{"@context":"https:\/\/schema.org\/","@type":"PublicationVolume","@id":"#volume773","volumeNumber":9421,"name":"Dagstuhl Seminar Proceedings, Volume 9421","dateCreated":"2010-01-19","datePublished":"2010-01-19","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":"#volume773"},"hasPart":[{"@type":"ScholarlyArticle","@id":"#article2608","name":"09421 Abstracts Collection \u2013 Algebraic Methods in Computational Complexity","abstract":"From 11.10. to 16.10.2009, the Dagstuhl Seminar 09421 ``Algebraic Methods in Computational Complexity '' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.\r\nDuring the seminar, several participants presented their current\r\nresearch, and ongoing work and open problems were discussed. Abstracts of\r\nthe presentations given during the seminar as well as abstracts of\r\nseminar results and ideas are put together in this paper. 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","Algebra"],"author":[{"@type":"Person","name":"Agrawal, Manindra","givenName":"Manindra","familyName":"Agrawal"},{"@type":"Person","name":"Fortnow, Lance","givenName":"Lance","familyName":"Fortnow"},{"@type":"Person","name":"Thierauf, Thomas","givenName":"Thomas","familyName":"Thierauf"},{"@type":"Person","name":"Umans, Christopher","givenName":"Christopher","familyName":"Umans"}],"position":1,"pageStart":1,"pageEnd":22,"dateCreated":"2010-01-20","datePublished":"2010-01-20","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Agrawal, Manindra","givenName":"Manindra","familyName":"Agrawal"},{"@type":"Person","name":"Fortnow, Lance","givenName":"Lance","familyName":"Fortnow"},{"@type":"Person","name":"Thierauf, Thomas","givenName":"Thomas","familyName":"Thierauf"},{"@type":"Person","name":"Umans, Christopher","givenName":"Christopher","familyName":"Umans"}],"copyrightYear":"2010","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.09421.1","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume773"},{"@type":"ScholarlyArticle","@id":"#article2609","name":"09421 Executive Summary \u2013 Algebraic Methods in Computational Complexity","abstract":"The seminar brought together more than 50 researchers covering\r\na wide spectrum of complexity theory. The focus on algebraic\r\nmethods showed once again the great importance of algebraic\r\ntechniques for theoretical computer science. We had almost 30\r\ntalks, most of them about 40 minutes leaving ample room for\r\ndiscussions. We also had a much appreciated open problem\r\nsession.\r\n\r\nThe talks ranged over a\r\nbroad assortment of subjects with the underlying theme of using\r\nalgebraic techniques. It was very fruitful and has hopefully\r\ninitiated new directions in research. Several participants\r\nspecifically mentioned that they appreciated the particular\r\nfocus on a common class of techniques (rather than end\r\nresults) as a unifying theme of the workshop. We look forward\r\nto our next meeting!","keywords":["Computational Complexity","Algebra"],"author":[{"@type":"Person","name":"Agrawal, Manindra","givenName":"Manindra","familyName":"Agrawal"},{"@type":"Person","name":"Fortnow, Lance","givenName":"Lance","familyName":"Fortnow"},{"@type":"Person","name":"Thierauf, Thomas","givenName":"Thomas","familyName":"Thierauf"},{"@type":"Person","name":"Umans, Christopher","givenName":"Christopher","familyName":"Umans"}],"position":2,"pageStart":1,"pageEnd":4,"dateCreated":"2010-01-19","datePublished":"2010-01-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Agrawal, Manindra","givenName":"Manindra","familyName":"Agrawal"},{"@type":"Person","name":"Fortnow, Lance","givenName":"Lance","familyName":"Fortnow"},{"@type":"Person","name":"Thierauf, Thomas","givenName":"Thomas","familyName":"Thierauf"},{"@type":"Person","name":"Umans, Christopher","givenName":"Christopher","familyName":"Umans"}],"copyrightYear":"2010","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.09421.2","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume773"},{"@type":"ScholarlyArticle","@id":"#article2610","name":"An Axiomatic Approach to Algebrization","abstract":"Non-relativization of complexity issues can be interpreted\r\nas giving some evidence that these issues cannot be resolved\r\nby \"black-box\" techniques. In the early 1990's, a sequence of\r\nimportant non-relativizing results was proved, mainly using\r\nalgebraic techniques. Two approaches have been proposed\r\nto understand the power and limitations of these algebraic\r\ntechniques: (1) Fortnow gives a construction of a class\r\nof oracles which have a similar algebraic and logical structure,\r\nalthough they are arbitrarily powerful. He shows that\r\nmany of the non-relativizing results proved using algebraic\r\ntechniques hold for all such oracles, but he does not