{"title":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","authors":"D. Wichs, Y. Mansour","doi":"10.1145/2897518","DOIUrl":"https://doi.org/10.1145/2897518","url":null,"abstract":"","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"49 9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115634844","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}
Existing n-process randomized wait-free (and obstruction-free) consensus protocols from registers all use at least n registers. In 1992, it was proved that such protocols must use Omega(sqrt(n)) registers. Recently, this was improved to Omega(n) registers in the anonymous setting, where processes do not have identifiers. Closing the gap in the general case, however, remained an open problem. We resolve this problem by proving that every randomized wait-free (or obstruction-free) consensus protocol for n processes must use at least n-1 registers.
{"title":"A tight space bound for consensus","authors":"Leqi Zhu","doi":"10.1145/2897518.2897565","DOIUrl":"https://doi.org/10.1145/2897518.2897565","url":null,"abstract":"Existing n-process randomized wait-free (and obstruction-free) consensus protocols from registers all use at least n registers. In 1992, it was proved that such protocols must use Omega(sqrt(n)) registers. Recently, this was improved to Omega(n) registers in the anonymous setting, where processes do not have identifiers. Closing the gap in the general case, however, remained an open problem. We resolve this problem by proving that every randomized wait-free (or obstruction-free) consensus protocol for n processes must use at least n-1 registers.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"52 5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126501746","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 interactive compression problem: Given a two-party communication protocol with small information cost, can it be compressed so that the total number of bits communicated is also small? We consider the case where the parties have inputs that are independent of each other, and give a simulation protocol that communicates I^2 * polylog(I) bits, where I is the information cost of the original protocol. Our protocol is the first simulation protocol whose communication complexity is bounded by a polynomial in the information cost of the original protocol.
{"title":"Interactive compression for product distributions","authors":"Gillat Kol","doi":"10.1145/2897518.2897537","DOIUrl":"https://doi.org/10.1145/2897518.2897537","url":null,"abstract":"We study the interactive compression problem: Given a two-party communication protocol with small information cost, can it be compressed so that the total number of bits communicated is also small? We consider the case where the parties have inputs that are independent of each other, and give a simulation protocol that communicates I^2 * polylog(I) bits, where I is the information cost of the original protocol. Our protocol is the first simulation protocol whose communication complexity is bounded by a polynomial in the information cost of the original protocol.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121023520","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}
In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph G and a source node s the goal is to maintain shortest paths between s and all other nodes in G under a sequence of online adversarial edge deletions. In their seminal work, Even and Shiloach [JACM 1981] presented an exact solution to the problem with only O(mn) total update time over all edge deletions. Their classic algorithm was the best known result for the decremental SSSP problem for three decades, even when approximate shortest paths are allowed. The first improvement over the Even-Shiloach algorithm was given by Bernstein and Roditty [SODA 2011], who for the case of an unweighted and undirected graph presented an approximate (1+) algorithm with constant query time and a total update time of O(n2+O(1/√logn)). This work triggered a series of new results, culminating in a recent breakthrough of Henzinger, Krinninger and Nanongkai [FOCS 14], who presented a -approximate algorithm whose total update time is near linear O(m1+ O(1/√logn)). In this paper they posed as a major open problem the question of derandomizing their result. In fact, all known improvements over the Even-Shiloach algorithm are randomized. All these algorithms maintain some truncated shortest path trees from a small subset of nodes. While in the randomized setting it is possible to “hide” these nodes from the adversary, in the deterministic setting this is impossible: the adversary can delete all edges touching these nodes, thus forcing the algorithm to choose a new set of nodes and incur a new computation of shortest paths. In this paper we present the first deterministic decremental SSSP algorithm that breaks the Even-Shiloach bound of O(mn) total update time, for unweighted and undirected graphs. Our algorithm is (1 + є) approximate and achieves a total update time of Õ(n2). Our algorithm can also achieve the same bounds in the incremental setting. It is worth mentioning that for dense instances where m = Ω(n2 − 1/√log(n)), our algorithm is also faster than all existing randomized algorithms.
