S. Chechik, Thomas Dueholm Hansen, G. Italiano, Jakub Lacki, Nikos Parotsidis
We present randomized algorithms with a total update time of Õ(m √n) for the problems of decremental single source reachability and decremental strongly connected components on directed graphs. This improves recent breakthrough results of Henzinger, Krinninger and Nanongkai [STOC 14, ICALP 15]. In addition, our algorithms are arguably simpler.
{"title":"Decremental Single-Source Reachability and Strongly Connected Components in Õ(m√n) Total Update Time","authors":"S. Chechik, Thomas Dueholm Hansen, G. Italiano, Jakub Lacki, Nikos Parotsidis","doi":"10.1109/FOCS.2016.42","DOIUrl":"https://doi.org/10.1109/FOCS.2016.42","url":null,"abstract":"We present randomized algorithms with a total update time of Õ(m √n) for the problems of decremental single source reachability and decremental strongly connected components on directed graphs. This improves recent breakthrough results of Henzinger, Krinninger and Nanongkai [STOC 14, ICALP 15]. In addition, our algorithms are arguably simpler.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117178922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Y. Azar, Niv Buchbinder, T-H. Hubert Chan, Shahar Chen, I. Cohen, Anupam Gupta, Zhiyi Huang, N. Kang, V. Nagarajan, J. Naor, Debmalya Panigrahi
We present online algorithms for covering and packing problems with (non-linear) convex objectives. The convex covering problem is defined as: minxϵR+nf(x) s.t. Ax ≥ 1, where f:R+n → R+ is a monotone convex function, and A is an m×n matrix with non-negative entries. In the online version, a new row of the constraint matrix, representing a new covering constraint, is revealed in each step and the algorithm is required to maintain a feasible and monotonically non-decreasing assignment x over time. We also consider a convex packing problem defined as: maxyϵR+m Σj=1m yj - g(AT y), where g:R+n→R+ is a monotone convex function. In the online version, each variable yj arrives online and the algorithm must decide the value of yj on its arrival. This represents the Fenchel dual of the convex covering program, when g is the convex conjugate of f. We use a primal-dual approach to give online algorithms for these generic problems, and use them to simplify, unify, and improve upon previous results for several applications.
{"title":"Online Algorithms for Covering and Packing Problems with Convex Objectives","authors":"Y. Azar, Niv Buchbinder, T-H. Hubert Chan, Shahar Chen, I. Cohen, Anupam Gupta, Zhiyi Huang, N. Kang, V. Nagarajan, J. Naor, Debmalya Panigrahi","doi":"10.1109/FOCS.2016.24","DOIUrl":"https://doi.org/10.1109/FOCS.2016.24","url":null,"abstract":"We present online algorithms for covering and packing problems with (non-linear) convex objectives. The convex covering problem is defined as: min<sub>xϵ</sub>R<sub>+</sub><sup>n</sup>f(x) s.t. Ax ≥ 1, where f:R<sub>+</sub><sup>n</sup> → R<sub>+</sub> is a monotone convex function, and A is an m×n matrix with non-negative entries. In the online version, a new row of the constraint matrix, representing a new covering constraint, is revealed in each step and the algorithm is required to maintain a feasible and monotonically non-decreasing assignment x over time. We also consider a convex packing problem defined as: max<sub>yϵR+</sub><sup>m</sup> Σ<sub>j=1</sub><sup>m</sup> yj - g(A<sup>T</sup> y), where g:R<sub>+</sub><sup>n</sup>→R<sub>+</sub> is a monotone convex function. In the online version, each variable yj arrives online and the algorithm must decide the value of yj on its arrival. This represents the Fenchel dual of the convex covering program, when g is the convex conjugate of f. We use a primal-dual approach to give online algorithms for these generic problems, and use them to simplify, unify, and improve upon previous results for several applications.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130581486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct two-message non-malleable commitments with respect to opening in the standard model, assuming only one-to-one one-way functions. Our protocol consists of two unidirectional messages by the committer (with no message from the receiver), and is secure against all polynomial-time adversaries in the standard synchronous setting. Pass (TCC 2013) proved that any commitment scheme with non-malleability with respect to commitment, using only 2 rounds of communication, cannot be proved secure via a black-box reduction to any "standard" intractability assumption. We extend this by showing a similar impossibility result for commitments with non-malleability with respect to opening, another standard notion of non-malleability for commitments, for any 2-message challenge-response protocol, as well. However, somewhat surprisingly, we show that this barrier breaks down in the setting of two unidirectional messages by the committer (with no message from the receiver), for non-malleability with respect to opening. ° Our protocol makes only black-box use of any non-interactive statistically binding commitment scheme. Such a scheme can be based on any one-to-one one-way function. ° Our techniques depart significantly from the commit-challenge-response structure followed by nearly all prior works on non-malleable protocols in the standard model. Our methods are combinatorial in nature. ° Our protocol resolves the round complexity of commitments with non-malleability with respect to opening via natural (non-embedding) black-box security reductions. We show that completely non-interactive non-malleable commitments w.r.t. opening cannot be proved secure via most natural black-box reductions. This result extends to also rule out bi-directional two-message non-malleable commitments w.r.t. opening in the synchronous or asynchronous setting. ° Our protocol, together with our impossibility result, also resolves the round complexity of block-wise non-malleable codes (Chandran et al) w.r.t. natural black-box reductions.