show,\r\ne.g., that the outcome of the \"P vs. NP\" question differs\r\nbetween different oracles in that class. (2) Aaronson and\r\nWigderson give definitions of algebrizing separations and\r\ncollapses of complexity classes, by comparing classes relative\r\nto one oracle to classes relative to an algebraic extension of\r\nthat oracle. Using these definitions, they show both that\r\nthe standard collapses and separations \"algebrize\" and that\r\nmany of the open questions in complexity fail to \"algebrize\",\r\nsuggesting that the arithmetization technique is close to its\r\nlimits. However, it is unclear how to formalize algebrization\r\nof more complicated complexity statements than collapses\r\nor separations, and whether the algebrizing statements are,\r\ne.g., closed under modus ponens; so it is conceivable that\r\nseveral algebrizing premises could imply (in a relativizing\r\nway) a non-algebrizing conclusion.\r\n\r\nHere, building on the work of Arora, Impagliazzo,\r\nand Vazirani [4], we propose an axiomatic approach to \"algebrization\",\r\nwhich complements and clarifies the approaches\r\nof Fortnow and Aaronso&Wigderson. We present logical theories formalizing the notion of algebrizing techniques so that most algebrizing results\r\nare provable within our theories and separations requiring\r\nnon-algebrizing techniques are independent of them.\r\n\r\nOur theories extend the [AIV] theory formalizing relativization\r\nby adding an Arithmetic Checkability axiom.\r\n\r\nWe show the following: (i) Arithmetic checkability holds\r\nrelative to arbitrarily powerful oracles (since Fortnow's algebraic oracles all satisfy Arithmetic Checkability\r\naxiom); by contrast, Local Checkability of [AIV] restricts the\r\noracle power to NP cap co-NP. (ii) Most of the algebrizing\r\ncollapses and separations from [AW], such as IP = PSPACE,\r\nNP subset ZKIP if one-way functions exist, MA-EXP not in P\/poly,\r\netc., are provable from Arithmetic Checkability. (iii) Many\r\nof the open complexity questions (shown to require nonalgebrizing\r\ntechniques in [AW]), such as \"P vs. NP\", \"NP vs.\r\nBPP\", etc., cannot be proved from Arithmetic Checkability.\r\n(iv) Arithmetic Checkability is also insufficient to prove one\r\nknown result, NEXP = MIP.","keywords":["Oracles","arithmetization","algebrization"],"author":[{"@type":"Person","name":"Impagliazzo, Russell","givenName":"Russell","familyName":"Impagliazzo"},{"@type":"Person","name":"Kabanets, Valentine","givenName":"Valentine","familyName":"Kabanets"},{"@type":"Person","name":"Kolokolova, Antonina","givenName":"Antonina","familyName":"Kolokolova"}],"position":3,"pageStart":1,"pageEnd":19,"dateCreated":"2010-01-19","datePublished":"2010-01-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Impagliazzo, Russell","givenName":"Russell","familyName":"Impagliazzo"},{"@type":"Person","name":"Kabanets, Valentine","givenName":"Valentine","familyName":"Kabanets"},{"@type":"Person","name":"Kolokolova, Antonina","givenName":"Antonina","familyName":"Kolokolova"}],"copyrightYear":"2010","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.09421.3","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume773"},{"@type":"ScholarlyArticle","@id":"#article2611","name":"Deterministic approximation algorithms for the nearest codeword problem","abstract":"The Nearest Codeword Problem (NCP) is a basic algorithmic question in the theory of error-correcting codes. Given a point v in F_2^n and a linear space L in F_2^n of dimension k NCP asks to find a point l in L that minimizes the (Hamming) distance from v. \r\n\r\nIt is well-known that the nearest codeword problem is NP-hard. Therefore approximation algorithms are of interest. The best effcient approximation algorithms for the NCP to date are due to Berman and Karpinski. They are a\r\ndeterministic algorithm that achieves an approximation ratio of O(k\/c)\r\nfor an arbitrary constant c; and a randomized algorithm that achieves\r\nan approximation ratio of O(k\/ log n).\r\n\r\nIn this paper we present new deterministic algorithms for approximating\r\nthe NCP that improve substantially upon the earlier work, (almost) de-randomizing the randomized algorithm of Berman and Karpinski.