{"@context":"https:\/\/schema.org\/","@type":"PublicationVolume","@id":"#volume717","volumeNumber":8381,"name":"Dagstuhl Seminar Proceedings, Volume 8381","dateCreated":"2008-12-11","datePublished":"2008-12-11","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"Periodical","@id":"#series119","name":"Dagstuhl Seminar Proceedings","issn":"1862-4405","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume717"},"hasPart":[{"@type":"ScholarlyArticle","@id":"#article2233","name":"08381 Abstracts Collection \u2013 Computational Complexity of Discrete Problems","abstract":"From the 14th of September to the 19th of September, the Dagstuhl Seminar\r\n08381 ``Computational Complexity of Discrete Problems'' was held in Schloss Dagstuhl - Leibniz Center for Informatics. \r\nDuring the seminar, several participants presented their current\r\nresearch, and ongoing work as well as open problems were discussed.\r\nAbstracts of the presentations given during the seminar as well as abstracts of\r\nseminar results and ideas are put together in this report. The first section\r\ndescribes the seminar topics and goals in general.\r\nLinks to extended abstracts or full papers are provided, if available.","keywords":["Computational complexity","discrete problems","Turing machines","circuits","proof complexity","pseudorandomness","derandomization","cryptography","computational learning","communication complexity","query complexity","hardness of approximation"],"author":[{"@type":"Person","name":"Miltersen, Peter Bro","givenName":"Peter Bro","familyName":"Miltersen"},{"@type":"Person","name":"Reischuk, R\u00fcdiger","givenName":"R\u00fcdiger","familyName":"Reischuk"},{"@type":"Person","name":"Schnitger, Georg","givenName":"Georg","familyName":"Schnitger"},{"@type":"Person","name":"van Melkebeek, Dieter","givenName":"Dieter","familyName":"van Melkebeek"}],"position":1,"pageStart":1,"pageEnd":18,"dateCreated":"2008-12-11","datePublished":"2008-12-11","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Miltersen, Peter Bro","givenName":"Peter Bro","familyName":"Miltersen"},{"@type":"Person","name":"Reischuk, R\u00fcdiger","givenName":"R\u00fcdiger","familyName":"Reischuk"},{"@type":"Person","name":"Schnitger, Georg","givenName":"Georg","familyName":"Schnitger"},{"@type":"Person","name":"van Melkebeek, Dieter","givenName":"Dieter","familyName":"van Melkebeek"}],"copyrightYear":"2008","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08381.1","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume717"},{"@type":"ScholarlyArticle","@id":"#article2234","name":"08381 Executive Summary \u2013 Computational Complexity of Discrete Problems","abstract":"Estimating the computational complexity of discrete problems constitutes one of the central and classical topics in the theory of computation. Mathematicians and computer scientists have long tried to classify natural families of Boolean relations according to fundamental complexity measures like time and space, both in the uniform and in the nonuniform setting. Several models of computation have been developed in order to capture the various capabilities of digital computing devices, including parallelism, randomness, and quantum interference.","keywords":["Computational complexity","discrete problems","Turing machines","circuits","proof complexity","pseudorandomness","derandomization","cryptography","computational learning","communication complexity","query complexity","hardness of approximation"],"author":[{"@type":"Person","name":"Miltersen, Peter Bro","givenName":"Peter Bro","familyName":"Miltersen"},{"@type":"Person","name":"Reischuk, R\u00fcdiger","givenName":"R\u00fcdiger","familyName":"Reischuk"},{"@type":"Person","name":"Schnitger, Georg","givenName":"Georg","familyName":"Schnitger"},{"@type":"Person","name":"van Melkebeek, Dieter","givenName":"Dieter","familyName":"van Melkebeek"}],"position":2,"pageStart":1,"pageEnd":7,"dateCreated":"2008-12-11","datePublished":"2008-12-11","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Miltersen, Peter Bro","givenName":"Peter Bro","familyName":"Miltersen"},{"@type":"Person","name":"Reischuk, R\u00fcdiger","givenName":"R\u00fcdiger","familyName":"Reischuk"},{"@type":"Person","name":"Schnitger, Georg","givenName":"Georg","familyName":"Schnitger"},{"@type":"Person","name":"van Melkebeek, Dieter","givenName":"Dieter","familyName":"van Melkebeek"}],"copyrightYear":"2008","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08381.2","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume717"},{"@type":"ScholarlyArticle","@id":"#article2235","name":"Approximation norms and duality for communication complexity lower bounds","abstract":"Abstract: We will discuss a general norm based framework for showing\r\nlower bounds on communication complexity. An advantage of this approach is that one can use duality theory to obtain a lower bound quantity phrased as a\r\nmaximization problem, which can be more convenient to work with in showing lower bounds.