Mathematisches Forschungsinstitut Oberwolfach Rep ort No Complexity Theory November th November th Complexity theory is concerned with the study of the intrinsic diculty of computational th tasks It is a central eld of theoretical computer science The Ob erwolfach Conference on Complexity Theory was organized by Joachim von zur Gathen Paderborn Oded Goldreich Rehovot and Claus Peter Schnorr Frankfurt The meeting consisted of ve general sessions and in addition sp ecial sessions on the following topics Algebraic Complexity Cryptography Lattices Pseudorandomness Another event that to ok place in the meeting was the awarding of the Ob erwolfach prize to Luca Trevisan who was one of the participants Abstracts of General Session Talks A Few Facts ab out Division Eric Allender Chiu Davida and Litow recently solved a decadesold problem by showing that there are logspaceuniform constantdepth threshold circuits for division It remains op en if the uniformity condition can b e improved to obtain Dlogtimeuniform circuits We precisely characterize the uniformity requirements by showing that Division is com plete under rstorder reductions for the class FOMPOW where FOM is an equivalent i formalization of Dlogtimeuniform TC and POW is the predicate a b mo d p for primes p of O log n bits We also show that FOM and FOMPOW coincide if a wellknown conjecture ab out smo oth primes holds In the talk I also mention a recent lower b ound joint work with Koucky Ronneburger Roy and Vinay showing that the lower b ound techniques of Fortnow can b e extended to the probabilistic mo del if division has uniform TC circuits Other consequences of the new division algorithm include a new translational lemma for very small spaceb ounded complexity classes Joint work with David Mix Barrington and William Hesse Sup erlinear timespace tradeo lower b ounds for randomized computation Paul Beame We prove the rst timespace lower b ound tradeos for randomized computation of de cision problems The b ounds hold even in the case that the computation is allowed to have arbitrary probability of error on a small fraction of inputs Our techniques are an extension of those used by Ajtai in his timespace tradeos for deterministic RAM algorithms computing element distinctness and for deterministic Bo olean branching programs computing an explicit function based on quadratic forms over GF Our results also give a quantitative improvement over those given by Ajtai Ajtai shows for certain sp ecic functions that any branching program using space S on requires time T that is sup erlinear The functional form of the sup erlinear b ound is not given in his pap er but optimizing the parameters in his arguments gives T n log log n log log log n for S O n For the same functions considered by Ajtai we prove a timespace tradeo of p log nS log log nS In particular for space O n this improves the form T n p the lower b ound on time to n log n log log n Joint work with Mike Saks Xiadong Sun and Erik Vee Expansion in Prop ositional Pro of Complexity Eli BenSasson In this survey talk we describ e the main technique used in recent years to prove lower b ounds in pro of complexity for simple pro of systems such as resolution and the p olynomial calculus We dene a certain form of expansion Boundary Expansion on bipartite graphs We dene a reduction of CNF formulas to bipartite graphs and claim the following For F an unsatisable CNF formula and GF its corresp onding bipartite graph if GF is an expander then The minimal width of refuting F in resolution is large linear The minimal size of refuting F in resolution is large exp onential The minimal space needed to refute F in resolution is large linear Similar lower b ounds hold for degree of refutation in the Polynomial Calculus This basic idea allows us to show nontrivial and often optimal lower b ounds for the Pigeonhole Principles Tseitin Graph formulas random k CNFs PseudorandomGenerator based formulas and many others Based on works by Alekhnovich Beame BenSasson Clegg Edmonds Grigoriev Im pagliazzo Pitassi Pudlak Razb orov Sgall and Wigderson Lower b ounds for the complexity of asso ciative algebras Markus Blaser Let C A resp R A denote the multiplicative resp bilinear complexity of a nite dimen sional asso ciative algebra A dim A n n if the decomp osition of Arad A We prove that R A t n n A A into simple algebras A D contains only noncommutative factors that t is the division algebra D is noncommutative or n If A is in addition semisimple then the same b ound holds for the multiplicative complexity ie C A dim A n n t n n essential multiplications In particular n nmatrix multiplication requires at least Approximating the Minimum