{"@context":"https:\/\/schema.org\/","@type":"PublicationVolume","@id":"#volume669","volumeNumber":7411,"name":"Dagstuhl Seminar Proceedings, Volume 7411","dateCreated":"2008-01-15","datePublished":"2008-01-15","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":"#volume669"},"hasPart":[{"@type":"ScholarlyArticle","@id":"#article1932","name":"07411 Abstracts Collection \u2013 Algebraic Methods in Computational Complexity","abstract":"From 07.10. to 12.10., the Dagstuhl Seminar 07411 ``Algebraic Methods in Computational Complexity'' was held in the International Conference and Research Center (IBFI),\r\nSchloss Dagstuhl.\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","quantum computing","(de-) randomization"],"author":[{"@type":"Person","name":"Agrawal, Manindra","givenName":"Manindra","familyName":"Agrawal"},{"@type":"Person","name":"Buhrman, Harry","givenName":"Harry","familyName":"Buhrman"},{"@type":"Person","name":"Fortnow, Lance","givenName":"Lance","familyName":"Fortnow"},{"@type":"Person","name":"Thierauf, Thomas","givenName":"Thomas","familyName":"Thierauf"}],"position":1,"pageStart":1,"pageEnd":13,"dateCreated":"2008-01-16","datePublished":"2008-01-16","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Agrawal, Manindra","givenName":"Manindra","familyName":"Agrawal"},{"@type":"Person","name":"Buhrman, Harry","givenName":"Harry","familyName":"Buhrman"},{"@type":"Person","name":"Fortnow, Lance","givenName":"Lance","familyName":"Fortnow"},{"@type":"Person","name":"Thierauf, Thomas","givenName":"Thomas","familyName":"Thierauf"}],"copyrightYear":"2008","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.07411.1","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume669"},{"@type":"ScholarlyArticle","@id":"#article1933","name":"07411 Executive Summary \u2013 Algebraic Methods in Computational Complexity","abstract":"The seminar brought together almost 50 researchers covering a wide\r\nspectrum of complexity theory. The focus on algebraic methods showed\r\nonce again the great importance of algebraic techniques for\r\ntheoretical computer science. We had almost 30 talks of length\r\nbetween 15 and 45 minutes. This left enough room for discussions. We\r\nhad an open problem session that was very much appreciated.","keywords":["Computational complexity","algebra","quantum computing","(de-) randomization"],"author":[{"@type":"Person","name":"Agrawal, Manindra","givenName":"Manindra","familyName":"Agrawal"},{"@type":"Person","name":"Buhrman, Harry","givenName":"Harry","familyName":"Buhrman"},{"@type":"Person","name":"Fortnow, Lance","givenName":"Lance","familyName":"Fortnow"},{"@type":"Person","name":"Thierauf, Thomas","givenName":"Thomas","familyName":"Thierauf"}],"position":2,"pageStart":1,"pageEnd":3,"dateCreated":"2008-01-16","datePublished":"2008-01-16","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Agrawal, Manindra","givenName":"Manindra","familyName":"Agrawal"},{"@type":"Person","name":"Buhrman, Harry","givenName":"Harry","familyName":"Buhrman"},{"@type":"Person","name":"Fortnow, Lance","givenName":"Lance","familyName":"Fortnow"},{"@type":"Person","name":"Thierauf, Thomas","givenName":"Thomas","familyName":"Thierauf"}],"copyrightYear":"2008","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.07411.2","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume669"},{"@type":"ScholarlyArticle","@id":"#article1934","name":"Classical Simulation Complexity of Quantum Branching Programs","abstract":"We present classical simulation techniques for measure once quantum\r\n branching programs.\r\n \r\n For bounded error syntactic quantum branching program of width $w$\r\n that computes a function with error $delta$ we present a classical\r\n deterministic branching program of the same length and width at most\r\n $(1+2\/(1-2delta))^{2w}$ that computes the same function.