2009 24th Annual IEEE Conference on Computational Complexity最新文献_第2页

The Complexity of the Annihilating Polynomial 湮灭多项式的复杂度

2009 24th Annual IEEE Conference on Computational Complexity Pub Date : 2009-07-15 DOI: 10.1109/CCC.2009.37

N. Kayal

{"title":"The Complexity of the Annihilating Polynomial","authors":"N. Kayal","doi":"10.1109/CCC.2009.37","DOIUrl":"https://doi.org/10.1109/CCC.2009.37","url":null,"abstract":"Let F be a field and f_1, ..., f_k in F[x_1, ..., x_n] be a set of k polynomials of degree d in n variables over the field F. These polynomials are said to be algebraically dependent if there exists a nonzero k-variate polynomial A(t_1, ..., t_k) in F[t_1, ..., t_k] such that A(f_1, ..., f_k) = 0. A is then called an (f_1, ..., f_k)-annihilating polynomial. Within computer science, the notion of algebraic dependence was used in Dvir, Gabizon and Wigderson to construct explicit deterministic extractors from low-degree polynomial sources. They also observed that given (f_1, ..., f_k) as arithmetic circuits, there exists an efficient randomized algorithm for testing their algebraic independence. The problems of determining good bounds on the degree of the annihilating polynomial and of computing it explicitly were posed as open questions. We solve the two posed problems in the following way: ≫≫ We give closely matching upper and lower bounds for the degree of the annihilating polynomial. ≫≫ We show that it is NP-hard to decide if A(0, .. ,0) equals zero. Indeed the annihilating polynomial A(t_1, .., t_k)$ does not even admit a small circuit representation unless the polynomial hierarchy collapses. This then, to the best of our knowledge, is the only natural computational problem where determining the existence of an object (the annihilating polynomial in our case) can be done efficiently but the actual computation of the object is provably hard.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133600753","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}

引用次数: 34

Quantum Copy-Protection and Quantum Money 量子复制保护和量子货币

2009 24th Annual IEEE Conference on Computational Complexity Pub Date : 2009-07-15 DOI: 10.1109/CCC.2009.42

S. Aaronson

{"title":"Quantum Copy-Protection and Quantum Money","authors":"S. Aaronson","doi":"10.1109/CCC.2009.42","DOIUrl":"https://doi.org/10.1109/CCC.2009.42","url":null,"abstract":"Forty years ago, Wiesner proposed using quantum states to create money that is physically impossible to counterfeit, something that cannot be done in the classical world. However, Wiesner's scheme required a central bank to verify the money, and the question of whether there can be unclonable quantum money that anyone can verify has remained open since. One can also ask a related question, which seems to be new: can quantum states be used as copy-protected programs, which let the user evaluate some function f, but not create more programs for f? This paper tackles both questions using the arsenal of modern computational complexity. Our main result is that there exist quantum oracles relative to which publicly-verifiable quantum money is possible, and any family of functions that cannot be efficiently learned from its input-output behavior can be quantumly copy-protected. This provides the first formal evidence that these tasks are achievable. The technical core of our result is a \"Complexity-Theoretic No-Cloning Theorem,\" which generalizes both the standard No-Cloning Theorem and the optimality of Grover search, and might be of independent interest. Our security argument also requires explicit constructions of quantum t-designs. Moving beyond the oracle world, we also present an explicit candidate scheme for publicly-verifiable quantum money, based on random stabilizer states; as well as two explicit schemes for copy-protecting the family of point functions. We do not know how to base the security of these schemes on any existing cryptographic assumption. (Note that without an oracle, we can only hope for security under some computational assumption.)","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130090750","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}

引用次数: 152

Extractors for Low-Weight Affine Sources 低权重仿射源的提取器

2009 24th Annual IEEE Conference on Computational Complexity Pub Date : 2009-07-15 DOI: 10.1109/CCC.2009.36

Anup Rao

引用次数: 64

Worst-Case Running Times for Average-Case Algorithms 平均情况算法的最坏情况运行时间

2009 24th Annual IEEE Conference on Computational Complexity Pub Date : 2009-07-15 DOI: 10.1109/CCC.2009.12

L. Antunes, L. Fortnow

引用次数: 26

Every Permutation CSP of arity 3 is Approximation Resistant 每个密度为3的排列CSP都是抗逼近的

2009 24th Annual IEEE Conference on Computational Complexity Pub Date : 2009-07-15 DOI: 10.1109/CCC.2009.29