{"title":"Breaking the Three Round Barrier for Non-malleable Commitments","authors":"Vipul Goyal, Dakshita Khurana, A. Sahai","doi":"10.1109/FOCS.2016.12","DOIUrl":"https://doi.org/10.1109/FOCS.2016.12","url":null,"abstract":"We construct two-message non-malleable commitments with respect to opening in the standard model, assuming only one-to-one one-way functions. Our protocol consists of two unidirectional messages by the committer (with no message from the receiver), and is secure against all polynomial-time adversaries in the standard synchronous setting. Pass (TCC 2013) proved that any commitment scheme with non-malleability with respect to commitment, using only 2 rounds of communication, cannot be proved secure via a black-box reduction to any \"standard\" intractability assumption. We extend this by showing a similar impossibility result for commitments with non-malleability with respect to opening, another standard notion of non-malleability for commitments, for any 2-message challenge-response protocol, as well. However, somewhat surprisingly, we show that this barrier breaks down in the setting of two unidirectional messages by the committer (with no message from the receiver), for non-malleability with respect to opening. ° Our protocol makes only black-box use of any non-interactive statistically binding commitment scheme. Such a scheme can be based on any one-to-one one-way function. ° Our techniques depart significantly from the commit-challenge-response structure followed by nearly all prior works on non-malleable protocols in the standard model. Our methods are combinatorial in nature. ° Our protocol resolves the round complexity of commitments with non-malleability with respect to opening via natural (non-embedding) black-box security reductions. We show that completely non-interactive non-malleable commitments w.r.t. opening cannot be proved secure via most natural black-box reductions. This result extends to also rule out bi-directional two-message non-malleable commitments w.r.t. opening in the synchronous or asynchronous setting. ° Our protocol, together with our impossibility result, also resolves the round complexity of block-wise non-malleable codes (Chandran et al) w.r.t. natural black-box reductions.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126152112","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Establishing the complexity of Bounded Distance Decoding for Reed-Solomon codes is a fundamental open problem in coding theory, explicitly asked by Guruswami and Vardy (IEEE Trans. Inf. Theory, 2005). The problem is motivated by the large current gap between the regime when it is NP-hard, and the regime when it is efficiently solvable (i.e., the Johnson radius). We show the first NP-hardness results for asymptotically smaller decoding radii than the maximum likelihood decoding radius of Guruswami and Vardy. Specifically, for Reed-Solomon codes of length N and dimension K = O(N), we show that it is NP-hard to decode more than N-K-O/log N log log N) errors. Moreover, we show that the problem is NP-hard under quasipolynomial-time reductions for an error amount > N-K-c log N (with c > 0 an absolute constant). An alternative natural reformulation of the Bounded Distance Decoding problem for Reed-Solomon codes is as a Polynomial Reconstruction problem. In this view, our results show that it is NP-hard to decide whether there exists a degree K polynomial passing through K + O(log N / log log N) points from a given set of points (a1, b1), (a2, b2) ..., (aN, bN). Furthermore, it is NP-hard under quasipolynomial-time reductions to decide whether there is a degree K polynomial passing through K + c log N many points (with c > 0 an absolute constant). These results follow from the NP-hardness of a generalization of the classical Subset Sum problem to higher moments, called Moments Subset Sum, which has been a known open problem, and which may be of independent interest. We further reveal a strong connection with the well-studied Prouhet-Tarry-Escott problem in Number Theory, which turns out to capture a main barrier in extending our techniques. We believe the Prouhet-Tarry-Escott problem deserves further study in the theoretical computer science community.
