The Gap-Hamming-Distance problem arose in the context of proving space lower bounds for a number of key problems in the data stream model. In this problem, Alice and Bob have to decide whether the Hamming distance between their $n$-bit input strings is large (i.e., at least $n/2 + sqrt n$) or small (i.e., at most $n/2 - sqrt n$); they do not care if it is neither large nor small. This $Theta(sqrt n)$ gap in the problem specification is crucial for capturing the approximation allowed to a data stream algorithm. Thus far, for randomized communication, an $Omega(n)$ lower bound on this problem was known only in the one-way setting. We prove an $Omega(n)$ lower bound for randomized protocols that use any constant number of rounds. As a consequence we conclude, for instance, that $epsilon$-approximately counting the number of distinct elements in a data stream requires $Omega(1/epsilon^2)$ space, even with multiple (a constant number of) passes over the input stream. This extends earlier one-pass lower bounds, answering a long-standing open question. We obtain similar results for approximating the frequency moments and for approximating the empirical entropy of a data stream. In the process, we also obtain tight $n - Theta(sqrt{n}log n)$ lower and upper bounds on the one-way deterministic communication complexity of the problem. Finally, we give a simple combinatorial proof of an $Omega(n)$ lower bound on the one-way randomized communication complexity.
{"title":"A Multi-Round Communication Lower Bound for Gap Hamming and Some Consequences","authors":"Joshua Brody, Amit Chakrabarti","doi":"10.1109/CCC.2009.31","DOIUrl":"https://doi.org/10.1109/CCC.2009.31","url":null,"abstract":"The Gap-Hamming-Distance problem arose in the context of proving space lower bounds for a number of key problems in the data stream model. In this problem, Alice and Bob have to decide whether the Hamming distance between their $n$-bit input strings is large (i.e., at least $n/2 + sqrt n$) or small (i.e., at most $n/2 - sqrt n$); they do not care if it is neither large nor small. This $Theta(sqrt n)$ gap in the problem specification is crucial for capturing the approximation allowed to a data stream algorithm. Thus far, for randomized communication, an $Omega(n)$ lower bound on this problem was known only in the one-way setting. We prove an $Omega(n)$ lower bound for randomized protocols that use any constant number of rounds. As a consequence we conclude, for instance, that $epsilon$-approximately counting the number of distinct elements in a data stream requires $Omega(1/epsilon^2)$ space, even with multiple (a constant number of) passes over the input stream. This extends earlier one-pass lower bounds, answering a long-standing open question. We obtain similar results for approximating the frequency moments and for approximating the empirical entropy of a data stream. In the process, we also obtain tight $n - Theta(sqrt{n}log n)$ lower and upper bounds on the one-way deterministic communication complexity of the problem. Finally, we give a simple combinatorial proof of an $Omega(n)$ lower bound on the one-way randomized communication complexity.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"168 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127198841","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}
{"title":"A Superpolynomial Lower Bound on the Size of Uniform Non-constant-depth Threshold Circuits for the Permanent","authors":"P. Koiran, Sylvain Perifel","doi":"10.1109/CCC.2009.19","DOIUrl":"https://doi.org/10.1109/CCC.2009.19","url":null,"abstract":"We show that the permanent cannot be computed by DLOGTIME-uniform threshold or arithmetic circuits of depth o(log log n) and polynomial size.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116321801","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 show that the rank of a depth-$3$ circuit (over any field) that is simple, minimal and zero is at most $O(k^3log d)$. The previous best rank bound known was $2^{O(k^2)}(log d)^{k-2}$ by Dvir and Shpilka (STOC 2005). This almost resolves the rank question first posed by Dvir and Shpilka (as we also provide a simple and minimal identity of rank $Omega(klog d)$). Our rank bound significantly improves (dependence on $k$ exponentially reduced) the best known deterministic black-box identity tests for depth-$3$ circuits by Karnin and Shpilka (CCC 2008). Our techniques also shed light on the factorization pattern of nonzero depth-$3$ circuits, most strikingly: the rank of linear factors of a simple, minimal and nonzero depth-$3$ circuit (over any field) is at most $O(k^3log d)$. The novel feature of this work is a new notion of maps between sets of linear forms, called emph{ideal matchings}, used to study depth-$3$ circuits. We prove interesting structural results about depth-$3$ identities using these techniques. We believe that these can lead to the goal of a deterministic polynomial time identity test for these circuits.
