu, thanks for all the encouragement and enlightened advice you gave me on so many occasions. To my thesis committee-Scott Aaronson, Neil Immerman and Joel Spencer-thanks for your feedback, especially a few insights that have improved the results in this thesis. I am eternally grateful to my early mentors, Scott Weinstein and Yuri Gurevich. Scott, with contagious enthusiasm, introduced me to the subject of finite model theory and pointed me in many fruitful directions. Yuri has been a valuable teacher, who guided me through early beginnings in research with patience and the most exceptional kindness. My sincere thanks to all those who hosted me with warm hospitality during various visits and internships: Albert Atserias, Eli Ben-Sasson, Ron Fagin, Phokion Kolaitis, Janos Makowsky, Sasha Razborov, Michel de Rougemont, Saharon Shelah and Osamu Watanabe. In addition to this list, I thank Andreas Blass, Anuj Dawar, Martin Grohe, Leonid Libkin, Patrice Ossona de Mendez and Mike Sipser for stimulating conversations. Special thanks to Rahul Santhanam for discussions that helped shape this thesis. Let me not fail to mention that I am a member of the Jarik Nesetril Appreciation Society. Many thanks to the friends at MIT who made this experience so great: Anna, Andy, Bernhard, Brendan, Chih-yu, Dan, Elena, Eriko, Erin, Jing, Krzysztof, Nadia, Oren, Petar, Rotem, Shubhangi, Swastik, Qinwen and Yulan (to name just a few). And two friends in particular, Annie Liu and Costin Alamariu, whose kindness and generosity are more than I deserve. (Annie, here is your own sentence.) Finally and most of all, I thank my family-David, Lynne, Jenny and Zayna-for their love and support. I dedicate this thesis to my parents, David and Lynne Rossman. Bibliographic note The main results in this thesis appear in two conference papers: " "On the constant-depth complexity of k-CLIQUE" [64], containing preliminary versions of the results in Chapter 3, and * "The monotone complexity of k-CLIQUE on random graphs" [66] (forthcoming), which constitutes Chapter 4. I thank Christoph Berkholz for pointing out a minor mistake in [64] (fixed in this thesis) and Tomoyuki Hayasaka and Koutaro Nakagawa for helpful feedback on [66]. Chapter 6 of this thesis (on descriptive complexity) includes an unpublished result due to Neil Immerman (Theorem 6.11), as well as the fruits of a collaboration with Joel Spencer (Theorem 6.23). Contents 1 Introduction 9 1.1 Lower bounds in computational complexity .................. 9 1.1.1 Bounded-depth circuits ......................... 10 1.1.2 Monotone circuits ... ......................... 11 1.2 Average-case lower bounds ........... ............. 11 1.2.1 Karp's question ......................... ..... 12 1.2.2 Scaling Karp's question to G(n, n-) ... ............... 12 1.3 Bounded-variable logics ........ ..................... 13 1.4 O ur results ............... ..................... 15 1.5 Applications in complexity and logic ...................... 16 1.6 Techniques ................... 17 2 Definitions and Preliminaries 19 2.1 Basic notation .................................. 19 2.2 C ircuits ...................................... 20 2.2.1 Circuit parameters ............. ............... 20 2.2.2 Complexity ...................... .21 2.3 Graphs and patterns .......................... ..... 21 2.3.1 Threshold exponent ............ ............... 21 2.3.2 Graph functions, monotonicity and minterms ............. 21 2.4 Probability .... ........................... ..... 22 2.4.1 Random graphs .................... .......... 22 2.4.2 Background lemmas .. ......................... 22 2.4.3 Janson's inequality ............ ............. ... 23 2.5 Small, medium, large .... ........................... 24 3 Lower Bound for Bounded-Depth Circuits 27 3.1 A lemma on random restrictions. ...................... 28 3.2 The f-sensitive subgraph......... .... ............... 31 3.3 Preliminary result: lower bound on wires ................... 35 3.4 Main result: lower bound on size ....... ............. .... 37 4 Lower Bound for Monotone Circuits 43 4.1 Results of this chapter ............. ............. .... 43 4.2 Razborov's approximation method ...... ............. .... 45 4.3 Quasi-sunflowers .. .............. ............. .... 46 4.4 The approximation via a closure operator...... ......... .... 48 4.5 K vs. G . ...... ... .................................. 51 4.6 GUKvs.GUG- ................................. 53 4.7 Removing the fan-in restriction ......................... 54 5 Matching Upper Bound 59 5.1 Auxiliary subcircuits. .................... .. 59 5.2 Som e random sets .......... ...................... 61 5.3 The full construction.. ........................... 63 6 Descriptive Complexity 65 6.1 k/4 variable lower bound ....................... ..... 66 6.2 Strictness of the variable hierarchy .. ..................... 66 6.3 Average-case definability of k-CLIQUE without order . ............ 69 6.3.1 k/2 variables are necessary ....................... 69 6.3.2 k/2+log k + O(1) variables suffice ................... 70 6.3.3 k/2 + 0(1) variables suffice ............. .......... 72 7 Extensions and Open Problems 75 7.1 Extensions of our results ............................. 75 7.1.1 Size-depth and size-gap tradeoffs .................... 75 7.1.2 Planting a larger clique ......................... 76 7.1.3 Subgraph isomorphism problem ...... ............... 76 7.2 Open problem s ....... ........................... 77 7.2.1 Karp's question at the k-clique threshold .......... ..... 77 7.2.2 Monotone lower bound at a single threshold .............. 77 7.2.3 Additional questions ........................... 77 Chapter 1 Introduction 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. That is, our aim is to show that k-CLIQUE cannot be solved with certain limited computational resources. Understanding the complexity of k-CLIQUE is key to unlocking the relationship between P and NP. In the traditional view where k is part of the input, the problem of detecting k- cliques is famously NP-complete (one Karp's 21 NP-complete problems [45]). In this thesis, however, we consider the problem for fixed but arbitrary values of k. Here the brute-force O(nk) algorithm places k-CLIQUE in P. A crucial observation is that to separate
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