确定Reed-Solomon码的有界距离解码的复杂性是编码理论中的一个基本开放问题，由Guruswami和Vardy (IEEE Trans.)明确提出。Inf. Theory, 2005)。该问题的动机是np困难状态与有效可解状态(即约翰逊半径)之间存在较大的电流间隙。我们展示了第一个np -硬度结果，其解码半径渐近小于Guruswami和Vardy的最大似然解码半径。具体来说，对于长度为N，维数为K = O(N)的Reed-Solomon码，我们证明了它是NP-hard解码超过N-K-O/log N log log N)个错误。此外，我们证明了在准多项式时间约简下，当误差量> N- k -c log N (c > 0是绝对常数)时，问题是np困难的。Reed-Solomon码的有界距离译码问题的另一种自然重构是多项式重构问题。在这种观点下，我们的结果表明，在给定的点(a1, b1)， (a2, b2)…的集合中，是否存在K次多项式经过K + O(log N / log log N)个点是np困难的。， (aN, bN)。此外，在拟多项式时间约简下，判定是否存在K次多项式经过K + c log N个点(c > 0为绝对常数)是np困难的。这些结果来自于将经典子集和问题推广到更高矩的np -硬度，称为矩子集和，这是一个已知的开放问题，并且可能具有独立的兴趣。我们进一步揭示了与数论中得到充分研究的Prouhet-Tarry-Escott问题的密切联系，该问题被证明抓住了扩展我们技术的主要障碍。我们认为prouet - tarry - escott问题值得在理论计算机科学界进一步研究。
{"title":"NP-Hardness of Reed-Solomon Decoding and the Prouhet-Tarry-Escott Problem","authors":"V. Gandikota, Badih Ghazi, Elena Grigorescu","doi":"10.1109/FOCS.2016.86","DOIUrl":"https://doi.org/10.1109/FOCS.2016.86","url":null,"abstract":"Establishing the complexity of Bounded Distance Decoding for Reed-Solomon codes is a fundamental open problem in coding theory, explicitly asked by Guruswami and Vardy (IEEE Trans. Inf. Theory, 2005). The problem is motivated by the large current gap between the regime when it is NP-hard, and the regime when it is efficiently solvable (i.e., the Johnson radius). We show the first NP-hardness results for asymptotically smaller decoding radii than the maximum likelihood decoding radius of Guruswami and Vardy. Specifically, for Reed-Solomon codes of length N and dimension K = O(N), we show that it is NP-hard to decode more than N-K-O/log N log log N) errors. Moreover, we show that the problem is NP-hard under quasipolynomial-time reductions for an error amount > N-K-c log N (with c > 0 an absolute constant). An alternative natural reformulation of the Bounded Distance Decoding problem for Reed-Solomon codes is as a Polynomial Reconstruction problem. In this view, our results show that it is NP-hard to decide whether there exists a degree K polynomial passing through K + O(log N / log log N) points from a given set of points (a1, b1), (a2, b2) ..., (aN, bN). Furthermore, it is NP-hard under quasipolynomial-time reductions to decide whether there is a degree K polynomial passing through K + c log N many points (with c > 0 an absolute constant). These results follow from the NP-hardness of a generalization of the classical Subset Sum problem to higher moments, called Moments Subset Sum, which has been a known open problem, and which may be of independent interest. We further reveal a strong connection with the well-studied Prouhet-Tarry-Escott problem in Number Theory, which turns out to capture a main barrier in extending our techniques. We believe the Prouhet-Tarry-Escott problem deserves further study in the theoretical computer science community.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121014257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend the recent hierarchy results of Rossman, Servedio and Tan [1] to address circuits of almost logarithmic depth. Our proof uses the same basic approach as [1] but a number of small differences enables us to obtain a stronger result by a significantly shorter proof.