\r\n \r\n Second technique is a classical stochastic simulation technique for\r\n bounded error and unbounded error quantum branching programs. Our\r\n result is that it is possible stochastically-classically simulate\r\n quantum branching programs with the same length and almost the same\r\n width, but we lost bounded error acceptance property.","keywords":["Quantum algorithms","Branching Programs","Complexity"],"author":{"@type":"Person","name":"Ablayev, Farid","givenName":"Farid","familyName":"Ablayev"},"position":3,"pageStart":1,"pageEnd":10,"dateCreated":"2008-01-23","datePublished":"2008-01-23","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Ablayev, Farid","givenName":"Farid","familyName":"Ablayev"},"copyrightYear":"2008","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.07411.3","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume669"},{"@type":"ScholarlyArticle","@id":"#article1935","name":"Diagonal Circuit Identity Testing and Lower Bounds","abstract":"In this talk we give a deterministic polynomial time algorithm for testing whether a {em diagonal}\r\ndepth-$3$ circuit $C(arg{x}{n})$ (i.e. $C$ is a sum of powers of linear functions) is \r\nzero.","keywords":["Arithmetic circuit","identity testing","depth 3","depth 4","determinant","permanent","lower bound"],"author":{"@type":"Person","name":"Saxena, Nitin","givenName":"Nitin","familyName":"Saxena"},"position":4,"pageStart":1,"pageEnd":1,"dateCreated":"2008-01-22","datePublished":"2008-01-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Saxena, Nitin","givenName":"Nitin","familyName":"Saxena"},"copyrightYear":"2008","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.07411.4","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume669"},{"@type":"ScholarlyArticle","@id":"#article1936","name":"High Entropy Random Selection Protocols","abstract":"We study the two party problem of randomly selecting a string among\r\n all the strings of length n. We want the protocol to have the\r\n property that the output distribution has high entropy, even\r\n when one of the two parties is dishonest and deviates from the\r\n protocol. We develop protocols that achieve high, close to n,\r\n entropy.\r\n\r\n In the literature the randomness guarantee is usually expressed as\r\n being close to the uniform distribution or in terms of resiliency.\r\n The notion of entropy is not directly comparable to that of\r\n resiliency, but we establish a connection between the two that\r\n allows us to compare our protocols with the existing ones.\r\n\r\nWe construct an\r\n explicit protocol that yields entropy n - O(1) and has 4log^* n\r\n rounds, improving over the protocol of Goldwasser\r\n et al. that also achieves this entropy but needs O(n)\r\n rounds. Both these protocols need O(n^2) bits of communication.\r\n\r\n Next we reduce the communication in our protocols. We show the existence,\r\n non-explicitly, of a protocol that has 6-rounds, 2n + 8log n bits\r\n of communication and yields entropy n- O(log n) and min-entropy\r\n n\/2 - O(log n). Our protocol achieves the same entropy bound as\r\n the recent, also non-explicit, protocol of Gradwohl\r\n et al., however achieves much higher min-entropy: n\/2 -\r\n O(log n) versus O(log n).\r\n\r\n Finally we exhibit very simple explicit protocols. We connect the\r\n security parameter of these geometric protocols with the well\r\n studied Kakeya problem motivated by harmonic analysis and analytical\r\n number theory. We are only able to prove that these protocols have\r\n entropy 3n\/4 but still n\/2 - O(log n) min-entropy. Therefore\r\n they do not perform as well with respect to the explicit\r\n constructions of Gradwohl et al. entropy-wise, but still\r\n have much better min-entropy. We conjecture that these simple\r\n protocols achieve n -o(n) entropy. Our geometric\r\n construction and its relation to the Kakeya problem follows a new and\r\n different approach to the random selection problem than any of the\r\n previously known protocols.","keywords":["Shannon entropy","Random string ds"],"author":[{"@type":"Person","name":"Vereshchagin, Nikolai K.","givenName":"Nikolai K.","familyName":"Vereshchagin"},{"@type":"Person","name":"Buhrman, Harry","givenName":"Harry","familyName":"Buhrman"},{"@type":"Person","name":"Cristandl, Matthias","givenName":"Matthias","familyName":"Cristandl"},{"@type":"Person","name":"Koucky, Michal","givenName":"Michal","familyName":"Koucky"},{"@type":"Person","name":"Lotker, Zvi","givenName":"Zvi","familyName":"Lotker"},{"@type":"Person","name":"Patt-Shamir, Boaz","givenName":"Boaz","familyName":"Patt-Shamir"}],"position":5,"pageStart":1,"pageEnd":0,"dateCreated":"2008-01-23","datePublished":"2008-01-23","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Vereshchagin, Nikolai K.","givenName":"Nikolai