\r\n\r\nWe also initiate a study of the following Remote Point Problem (RPP). Given a linear space L in F_2^n of dimension k RPP asks to find a point v in F_2^n that is far from L. We say that an algorithm achieves a remoteness of r for the RPP if it always outputs a point v that is at least r-far from L. In this paper we present a deterministic polynomial time algorithm that achieves a remoteness of Omega(n log k \/ k) for all k < n\/2. \r\n\r\nWe motivate the remote point problem by relating it to both the nearest codeword problem and the matrix rigidity approach to circuit lower bounds in\r\ncomputational complexity theory.","author":[{"@type":"Person","name":"Alon, Noga","givenName":"Noga","familyName":"Alon"},{"@type":"Person","name":"Panigrahy, Rina","givenName":"Rina","familyName":"Panigrahy"},{"@type":"Person","name":"Yekhanin, Sergey","givenName":"Sergey","familyName":"Yekhanin"}],"position":4,"pageStart":1,"pageEnd":13,"dateCreated":"2010-01-19","datePublished":"2010-01-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Alon, Noga","givenName":"Noga","familyName":"Alon"},{"@type":"Person","name":"Panigrahy, Rina","givenName":"Rina","familyName":"Panigrahy"},{"@type":"Person","name":"Yekhanin, Sergey","givenName":"Sergey","familyName":"Yekhanin"}],"copyrightYear":"2010","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.09421.4","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume773"},{"@type":"ScholarlyArticle","@id":"#article2612","name":"Learning Parities in the Mistake-Bound model","abstract":"We study the problem of learning parity functions that depend on at most $k$ variables ($k$-parities) attribute-efficiently in the mistake-bound model.\r\nWe design a simple, deterministic, polynomial-time algorithm for learning $k$-parities with mistake bound $O(n^{1-frac{c}{k}})$, for any constant $c > 0$. This is the first polynomial-time algorithms that learns $omega(1)$-parities in the mistake-bound model with mistake bound $o(n)$.\r\n\r\nUsing the standard conversion techniques from the mistake-bound model to the PAC model, our algorithm can also be used for learning $k$-parities in the PAC model. In particular, this implies a slight improvement on the results of Klivans and Servedio\r\ncite{rocco} for learning $k$-parities in the PAC model.\r\n\r\nWe also show that the $widetilde{O}(n^{k\/2})$ time algorithm from cite{rocco} that PAC-learns $k$-parities with optimal sample complexity can be extended to the mistake-bound model.","keywords":["Attribute-efficient learning","parities","mistake-bound"],"author":[{"@type":"Person","name":"Buhrman, Harry","givenName":"Harry","familyName":"Buhrman"},{"@type":"Person","name":"Garcia-Soriano, David","givenName":"David","familyName":"Garcia-Soriano"},{"@type":"Person","name":"Matsliah, Arie","givenName":"Arie","familyName":"Matsliah"}],"position":5,"pageStart":1,"pageEnd":9,"dateCreated":"2010-01-19","datePublished":"2010-01-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Buhrman, Harry","givenName":"Harry","familyName":"Buhrman"},{"@type":"Person","name":"Garcia-Soriano, David","givenName":"David","familyName":"Garcia-Soriano"},{"@type":"Person","name":"Matsliah, Arie","givenName":"Arie","familyName":"Matsliah"}],"copyrightYear":"2010","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.09421.5","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume773"},{"@type":"ScholarlyArticle","@id":"#article2613","name":"Planar Graph Isomorphism is in Log-Space","abstract":"Graph Isomorphism is the prime example of a computational problem with a wide\r\ndifference between the best known lower and upper bounds on its complexity. There\r\nis a significant gap between extant lower and upper bounds for planar graphs as well.\r\nWe bridge the gap for this natural and important special case by presenting an upper\r\nbound that matches the known log-space hardness [JKMT03]. In fact, we show the\r\nformally stronger result that planar graph canonization is in log-space. This improves the\r\npreviously known upper bound of AC1 [MR91].\r\nOur algorithm first constructs the biconnected component tree of a connected planar\r\ngraph and then refines each biconnected component into a triconnected component\r\ntree. The next step is to log-space reduce the biconnected planar graph isomorphism and\r\ncanonization problems to those for 3-connected planar graphs, which are known to be in\r\nlog-space by [DLN08]. This is achieved by using the above decomposition, and by making\r\nsignificant modifications to Lindell\u2019s algorithm for tree canonization, along with changes\r\nin the space complexity analysis.