\r\n\r\nWe discuss two applications of this approach.\r\n\r\n1. The approximation rank of a matrix A is the minimum rank of a\r\nmatrix close to A in ell_infty norm. The logarithm of approximation rank lower bounds quantum communication complexity and is one of the most powerful techniques available, albeit difficult to compute in practice. We\r\nshow that an approximation norm known as gamma_2 is polynomially\r\nrelated to approximation rank.\r\nThis results in a polynomial time algorithm to approximate\r\napproximation rank, and also shows that the logarithm of approximation rank lower bounds quantum communication complexity even with entanglement which was previously not known.\r\n\r\n2. By means of an approximation norm which lower bounds multiparty\r\nnumber-on-the-forehead complexity, we show non-trivial lower bounds on the complexity of the disjointness function for up to c log log n players, c <1.","keywords":["Communication complexity","lower bounds"],"author":[{"@type":"Person","name":"Lee, Troy","givenName":"Troy","familyName":"Lee"},{"@type":"Person","name":"Shraibman, Adi","givenName":"Adi","familyName":"Shraibman"}],"position":3,"pageStart":1,"pageEnd":9,"dateCreated":"2008-12-11","datePublished":"2008-12-11","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Lee, Troy","givenName":"Troy","familyName":"Lee"},{"@type":"Person","name":"Shraibman, Adi","givenName":"Adi","familyName":"Shraibman"}],"copyrightYear":"2008","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08381.3","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume717"},{"@type":"ScholarlyArticle","@id":"#article2236","name":"Depth Reduction for Circuits with a Single Layer of Modular Counting Gates","abstract":"We consider the class of constant depth AND\/OR circuits augmented with\r\na layer of modular counting gates at the bottom layer, i.e ${AC}^0 circ {MOD}_m$ circuits. We show that the following\r\nholds for several types of gates $G$: by adding a gate of type $G$ at\r\nthe output, it is possible to obtain an equivalent randomized depth 2\r\ncircuit of quasipolynomial size consisting of a gate of type $G$ at\r\nthe output and a layer of modular counting gates, i.e $G circ {MOD}_m$ circuits. The types of gates $G$ we consider are modular\r\ncounting gates and threshold-style gates. For all of these, strong\r\nlower bounds are known for (deterministic) $G circ {MOD}_m$\r\ncircuits.","keywords":["Boolean Circuits","Randomized Polynomials","Fourier sums"],"author":{"@type":"Person","name":"Hansen, Kristoffer Arnsfelt","givenName":"Kristoffer Arnsfelt","familyName":"Hansen"},"position":4,"pageStart":1,"pageEnd":11,"dateCreated":"2008-12-17","datePublished":"2008-12-17","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Hansen, Kristoffer Arnsfelt","givenName":"Kristoffer Arnsfelt","familyName":"Hansen"},"copyrightYear":"2008","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08381.4","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume717"},{"@type":"ScholarlyArticle","@id":"#article2237","name":"Fast polynomial factorization and modular composition","abstract":"We obtain randomized algorithms for factoring degree $n$\r\nunivariate polynomials over $F_q$ requiring $O(n^{1.5 +\r\no(1)} log^{1+o(1)} q+ n^{1 + o(1)}log^{2+o(1)} q)$ bit operations.\r\nWhen $log q < n$, this is asymptotically faster than the best previous algorithms (von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998)); for\r\n$log q ge n$, it matches the asymptotic running time of the best\r\nknown algorithms.\r\n\r\nThe improvements come from new algorithms for modular composition\r\nof degree $n$ univariate polynomials, which is the asymptotic\r\nbottleneck in fast algorithms for factoring polynomials over\r\nfinite fields. The best previous algorithms for modular\r\ncomposition use $O(n^{(omega + 1)\/2})$ field operations, where\r\n$omega$ is the exponent of matrix multiplication (Brent & Kung\r\n(1978)), with a slight improvement in the exponent achieved by\r\nemploying fast rectangular matrix multiplication (Huang & Pan\r\n(1997)).