M. Charikar, V. Guruswami, R. Manokaran

{"title":"Every Permutation CSP of arity 3 is Approximation Resistant","authors":"M. Charikar, V. Guruswami, R. Manokaran","doi":"10.1109/CCC.2009.29","DOIUrl":"https://doi.org/10.1109/CCC.2009.29","url":null,"abstract":"A permutation constraint satisfaction problem (permCSP) of arity k is specified by a subset Lambda of permutations on ${1,2,dots,k}$. An instance of such a permCSP consists of a set of variables $V$ and a collection of constraints each of which is an ordered $k$-tuple of $V$. The objective is to find a global ordering $sigma$ of the variables that maximizes the number of constraint tuples whose ordering (under $sigma$) follows a permutation in $Lambda$. This is just the natural extension of constraint satisfaction problems over finite domains (such as Boolean CSPs) to the world of ordering problems. The simplest permCSP corresponds to the case when $Lambda$ consists of the identity permutation on two variables. This is just the Maximum Acyclic Subgraph (mas) problem. It was recently shown that the mas problem is Unique-Games hard to approximate within a factor better than the trivial $1/2$ achieved by a random ordering [GMR08]. Building on this work, in this paper we show that for *every* permCSP of arity $3$, beating the random ordering is Unique-Games hard. The result is in fact stronger: we show that for every $Lambda subseteq Pi subseteq S_3$, given an instance of permCSP$(Lambda)$ that is almost-satisfiable, it is hard to find an ordering that satisfies more than $frac{|Pi|}{6} +eps$ of the constraints even under the relaxed constraint $Pi$ (for arbitrary $eps ≫ 0$). A special case of our result is that the *Betweenness* problem is hard to approximate beyond a factor $1/3$. Interestingly, for *satisfiable* instances of Betweenness, a factor $1/2$ approximation algorithm is known. Thus, every permutation CSP of arity up to $3$ resists approximation beyond the trivial random ordering threshold. In contrast, for Boolean CSPs, there are both approximation resistant and non-trivially approximable CSPs of arity $3$.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116961965","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}

引用次数: 39

Locally Testable Codes Require Redundant Testers 本地可测试代码需要冗余的测试人员

2009 24th Annual IEEE Conference on Computational Complexity Pub Date : 2009-07-15 DOI: 10.1137/090779875

Eli Ben-Sasson, V. Guruswami, T. Kaufman, M. Sudan, M. Viderman

引用次数: 33

On the Communication Complexity of Read-Once AC^0 Formulae 关于一次读AC^0公式的通信复杂度

2009 24th Annual IEEE Conference on Computational Complexity Pub Date : 2009-07-15 DOI: 10.1109/CCC.2009.39

T. S. Jayram, Swastik Kopparty, P. Raghavendra

{"title":"On the Communication Complexity of Read-Once AC^0 Formulae","authors":"T. S. Jayram, Swastik Kopparty, P. Raghavendra","doi":"10.1109/CCC.2009.39","DOIUrl":"https://doi.org/10.1109/CCC.2009.39","url":null,"abstract":"We study the 2-party randomized communication complexity of read-once AC0 formulae. For balanced AND-OR trees T with n inputs and depth d, we show that the communication complexity of the function f(x, y) = T(x circ y) is Omega(n/4^d) where (x circ y) is defined so that the resulting tree also has alternating levels of AND and OR gates. For each bit of x circ y, the operation circ is either AND or OR depending on the gate in T to which it is an input. Using this, we show that for general AND-OR trees T with n inputs and depth d, the communication complexity of f (x circ y) is n/2^{O(d log d)}. These results generalize classical results on the communication complexity of set-disjointness [1], [2] (where T is an OR -gate) and recent results on the communication complexity of the TRIBES functions [3] (where T is a depth-2 read-once formula). Our techniques build on and extend the information complexity methodology [4], [5], [3] for proving lower bounds on randomized communication complexity. Our analysis for trees of depth d proceeds in two steps: (1) reduction to measuring the information complexity of binary depth-d trees, and (2) proving lower bounds on the information complexity of binary trees. In order to execute this program, we carefully construct input distributions under which both these steps can be carried out simultaneously. We believe the tools we develop will prove useful in further studies of information complexity in particular, and communication complexity in general.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122133623","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}

引用次数: 16

Inapproximability of Vertex Cover and Independent Set in Bounded Degree Graphs 有界度图中顶点覆盖与独立集的不可逼近性

2009 24th Annual IEEE Conference on Computational Complexity Pub Date : 2009-07-15 DOI: 10.4086/toc.2011.v007a003