{"title":"An Almost Optimal Rank Bound for Depth-3 Identities","authors":"Nitin Saxena, C. Seshadhri","doi":"10.1137/090770679","DOIUrl":"https://doi.org/10.1137/090770679","url":null,"abstract":"We show that the rank of a depth-$3$ circuit (over any field) that is simple, minimal and zero is at most $O(k^3log d)$. The previous best rank bound known was $2^{O(k^2)}(log d)^{k-2}$ by Dvir and Shpilka (STOC 2005). This almost resolves the rank question first posed by Dvir and Shpilka (as we also provide a simple and minimal identity of rank $Omega(klog d)$). Our rank bound significantly improves (dependence on $k$ exponentially reduced) the best known deterministic black-box identity tests for depth-$3$ circuits by Karnin and Shpilka (CCC 2008). Our techniques also shed light on the factorization pattern of nonzero depth-$3$ circuits, most strikingly: the rank of linear factors of a simple, minimal and nonzero depth-$3$ circuit (over any field) is at most $O(k^3log d)$. The novel feature of this work is a new notion of maps between sets of linear forms, called emph{ideal matchings}, used to study depth-$3$ circuits. We prove interesting structural results about depth-$3$ identities using these techniques. We believe that these can lead to the goal of a deterministic polynomial time identity test for these circuits.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128684792","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}
This paper presents three results on the power of two-prover one-round interactive proof systems based on oracularization under the existence of prior entanglement between dishonest provers. It is proved that the two-prover one-round interactive proof system for PSPACE by Cai, Condon, and Lipton [JCSS 48:183-193, 1994] still achieves exponentially small soundness error in the existence of prior entanglement between dishonest provers (and more strongly, even if dishonest provers are allowed to use arbitrary no-signaling strategies). It follows that, unless the polynomial-time hierarchy collapses to the second level, two-prover systems are still advantageous to single-prover systems even when only malicious provers can use quantum information. It is also shown that a "dummy" question may be helpful when constructing an entanglement-resistant multi-prover system via oracularization. This affirmatively settles a question posed by Kempe et al. [FOCS 2008, pp. 447-456] and every language in NEXP is proved to have a two-prover one-round interactive proof system even against entangled provers, albeit with exponentially small gap between completeness and soundness. In other words, it is NP-hard to approximate within an inverse-polynomial the value of a classical two-prover one-round game against entangled provers. Finally, both for the above proof system for NEXP and for the quantum two-prover one-round proof system for NEXP proposed by Kempe et al., it is proved that exponentially small completeness-soundness gaps are best achievable unless soundness analysis uses the structure of the underlying system with unentangled provers.
本文给出了在不诚实证明者之间存在先验纠缠的情况下，基于神谕化的双证明者一轮交互证明系统的力的三个结果。Cai, Condon, and Lipton [JCSS 48:183- 193,1994]证明了PSPACE的两个证明者一轮交互证明系统在不诚实证明者之间存在先验纠缠的情况下仍然可以实现指数级小的健全误差(并且更强，即使允许不诚实证明者使用任意的无信令策略)。由此可见，除非多项式时间层次结构崩溃到第二级，否则即使只有恶意的证明者可以使用量子信息，双证明者系统仍然比单证明者系统有利。通过oracle化构造一个抗纠缠的多证明者系统时，“虚拟”问题可能会有所帮助。这肯定地解决了Kempe等人提出的问题[FOCS 2008，第447-456页]，并且NEXP中的每种语言都被证明具有两个证明者的一轮交互式证明系统，即使是针对纠缠的证明者，尽管完整性和稳健性之间的差距很小。换句话说，在一个逆多项式中近似经典的两个证明者对抗纠缠证明者的一轮博弈的值是np困难的。最后，对于上述NEXP证明系统和Kempe等人提出的NEXP量子双证明者一轮证明系统，证明了除非健全性分析使用无纠缠证明者的底层系统结构，否则指数级小的完备性-健全性间隙是最好实现的。