\r\nThe reduction from the connected case to the biconnected case requires further new\r\nideas including a non-trivial case analysis and a group theoretic lemma to bound the\r\nnumber of automorphisms of a colored 3-connected planar graph.","keywords":["Planar Graphs","Graph Isomorphism","Logspace"],"author":[{"@type":"Person","name":"Datta, Samir","givenName":"Samir","familyName":"Datta"},{"@type":"Person","name":"Limaye, Nutan","givenName":"Nutan","familyName":"Limaye"},{"@type":"Person","name":"Nimbhorkar, Prajakta","givenName":"Prajakta","familyName":"Nimbhorkar"},{"@type":"Person","name":"Thierauf, Thomas","givenName":"Thomas","familyName":"Thierauf"},{"@type":"Person","name":"Wagner, Fabian","givenName":"Fabian","familyName":"Wagner"}],"position":6,"pageStart":1,"pageEnd":32,"dateCreated":"2010-01-19","datePublished":"2010-01-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Datta, Samir","givenName":"Samir","familyName":"Datta"},{"@type":"Person","name":"Limaye, Nutan","givenName":"Nutan","familyName":"Limaye"},{"@type":"Person","name":"Nimbhorkar, Prajakta","givenName":"Prajakta","familyName":"Nimbhorkar"},{"@type":"Person","name":"Thierauf, Thomas","givenName":"Thomas","familyName":"Thierauf"},{"@type":"Person","name":"Wagner, Fabian","givenName":"Fabian","familyName":"Wagner"}],"copyrightYear":"2010","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.09421.6","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume773"},{"@type":"ScholarlyArticle","@id":"#article2614","name":"Small space analogues of Valiant's classes and the limitations of skew formula","abstract":"In the uniform circuit model of computation, the width of a boolean\r\ncircuit exactly characterises the ``space'' complexity of the\r\ncomputed function. Looking for a similar relationship in Valiant's\r\nalgebraic model of computation, we propose width of an arithmetic\r\ncircuit as a possible measure of space. We introduce the class\r\nVL as an algebraic variant of deterministic log-space L. In\r\nthe uniform setting, we show that our definition coincides with that\r\nof VPSPACE at polynomial width.\r\n\r\nFurther, to define algebraic variants of non-deterministic\r\nspace-bounded classes, we introduce the notion of ``read-once''\r\ncertificates for arithmetic circuits. We show that polynomial-size\r\nalgebraic branching programs can be expressed as a read-once\r\nexponential sum over polynomials in VL, ie\r\n$mbox{VBP}inSigma^R cdotmbox{VL}$.\r\nWe also show that $Sigma^R cdot mbox{VBP} =mbox{VBP}$, ie\r\nVBPs are stable under read-once exponential sums. Further, we\r\nshow that read-once exponential sums over a restricted class of\r\nconstant-width arithmetic circuits are within VQP, and this is the\r\nlargest known such subclass of poly-log-width circuits with this\r\nproperty.\r\n\r\nWe also study the power of skew formulas and show that exponential\r\nsums of a skew formula cannot represent the determinant polynomial.","keywords":["Algebraic circuits","space bounds","circuit width","nondeterministic circuits","skew formulas"],"author":[{"@type":"Person","name":"Mahajan, Meena","givenName":"Meena","familyName":"Mahajan"},{"@type":"Person","name":"Rao B. V., Raghavendra","givenName":"Raghavendra","familyName":"Rao B. V."}],"position":7,"pageStart":1,"pageEnd":23,"dateCreated":"2010-01-19","datePublished":"2010-01-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Mahajan, Meena","givenName":"Meena","familyName":"Mahajan"},{"@type":"Person","name":"Rao B. V., Raghavendra","givenName":"Raghavendra","familyName":"Rao B. V."}],"copyrightYear":"2010","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.09421.7","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume773"},{"@type":"ScholarlyArticle","@id":"#article2615","name":"Unconditional Lower Bounds against Advice","abstract":"We show several unconditional lower bounds for exponential time classes against polynomial time classes with advice, including: (1) For any constant c, NEXP not in P^{NP[n^c]} (2) For any constant c, MAEXP not in MA\/n^c (3) BPEXP not in BPP\/n^{o(1)}. \r\n\r\nIt was previously unknown even whether NEXP in NP\/n^{0.01}. For the probabilistic classes, no lower bounds for uniform exponential time against advice were known before. We also consider the question of whether these lower bounds can be made to work on almost all input lengths rather than on infinitely many. We give an oracle relative to which NEXP in i.o.NP, which provides evidence that this is not possible with current techniques.","keywords":["Advice","derandomization","diagonalization","lower bounds","semantic classes"],"author":[{"@type":"Person","name":"Buhrman, Harry","givenName":"Harry","familyName":"Buhrman"},{"@type":"Person","name":"Fortnow, Lance","givenName":"Lance","familyName":"Fortnow"},{"@type":"Person","name":"Santhanam, Rahul","givenName":"Rahul","familyName":"Santhanam"}],"position":8,"pageStart":1,"pageEnd":11,"dateCreated":"2010-01-19","datePublished":"2010-01-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Buhrman, Harry","givenName":"Harry","familyName":"Buhrman"},{"@type":"Person","name":"Fortnow, Lance","givenName":"Lance","familyName":"Fortnow"},{"@type":"Person","name":"Santhanam, Rahul","givenName":"Rahul","familyName":"Santhanam"}],"copyrightYear":"2010","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.09421.8","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume773"}]}