Bisection Uriel Feige A Bisection of a graph with n vertices is a partition of its vertices into two sets each of size n The bisection cost is the number of edges connecting the two sets Finding the minimum bisection cost is NPhard We present several approximation algorithms for bisection the b est of which nds a bisection whose cost is within a ratio of O log n from optimal The previously known approximation ratio for bisection was n Joint work with Rob ert Krauthgamer and in part with Kobbi Nissim In search of an easy witness Applications to Exp onential Time Valentine Kabanets Using the hardnessrandomness tradeos as well as the idea of easy witnesses we show several complexitytheoretic results involving exp onentialtime complexity classes First we prove that NEXP Pp oly i NEXPMA This can b e interpreted as saying that one cannot derandomize MA without proving sup erp olynomial circuit lower b ounds for NEXP We also establish several downward closure results for the probabilistic complexity classes ZPP RP BPP and MA In particular we prove that EXPBPP i EEBPE where EE is O n double exp onential time and BPE is the time analog of the class BPP Joint work with Russell Impagliazzo and Avi Wigderson On Rounds in Quantum Communication Hartmut Klauck We investigate the p ower of interaction in two player quantum communication proto cols Our main result is a roundscommunication hierarchy for the p ointer jumping function f k We show that f needs quantum communication n if Bob starts the communication k and the number of rounds is limited to k for any constant k Trivially if Alice starts O k log n communication in k rounds suces The lower b ound employs a result relating the relative von Neumann entropy b etween density matrices to their trace distance and uses a new measure of information We also describ e a classical probabilistic k round proto col for f with communication k O nk k log k in which Bob starts the communication for k at least log n Furthermore as a consequence of the lower b ound for p ointer jumping we show that any k k round quantum proto col for the disjointness problem needs communication n for k O A linear space algorithm for computing the Hermite Normal Form of an integer lattice Daniele Micciancio Computing the Hermite Normal Form of an n n matrix using the b est current algorithms typically requires O n log M space where M is a b ound on the length of the columns of the input matrix Although p olynomial in the input size which is O n log M this space blowup can easily b ecome a serious issue in practice when working on big integer matrices In this talk we present a new algorithm for computing the Hermite Normal Form which uses only O n log M space ie essentially the same as the input size When implemented using standard integer arithmetic our algorithm has the same time complexity of the asymptoti cally fastest but space inecient algorithms We also suggest simple heuristics that when incorp orated in our algorithm result in essentially the same asymptotic running time of the theoretically fastest solutions still maintaining our algorithm extremely practical Joint work with Bogdan Warinschi The ZigZag Graph Pro duct and Elementary Construction of Expander Graphs Omer Reingold Expander graphs are combinatorial ob jects which are fascinating and useful but seemed hard to construct The main result we present is an elementary way of constructing them The essential ingredient is a new type of graph pro duct which we call the zigzag pro duct Taking a pro duct of a large graph with a small graph the resulting graph inherits roughly its size from the large one its degree from the small one and its expansion prop erties from b oth Iteration yields simple explicit constructions of constant degree expanders of arbitrary size starting from one constantsize expander Crucial to our intuition and simple analysis of the prop erties of this graph pro duct is the view of expanders as functions which act as entropy wave propagators they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated In these terms the graph pro duct aords the constructive interference of two such waves No sp ecial background is assumed Joint work with Salil Vadhan and Avi Wigderson Variation of the BaurStrassen Theorem for Size and Depth Arnold Schonhage In this talk I present a simple pro of for the following Theorem Let a rational function f K x x b e computable by an arithmetical n circuit D of size s and depth d with the indeterminates x x and any constants K n as costfree inputs of D and op eration no des using f g at unit cost Then there exists also a circuit D of size s and depth d computing f plus its rst partial derivatives f