{"title":"An Average-Case Depth Hierarchy Theorem for Higher Depth","authors":"J. Håstad","doi":"10.1109/FOCS.2016.18","DOIUrl":"https://doi.org/10.1109/FOCS.2016.18","url":null,"abstract":"We extend the recent hierarchy results of Rossman, Servedio and Tan [1] to address circuits of almost logarithmic depth. Our proof uses the same basic approach as [1] but a number of small differences enables us to obtain a stronger result by a significantly shorter proof.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128395132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the two-party communication complexity of finding an approximate Brouwer fixed point of a composition of two Lipschitz functions g o f: [0,1]n → [0,1]n, where Alice holds f and Bob holds g. We prove an exponential (in n) lower bound on the deterministic communication complexity of this problem. Our technical approach is to adapt the Raz-McKenzie simulation theorem (FOCS 1999) into geometric settings, thereby "smoothly lifting" the deterministic query lower bound for finding an approximate fixed point (Hirsch, Papadimitriou and Vavasis, Complexity 1989) from the oracle model to the two-party model. Our results also suggest an approach to the well-known open problem of proving strong lower bounds on the communication complexity of computing approximate Nash equilibria. Specifically, we show that a slightly "smoother" version of our fixed-point computation lower bound (by an absolute constant factor) would imply that: The deterministic two-party communication complexity of finding an ∈ = Ω(1/log2 N)-approximate Nash equilibrium in an N × N bimatrix game (where each player knows only his own payoff matrix) is at least Nγ for some constant γ > 0. (In contrast, the nondeterministic communication complexity of this problem is only O(log6 N)). ; The deterministic (Number-In-Hand) multiparty communication complexity of finding an ∈ = Ω(1)-Nash equilibrium in a k-player constant-action game is at least 2Ω(k/log k) (while the nondeterministic communication complexity is only O(k)).
{"title":"On the Communication Complexity of Approximate Fixed Points","authors":"T. Roughgarden, Omri Weinstein","doi":"10.1109/FOCS.2016.32","DOIUrl":"https://doi.org/10.1109/FOCS.2016.32","url":null,"abstract":"We study the two-party communication complexity of finding an approximate Brouwer fixed point of a composition of two Lipschitz functions g o f: [0,1]n → [0,1]n, where Alice holds f and Bob holds g. We prove an exponential (in n) lower bound on the deterministic communication complexity of this problem. Our technical approach is to adapt the Raz-McKenzie simulation theorem (FOCS 1999) into geometric settings, thereby \"smoothly lifting\" the deterministic query lower bound for finding an approximate fixed point (Hirsch, Papadimitriou and Vavasis, Complexity 1989) from the oracle model to the two-party model. Our results also suggest an approach to the well-known open problem of proving strong lower bounds on the communication complexity of computing approximate Nash equilibria. Specifically, we show that a slightly \"smoother\" version of our fixed-point computation lower bound (by an absolute constant factor) would imply that: The deterministic two-party communication complexity of finding an ∈ = Ω(1/log2 N)-approximate Nash equilibrium in an N × N bimatrix game (where each player knows only his own payoff matrix) is at least Nγ for some constant γ > 0. (In contrast, the nondeterministic communication complexity of this problem is only O(log6 N)). ; The deterministic (Number-In-Hand) multiparty communication complexity of finding an ∈ = Ω(1)-Nash equilibrium in a k-player constant-action game is at least 2Ω(k/log k) (while the nondeterministic communication complexity is only O(k)).","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128576138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
K. Bringmann, F. Grandoni, B. Saha, V. V. Williams
It is a major open problem whether the (min,+)-product of two n by n matrices has a truly sub-cubic time algorithm, as it is equivalent to the famous All-Pairs-Shortest-Paths problem (APSP) in n-vertex graphs. There are some restrictions of the (min,+)-product to special types of matrices that admit truly sub-cubic algorithms, each giving rise to a special case of APSP that can be solved faster. In this paper we consider a new, different and powerful restriction in which one matrix can be arbitrary, as long as the other matrix has "bounded differences" in either its columns or rows, i.e. any two consecutive entries differ by only a small amount. We obtain the first truly sub-cubic algorithm for this Bounded Differences (min,+)-product (answering an open problem of Chan and Lewenstein). Our new algorithm, combined with a strengthening of an approach of L. Valiant for solving context-free grammar parsing with matrix multiplication, yields the first truly sub-cubic algorithms for the following problems: Language Edit Distance (a major problem in the parsing community), RNA-folding (a major problem in bioinformatics) and Optimum Stack Generation (answering an open problem of Tarjan).