{"title":"Oracularization and Two-Prover One-Round Interactive Proofs against Nonlocal Strategies","authors":"Tsuyoshi Ito, Hirotada Kobayashi, Keiji Matsumoto","doi":"10.1109/CCC.2009.22","DOIUrl":"https://doi.org/10.1109/CCC.2009.22","url":null,"abstract":"This paper presents three results on the power of two-prover one-round interactive proof systems based on oracularization under the existence of prior entanglement between dishonest provers. It is proved that the two-prover one-round interactive proof system for PSPACE by Cai, Condon, and Lipton [JCSS 48:183-193, 1994] still achieves exponentially small soundness error in the existence of prior entanglement between dishonest provers (and more strongly, even if dishonest provers are allowed to use arbitrary no-signaling strategies). It follows that, unless the polynomial-time hierarchy collapses to the second level, two-prover systems are still advantageous to single-prover systems even when only malicious provers can use quantum information. It is also shown that a \"dummy\" question may be helpful when constructing an entanglement-resistant multi-prover system via oracularization. This affirmatively settles a question posed by Kempe et al. [FOCS 2008, pp. 447-456] and every language in NEXP is proved to have a two-prover one-round interactive proof system even against entangled provers, albeit with exponentially small gap between completeness and soundness. In other words, it is NP-hard to approximate within an inverse-polynomial the value of a classical two-prover one-round game against entangled provers. Finally, both for the above proof system for NEXP and for the quantum two-prover one-round proof system for NEXP proposed by Kempe et al., it is proved that exponentially small completeness-soundness gaps are best achievable unless soundness analysis uses the structure of the underlying system with unentangled provers.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115218388","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}
One of the strongest techniques available for showing lower bounds on bounded-error communication complexity is the logarithm of the approximation rank of the communication matrix---the minimum rank of a matrix which is entrywise close to the communication matrix. Krause showed that the logarithm of approximation rank is a lower bound in the randomized case, and later Buhrman and de Wolf showed it could also be used for quantum communication complexity. As a lower bound technique, approximation rank has two main drawbacks: it is difficult to compute, and it is not known to lower bound the model of quantum communication complexity with entanglement. We give a polynomial time constant factor approximation algorithm to the logarithm of approximation rank, and show that the logarithm of approximation rank is a lower bound on quantum communication complexity with entanglement.
{"title":"An Approximation Algorithm for Approximation Rank","authors":"Troy Lee, A. Shraibman","doi":"10.1109/CCC.2009.25","DOIUrl":"https://doi.org/10.1109/CCC.2009.25","url":null,"abstract":"One of the strongest techniques available for showing lower bounds on bounded-error communication complexity is the logarithm of the approximation rank of the communication matrix---the minimum rank of a matrix which is entrywise close to the communication matrix. Krause showed that the logarithm of approximation rank is a lower bound in the randomized case, and later Buhrman and de Wolf showed it could also be used for quantum communication complexity. As a lower bound technique, approximation rank has two main drawbacks: it is difficult to compute, and it is not known to lower bound the model of quantum communication complexity with entanglement. We give a polynomial time constant factor approximation algorithm to the logarithm of approximation rank, and show that the logarithm of approximation rank is a lower bound on quantum communication complexity with entanglement.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132689975","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}
This paper studies a simple class of zero-sum games played by two competing quantum players: each player sends a mixed quantum state to a referee, who performs a joint measurement on the two states to determine the players' payoffs. We prove that an equilibrium point of any such game can be approximated by means of an efficient parallel algorithm, which implies that one-turn quantum refereed games, wherein the referee is specified by a quantum circuit, can be simulated in polynomial space.