\r\n\r\nWe show that modular composition and multipoint evaluation of\r\nmultivariate polynomials are essentially equivalent, in the sense\r\nthat an algorithm for one achieving exponent $alpha$ implies an\r\nalgorithm for the other with exponent $alpha + o(1)$, and vice\r\nversa. We then give two new algorithms that solve the problem\r\noptimally (up to lower order terms): an algebraic algorithm for\r\nfields of characteristic at most $n^{o(1)}$, and a\r\nnonalgebraic algorithm that works in arbitrary characteristic.\r\nThe latter algorithm works by lifting to characteristic 0,\r\napplying a small number of rounds of {em multimodular reduction},\r\nand finishing with a small number of multidimensional FFTs. The\r\nfinal evaluations are reconstructed using the Chinese Remainder\r\nTheorem. As a bonus, this algorithm produces a very efficient data\r\nstructure supporting polynomial evaluation queries, which is of\r\nindependent interest.\r\n\r\nOur algorithms use techniques which are commonly employed in\r\npractice, so they may be competitive for real problem sizes. This\r\ncontrasts with all previous subquadratic algorithsm for these\r\nproblems, which rely on fast matrix multiplication.\r\n\r\nThis is joint work with Kiran Kedlaya.","keywords":"Modular composition; polynomial factorization; multipoint evaluation; Chinese Remaindering","author":[{"@type":"Person","name":"Kedlaya, Kiran","givenName":"Kiran","familyName":"Kedlaya"},{"@type":"Person","name":"Umans, Christopher","givenName":"Christopher","familyName":"Umans"}],"position":5,"pageStart":1,"pageEnd":33,"dateCreated":"2008-12-11","datePublished":"2008-12-11","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kedlaya, Kiran","givenName":"Kiran","familyName":"Kedlaya"},{"@type":"Person","name":"Umans, Christopher","givenName":"Christopher","familyName":"Umans"}],"copyrightYear":"2008","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08381.5","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume717"},{"@type":"ScholarlyArticle","@id":"#article2238","name":"Understanding space in resolution: optimal lower bounds and exponential trade-offs","abstract":"We continue the study of tradeoffs between space and length of\r\nresolution proofs and focus on two new results:\r\n\r\nbegin{enumerate}\r\nitem \r\nWe show that length and space in resolution are uncorrelated. This\r\nis proved by exhibiting families of CNF formulas of size $O(n)$ that\r\nhave proofs of length $O(n)$ but require space $Omega(n \/ log n)$. Our\r\nseparation is the strongest possible since any proof of length $O(n)$\r\ncan always be transformed into a proof in space $O(n \/ log n)$, and\r\nimproves previous work reported in [Nordstr\"{o}m 2006, Nordstr\"{o}m and\r\nH{aa}stad 2008].\r\n\r\nitem We prove a number of trade-off results for space in the range\r\nfrom constant to $O(n \/ log n)$, most of them superpolynomial or even\r\nexponential. This is a dramatic improvement over previous results in\r\n[Ben-Sasson 2002, Hertel and Pitassi 2007, Nordstr\"{o}m 2007].\r\nend{enumerate}\r\n\r\nThe key to our results is the following, somewhat surprising, theorem:\r\n\r\nAny CNF formula $F$ can be transformed by simple substitution\r\ntransformation into a new formula $F'$ such that if $F$ has the right\r\nproperties, $F'$ can be proven in resolution in essentially the same\r\nlength as $F$ but the minimal space needed for $F'$ is lower-bounded\r\nby the number of variables that have to be mentioned simultaneously in\r\nany proof for $F$. Applying this theorem to so-called pebbling\r\nformulas defined in terms of pebble games over directed acyclic graphs\r\nand analyzing black-white pebbling on these graphs yields our results.","keywords":["Proof complexity","Resolution","Pebbling."],"author":[{"@type":"Person","name":"Ben-Sasson, Eli","givenName":"Eli","familyName":"Ben-Sasson"},{"@type":"Person","name":"Nordstr\u00f6m, Jakob","givenName":"Jakob","familyName":"Nordstr\u00f6m"}],"position":6,"pageStart":1,"pageEnd":0,"dateCreated":"2008-12-17","datePublished":"2008-12-17","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Ben-Sasson, Eli","givenName":"Eli","familyName":"Ben-Sasson"},{"@type":"Person","name":"Nordstr\u00f6m, Jakob","givenName":"Jakob","familyName":"Nordstr\u00f6m"}],"copyrightYear":"2008","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08381.6","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume717"}]}