{"title":"Truly Sub-cubic Algorithms for Language Edit Distance and RNA-Folding via Fast Bounded-Difference Min-Plus Product","authors":"K. Bringmann, F. Grandoni, B. Saha, V. V. Williams","doi":"10.1109/FOCS.2016.48","DOIUrl":"https://doi.org/10.1109/FOCS.2016.48","url":null,"abstract":"It is a major open problem whether the (min,+)-product of two n by n matrices has a truly sub-cubic time algorithm, as it is equivalent to the famous All-Pairs-Shortest-Paths problem (APSP) in n-vertex graphs. There are some restrictions of the (min,+)-product to special types of matrices that admit truly sub-cubic algorithms, each giving rise to a special case of APSP that can be solved faster. In this paper we consider a new, different and powerful restriction in which one matrix can be arbitrary, as long as the other matrix has \"bounded differences\" in either its columns or rows, i.e. any two consecutive entries differ by only a small amount. We obtain the first truly sub-cubic algorithm for this Bounded Differences (min,+)-product (answering an open problem of Chan and Lewenstein). Our new algorithm, combined with a strengthening of an approach of L. Valiant for solving context-free grammar parsing with matrix multiplication, yields the first truly sub-cubic algorithms for the following problems: Language Edit Distance (a major problem in the parsing community), RNA-folding (a major problem in bioinformatics) and Optimum Stack Generation (answering an open problem of Tarjan).","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"146 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116386902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
All constructions of general purpose indistinguishability obfuscation (IO) rely on either meta-assumptions that encapsulate an exponential family of assumptions (e.g., Pass, Seth and Telang, CRYPTO 2014 and Lin, EUROCRYPT 2016), or polynomial families of assumptions on graded encoding schemes with a high polynomial degree/multilinearity (e.g., Gentry, Lewko, Sahai and Waters, FOCS 2014). We present a new construction of IO, with a security reduction based on two assumptions: (a) a DDH-like assumption - called the sSXDH assumption - on constant degree graded encodings, and (b) the existence of polynomial-stretch pseudorandom generators (PRG) in NC0. Our assumption on graded encodings is simple, has constant size, and does not require handling composite-order rings. This narrows the gap between the mathematical objects that exist (bilinear maps, from elliptic curve groups) and ones that suffice to construct general purpose indistinguishability obfuscation.
{"title":"Indistinguishability Obfuscation from DDH-Like Assumptions on Constant-Degree Graded Encodings","authors":"Huijia Lin, V. Vaikuntanathan","doi":"10.1109/FOCS.2016.11","DOIUrl":"https://doi.org/10.1109/FOCS.2016.11","url":null,"abstract":"All constructions of general purpose indistinguishability obfuscation (IO) rely on either meta-assumptions that encapsulate an exponential family of assumptions (e.g., Pass, Seth and Telang, CRYPTO 2014 and Lin, EUROCRYPT 2016), or polynomial families of assumptions on graded encoding schemes with a high polynomial degree/multilinearity (e.g., Gentry, Lewko, Sahai and Waters, FOCS 2014). We present a new construction of IO, with a security reduction based on two assumptions: (a) a DDH-like assumption - called the sSXDH assumption - on constant degree graded encodings, and (b) the existence of polynomial-stretch pseudorandom generators (PRG) in NC0. Our assumption on graded encodings is simple, has constant size, and does not require handling composite-order rings. This narrows the gap between the mathematical objects that exist (bilinear maps, from elliptic curve groups) and ones that suffice to construct general purpose indistinguishability obfuscation.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131449323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop a paradigm for studying multi-player deterministic communication, based on a novel combinatorial concept that we call a strong fooling set. Our paradigm leads to optimal lower bounds on the per-player communication required for solving multi-player EQUALITY problems in a private-message setting. This in turn gives a very strong - O(1) versus Ω(n) - separation between private-message and one-way blackboard communication complexities. Applying our communication complexity results, we show that for deterministic data streaming algorithms, even loose estimations of some basic statistics of an input stream require large amounts of space. For instance, approximating the frequency moment Fk within a factor α requires Ω(n/α1/(1-k)) space for k > 1 and roughly Ω(n/αk/(k-1)) space for k > 1. In particular, approximation within any constant factor α, however large, requires linear space, with the trivial exception of k = 1. This is in sharp contrast to the situation for randomized streaming algorithms, which can approximate Fk to within (1±ε) factors using Õ(1) space for k ≤ 2 and o(n) space for all finite k and all constant ε > 0. Previous linear-space lower bounds for deterministic estimation were limited to small factors α, such as α <; 2 for approximating F0 or F2. We also provide certain space/approximation tradeoffs in a deterministic setting for the problems of estimating the empirical entropy of a stream as well as the size of the maximum matching and the edge connectivity of a streamed graph.