{"title":"Parallel Approximation of Non-interactive Zero-sum Quantum Games","authors":"Rahul Jain, John Watrous","doi":"10.1109/CCC.2009.26","DOIUrl":"https://doi.org/10.1109/CCC.2009.26","url":null,"abstract":"This paper studies a simple class of zero-sum games played by two competing quantum players: each player sends a mixed quantum state to a referee, who performs a joint measurement on the two states to determine the players' payoffs. We prove that an equilibrium point of any such game can be approximated by means of an efficient parallel algorithm, which implies that one-turn quantum refereed games, wherein the referee is specified by a quantum circuit, can be simulated in polynomial space.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127373710","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}
Barrington, Straubing and Therien (1990) conjectured that the Boolean AND function can not be computed by polynomial size constant depth circuits built from modular counting gates, i.e., by CC^0 circuits. In this work we show that the AND function can be computed by uniform probabilistic CC^0 circuits that use only O(log n) random bits. This may be viewed as evidence contrary to the conjecture. As a consequence of our construction we get that all of ACC^0 can be computed by probabilistic CC^0 circuits that use only O(log n) random bits. Thus, if one were able to derandomize such circuits, we would obtain a collapse of circuit classes giving ACC^0=CC^0. We present a derandomization of probabilistic CC^0 circuits using AND and OR gates to obtain ACC^0 = AND o OR o CC^0 = OR o AND o CC^0. AND and OR gates of sublinear fan-in suffice. Both these results hold for uniform as well as non-uniform circuit classes. For non-uniform circuits we obtain the stronger conclusion that ACC^0 = rand-ACC^0 = rand-CC^0 = rand(log n)-CC^0, i.e., probabilistic ACC^0 circuits can be simulated by probabilistic CC^0 circuits using only O(log n) random bits. As an application of our results we obtain a characterization of ACC^0 by constant width planar nondeterministic branching programs, improving a previous characterization for the quasipolynomial size setting.
Barrington, Straubing和Therien(1990)推测，布尔与函数不能通过由模块化计数门构建的多项式大小的定深电路来计算，即通过CC^0电路。在这项工作中，我们证明了与函数可以通过只使用O(log n)个随机比特的均匀概率CC^0电路来计算。这可以看作是与猜想相反的证据。作为我们构造的结果，我们得到所有的ACC^0都可以通过只使用O(log n)个随机比特的概率CC^0电路来计算。因此，如果我们能够对这样的电路进行非随机化，我们将得到一个电路类的崩溃，给出ACC^0=CC^0。提出了一种利用与或门对概率CC^0电路进行非随机化的方法，得到ACC^0 = AND 0 OR或0 CC^0 = OR 0 AND 0 CC^0。亚线性扇入的与或门就足够了。这两个结果都适用于均匀和非均匀电路类。对于非均匀电路，我们得到了ACC^0 = rand-ACC^0 = rand-CC^0 = rand(log n)-CC^0的更强的结论，即概率ACC^0电路可以只用O(log n)个随机比特用概率CC^0电路来模拟。作为我们研究结果的一个应用，我们得到了用等宽平面不确定性分支规划表征ACC^0的方法，改进了先前对拟多项式尺寸设置的表征。
{"title":"A New Characterization of ACC^0 and Probabilistic CC^0","authors":"Kristoffer Arnsfelt Hansen, M. Koucký","doi":"10.1109/CCC.2009.15","DOIUrl":"https://doi.org/10.1109/CCC.2009.15","url":null,"abstract":"Barrington, Straubing and Therien (1990) conjectured that the Boolean AND function can not be computed by polynomial size constant depth circuits built from modular counting gates, i.e., by CC^0 circuits. In this work we show that the AND function can be computed by uniform probabilistic CC^0 circuits that use only O(log n) random bits. This may be viewed as evidence contrary to the conjecture. As a consequence of our construction we get that all of ACC^0 can be computed by probabilistic CC^0 circuits that use only O(log n) random bits. Thus, if one were able to derandomize such circuits, we would obtain a collapse of circuit classes giving ACC^0=CC^0. We present a derandomization of probabilistic CC^0 circuits using AND and OR gates to obtain ACC^0 = AND o OR o CC^0 = OR o AND o CC^0. AND and OR gates of sublinear fan-in suffice. Both these results hold for uniform as well as non-uniform circuit classes. For non-uniform circuits we obtain the stronger conclusion that ACC^0 = rand-ACC^0 = rand-CC^0 = rand(log n)-CC^0, i.e., probabilistic ACC^0 circuits can be simulated by probabilistic CC^0 circuits using only O(log n) random bits. As an application of our results we obtain a characterization of ACC^0 by constant width planar nondeterministic branching programs, improving a previous characterization for the quasipolynomial size setting.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122246457","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：