Per Austrin, Subhash Khot, S. Safra

引用次数: 120

Reconstruction of Generalized Depth-3 Arithmetic Circuits with Bounded Top Fan-in 具有有界顶部扇入的广义深度-3算术电路的重构

2009 24th Annual IEEE Conference on Computational Complexity Pub Date : 2009-07-15 DOI: 10.1109/CCC.2009.18

Zohar S. Karnin, Amir Shpilka

{"title":"Reconstruction of Generalized Depth-3 Arithmetic Circuits with Bounded Top Fan-in","authors":"Zohar S. Karnin, Amir Shpilka","doi":"10.1109/CCC.2009.18","DOIUrl":"https://doi.org/10.1109/CCC.2009.18","url":null,"abstract":"In this paper we give reconstruction algorithms for depth-3 arithmetic circuits with $k$ multiplication gates (also known as $SigmaPiSigma(k)$ circuits), where $k=O(1)$. Namely, we give an algorithm that when given a black box holding a $SigmaPiSigma(k)$ circuit $C$ over a field $F$ as input, makes queries to the black box (possibly over a polynomial sized extension field of $F$) and outputs a circuit $C'$ computing the same polynomial as $C$. In particular we obtain the following results. 1) When $C$ is a multilinear $SigmaPiSigma(k)$ circuit (i.e. each of its multiplication gates computes a multilinear polynomial) then our algorithm runs in polynomial time (when $k$ is a constant) and outputs a multilinear $SigmaPiSigma(k)$ circuits computing the same polynomial. 2) In the general case, our algorithm runs in quasi-polynomial time and outputs a generalized depth-3 circuit (as defined in cite{KarninShpilka08}) with $k$ multiplication gates. For example, the polynomials computed by generalized depth-3 circuits can be computed by quasi-polynomial sized depth-3 circuits. In fact, our algorithm works in the slightly more general case where the black box holds a generalized depth-3 circuits. Prior to this work there were reconstruction algorithms for several different models of bounded depth circuits: the well studied class of depth-2 arithmetic circuits (that compute sparse polynomials) and its close by model of depth-3 set-multilinear circuits. For the class of depth-3 circuits only the case of $k=2$ (i.e. $SigmaPiSigma(2)$ circuits) was known. Our proof technique combines ideas from [Shpilka09] and [KarninShpilka08] with some new ideas. Our most notable new ideas are: We prove the existence of a unique canonical representation of depth-3 circuits. This enables us to work with a specific representation in mind. Another technical contribution is an isolation lemma for depth-3 circuits that enables us to reconstruct a single multiplication gate of the circuit.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133263132","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}

引用次数: 51

New Results in the Simultaneous Message Passing Model via Information Theoretic Techniques 基于信息理论技术的同步消息传递模型的新结果

2009 24th Annual IEEE Conference on Computational Complexity Pub Date : 2009-02-18 DOI: 10.1109/CCC.2009.28

Rahul Jain, H. Klauck

{"title":"New Results in the Simultaneous Message Passing Model via Information Theoretic Techniques","authors":"Rahul Jain, H. Klauck","doi":"10.1109/CCC.2009.28","DOIUrl":"https://doi.org/10.1109/CCC.2009.28","url":null,"abstract":"Consider the following {em Simultaneous Message Passing} ($smp$) model for computing a relation $f subseteq cX times cY times cZ$. In this model $alice$, on input $x in cX$ and $bob$, on input $yincY$, send one message each to a third party $referee$ who then outputs a $z in cZ$ such that $(x,y,z)in f$. We first show optimal {em Direct sum} results for all relations $f$ in this model, both in the quantum and classical settings, in the situation where we allow shared resources (shared entanglement in quantum protocols and public coins in classical protocols) between $alice$ and $referee$ and $bob$ and $referee$ and no shared resource between $alice$ and $bob$. This implies that, in this model, the communication required to compute $k$ simultaneous instances of $f$, with constant success overall, is at least $k$-times the communication required to compute one instance with constant success. This in particular implies an earlier Direct sum result, shown by Chakrabarti, Shi, Wirth and Yao~cite{ChakrabartiSWY01} for the Equality function (and a class of other so-called robust functions), in the classical $smp$ model with no shared resources between any parties. Furthermore we investigate the gap between the $smp$ model and the one-way model in communication complexity and exhibit a partial function that is exponentially more expensive in the former if quantum communication with entanglement is allowed, compared to the latter even in the deterministic case.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"114 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124756232","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}

引用次数: 18