{"title":"Strong Fooling Sets for Multi-player Communication with Applications to Deterministic Estimation of Stream Statistics","authors":"Amit Chakrabarti, S. Kale","doi":"10.1109/FOCS.2016.14","DOIUrl":"https://doi.org/10.1109/FOCS.2016.14","url":null,"abstract":"We develop a paradigm for studying multi-player deterministic communication, based on a novel combinatorial concept that we call a strong fooling set. Our paradigm leads to optimal lower bounds on the per-player communication required for solving multi-player EQUALITY problems in a private-message setting. This in turn gives a very strong - O(1) versus Ω(n) - separation between private-message and one-way blackboard communication complexities. Applying our communication complexity results, we show that for deterministic data streaming algorithms, even loose estimations of some basic statistics of an input stream require large amounts of space. For instance, approximating the frequency moment Fk within a factor α requires Ω(n/α1/(1-k)) space for k > 1 and roughly Ω(n/αk/(k-1)) space for k > 1. In particular, approximation within any constant factor α, however large, requires linear space, with the trivial exception of k = 1. This is in sharp contrast to the situation for randomized streaming algorithms, which can approximate Fk to within (1±ε) factors using Õ(1) space for k ≤ 2 and o(n) space for all finite k and all constant ε > 0. Previous linear-space lower bounds for deterministic estimation were limited to small factors α, such as α <; 2 for approximating F0 or F2. We also provide certain space/approximation tradeoffs in a deterministic setting for the problems of estimating the empirical entropy of a stream as well as the size of the maximum matching and the edge connectivity of a streamed graph.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129067124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give new algorithms based on the sum-of-squares method for tensor decomposition. Our results improve the best known running times from quasi-polynomial to polynomial for several problems, including decomposing random overcomplete 3-tensors and learning overcomplete dictionaries with constant relative sparsity. We also give the first robust analysis for decomposing overcomplete 4-tensors in the smoothed analysis model. A key ingredient of our analysis is to establish small spectral gaps in moment matrices derived from solutions to sum-of-squares relaxations. To enable this analysis we augment sum-of-squaresrelaxations with spectral analogs of maximum entropy constraints.
{"title":"Polynomial-Time Tensor Decompositions with Sum-of-Squares","authors":"Tengyu Ma, Jonathan Shi, David Steurer","doi":"10.1109/FOCS.2016.54","DOIUrl":"https://doi.org/10.1109/FOCS.2016.54","url":null,"abstract":"We give new algorithms based on the sum-of-squares method for tensor decomposition. Our results improve the best known running times from quasi-polynomial to polynomial for several problems, including decomposing random overcomplete 3-tensors and learning overcomplete dictionaries with constant relative sparsity. We also give the first robust analysis for decomposing overcomplete 4-tensors in the smoothed analysis model. A key ingredient of our analysis is to establish small spectral gaps in moment matrices derived from solutions to sum-of-squares relaxations. To enable this analysis we augment sum-of-squaresrelaxations with spectral analogs of maximum entropy constraints.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116899530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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