K.","familyName":"Vereshchagin"},{"@type":"Person","name":"Buhrman, Harry","givenName":"Harry","familyName":"Buhrman"},{"@type":"Person","name":"Cristandl, Matthias","givenName":"Matthias","familyName":"Cristandl"},{"@type":"Person","name":"Koucky, Michal","givenName":"Michal","familyName":"Koucky"},{"@type":"Person","name":"Lotker, Zvi","givenName":"Zvi","familyName":"Lotker"},{"@type":"Person","name":"Patt-Shamir, Boaz","givenName":"Boaz","familyName":"Patt-Shamir"}],"copyrightYear":"2008","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.07411.5","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume669"},{"@type":"ScholarlyArticle","@id":"#article1937","name":"The Unique Games Conjecture with Entangled Provers is False","abstract":"We consider one-round games between a classical verifier and two provers who share entanglement. We show that\r\nwhen the constraints enforced by the verifier are `unique' constraints (i.e., permutations), the value of the\r\ngame can be well approximated by a semidefinite program. Essentially the only algorithm known previously was\r\nfor the special case of binary answers, as follows from the work of Tsirelson in 1980. Among other things,\r\nour result implies that the variant of the unique games conjecture where we allow the provers to share\r\nentanglement is false. Our proof is based on a novel `quantum rounding technique', showing how to take a\r\nsolution to an SDP and transform it to a strategy for entangled provers.","keywords":["Unique games","entanglement"],"author":[{"@type":"Person","name":"Kempe, Julia","givenName":"Julia","familyName":"Kempe"},{"@type":"Person","name":"Regev, Oded","givenName":"Oded","familyName":"Regev"},{"@type":"Person","name":"Toner, Ben","givenName":"Ben","familyName":"Toner"}],"position":6,"pageStart":1,"pageEnd":17,"dateCreated":"2008-01-15","datePublished":"2008-01-15","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kempe, Julia","givenName":"Julia","familyName":"Kempe"},{"@type":"Person","name":"Regev, Oded","givenName":"Oded","familyName":"Regev"},{"@type":"Person","name":"Toner, Ben","givenName":"Ben","familyName":"Toner"}],"copyrightYear":"2008","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.07411.6","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume669"},{"@type":"ScholarlyArticle","@id":"#article1938","name":"Uniqueness of Optimal Mod 3 Circuits for Parity","abstract":"We prove that the quadratic polynomials modulo $3$\r\n with the largest correlation with parity are unique up to\r\n permutation of variables and constant factors. As a consequence of\r\n our result, we completely characterize the smallest \r\nMAJ~$circ mbox{MOD}_3 circ {\r\nm AND}_2$ circuits that compute parity, where a\r\n MAJ~$circ mbox{MOD}_3 circ {\r\nm AND}_2$ circuit is one that has a\r\n majority gate as output, a middle layer of MOD$_3$ gates and a\r\n bottom layer of AND gates of fan-in $2$. We\r\n also prove that the sub-optimal circuits exhibit a stepped behavior:\r\n any sub-optimal circuits of this class that compute parity \r\n must have size at least a factor of $frac{2}{sqrt{3}}$ times the\r\n optimal size. This verifies, for the special case of $m=3$,\r\n two conjectures made\r\n by Due~{n}ez, Miller, Roy and Straubing (Journal of Number Theory, 2006) for general MAJ~$circ mathrm{MOD}_m circ\r\n {\r\nm AND}_2$ circuits for any odd $m$. The correlation\r\n and circuit bounds are obtained by studying the associated\r\n exponential sums, based on some of the techniques developed \r\n by Green (JCSS, 2004). We regard this as a step towards\r\n obtaining tighter bounds both for the $m \r\not = 3$ quadratic\r\n case as well as for\r\n higher degrees.","keywords":["Circuit complexity","correlations","exponential sums"],"author":[{"@type":"Person","name":"Green, Frederic","givenName":"Frederic","familyName":"Green"},{"@type":"Person","name":"Roy, Amitabha","givenName":"Amitabha","familyName":"Roy"}],"position":7,"pageStart":1,"pageEnd":15,"dateCreated":"2008-01-15","datePublished":"2008-01-15","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Green, Frederic","givenName":"Frederic","familyName":"Green"},{"@type":"Person","name":"Roy, Amitabha","givenName":"Amitabha","familyName":"Roy"}],"copyrightYear":"2008","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.07411.7","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume669"}]}