{"title":"Deterministic decremental single source shortest paths: beyond the o(mn) bound","authors":"A. Bernstein, S. Chechik","doi":"10.1145/2897518.2897521","DOIUrl":"https://doi.org/10.1145/2897518.2897521","url":null,"abstract":"In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph G and a source node s the goal is to maintain shortest paths between s and all other nodes in G under a sequence of online adversarial edge deletions. In their seminal work, Even and Shiloach [JACM 1981] presented an exact solution to the problem with only O(mn) total update time over all edge deletions. Their classic algorithm was the best known result for the decremental SSSP problem for three decades, even when approximate shortest paths are allowed. The first improvement over the Even-Shiloach algorithm was given by Bernstein and Roditty [SODA 2011], who for the case of an unweighted and undirected graph presented an approximate (1+) algorithm with constant query time and a total update time of O(n2+O(1/√logn)). This work triggered a series of new results, culminating in a recent breakthrough of Henzinger, Krinninger and Nanongkai [FOCS 14], who presented a -approximate algorithm whose total update time is near linear O(m1+ O(1/√logn)). In this paper they posed as a major open problem the question of derandomizing their result. In fact, all known improvements over the Even-Shiloach algorithm are randomized. All these algorithms maintain some truncated shortest path trees from a small subset of nodes. While in the randomized setting it is possible to “hide” these nodes from the adversary, in the deterministic setting this is impossible: the adversary can delete all edges touching these nodes, thus forcing the algorithm to choose a new set of nodes and incur a new computation of shortest paths. In this paper we present the first deterministic decremental SSSP algorithm that breaks the Even-Shiloach bound of O(mn) total update time, for unweighted and undirected graphs. Our algorithm is (1 + є) approximate and achieves a total update time of Õ(n2). Our algorithm can also achieve the same bounds in the incremental setting. It is worth mentioning that for dense instances where m = Ω(n2 − 1/√log(n)), our algorithm is also faster than all existing randomized algorithms.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124053749","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}
A. Cohen, Justin Holmgren, R. Nishimaki, V. Vaikuntanathan, D. Wichs
A watermarking scheme for programs embeds some information called a mark into a program while preserving its functionality. No adversary can remove the mark without damaging the functionality of the program. In this work, we study the problem of watermarking various cryptographic programs such as pseudorandom function (PRF) evaluation, decryption, and signing. For example, given a PRF key K, we create a marked program C that evaluates the PRF F(K,). An adversary that gets C cannot come up with any program C* in which the mark is removed but which still evaluates the PRF correctly on even a small fraction of the inputs. The work of Barak, Goldreich, Impagliazzo, Rudich, Sahai, Vadhan, and Yang (CRYPTO'01 and Journal of ACM 59(2)) shows that, assuming indistinguishability obfuscation (iO), such watermarking is impossible if the marked program C evaluates the original program with perfect correctness. In this work we show that, assuming iO, such watermarking is possible if the marked program C is allowed to err with even a negligible probability, which would be undetectable to the user. Our watermarking schemes are public key, namely we use a secret marking key to embed marks in programs, and a public detection key that allows anyone to detect marks in programs. Our schemes are secure against chosen program attacks, that is even if the adversary is given oracle access to the marking functionality. We emphasize that our security notion of watermark non-removability considers arbitrary adversarial strategies to modify the marked program, in contrast to the prior works (Nishimaki, EUROCRYPT '13).
用于程序的水印方案在保留程序功能的同时将一些称为标记的信息嵌入到程序中。任何对手都无法在不破坏程序功能的情况下移除标记。在这项工作中，我们研究了各种加密程序的水印问题，如伪随机函数(PRF)评估，解密和签名。例如，给定一个PRF键K，我们创建一个计算PRF F(K，)的标记程序C。一个得到C的对手不能想出任何程序C*，其中的标记被删除，但仍然正确地评估PRF，即使是一小部分输入。Barak, Goldreich, Impagliazzo, Rudich, Sahai, Vadhan和Yang (CRYPTO'01 and Journal of ACM 59(2))的工作表明，假设不可区分混淆(iO)，如果标记的程序C以完全正确的方式评估原始程序，则这种水印是不可能的。在这项工作中，我们表明，假设iO，这样的水印是可能的，如果被标记的程序C允许错误，甚至可以忽略不计的概率，这将是无法检测到的用户。我们的水印方案是公开密钥，即使用秘密标记密钥在程序中嵌入标记，以及使用公开检测密钥允许任何人检测程序中的标记。我们的方案对于选定的程序攻击是安全的，即使攻击者获得了对标记功能的oracle访问权限。我们强调，与之前的作品(Nishimaki, EUROCRYPT '13)相比，我们的水印不可移除性的安全概念考虑了任意对抗策略来修改标记程序。
{"title":"Watermarking cryptographic capabilities","authors":"A. Cohen, Justin Holmgren, R. Nishimaki, V. Vaikuntanathan, D. Wichs","doi":"10.1145/2897518.2897651","DOIUrl":"https://doi.org/10.1145/2897518.2897651","url":null,"abstract":"A watermarking scheme for programs embeds some information called a mark into a program while preserving its functionality. No adversary can remove the mark without damaging the functionality of the program. In this work, we study the problem of watermarking various cryptographic programs such as pseudorandom function (PRF) evaluation, decryption, and signing. For example, given a PRF key K, we create a marked program C that evaluates the PRF F(K,). An adversary that gets C cannot come up with any program C* in which the mark is removed but which still evaluates the PRF correctly on even a small fraction of the inputs. The work of Barak, Goldreich, Impagliazzo, Rudich, Sahai, Vadhan, and Yang (CRYPTO'01 and Journal of ACM 59(2)) shows that, assuming indistinguishability obfuscation (iO), such watermarking is impossible if the marked program C evaluates the original program with perfect correctness. In this work we show that, assuming iO, such watermarking is possible if the marked program C is allowed to err with even a negligible probability, which would be undetectable to the user. Our watermarking schemes are public key, namely we use a secret marking key to embed marks in programs, and a public detection key that allows anyone to detect marks in programs. Our schemes are secure against chosen program attacks, that is even if the adversary is given oracle access to the marking functionality. We emphasize that our security notion of watermark non-removability considers arbitrary adversarial strategies to modify the marked program, in contrast to the prior works (Nishimaki, EUROCRYPT '13).","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131094087","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}
Recently, Aissi et al. gave new counting and algorithmic bounds for parametric minimum cuts in a graph, where each edge cost is a linear combination of multiple cost criteria and different cuts become minimum as the coefficients of the linear combination are varied. In this article, we derive better bounds using a mathematically simpler argument. We provide faster algorithms for enumerating these cuts. We give a lower bound showing our upper bounds have roughly the right degree. Our results also immediately generalize to parametric versions of other problems solved by the Contraction Algorithm, including approximate min-cuts, multi-way cuts, and a matroid optimization problem. We also give a first generalization to nonlinear parametric minimum cuts.
{"title":"Enumerating parametric global minimum cuts by random interleaving","authors":"David R Karger","doi":"10.1145/2897518.2897578","DOIUrl":"https://doi.org/10.1145/2897518.2897578","url":null,"abstract":"Recently, Aissi et al. gave new counting and algorithmic bounds for parametric minimum cuts in a graph, where each edge cost is a linear combination of multiple cost criteria and different cuts become minimum as the coefficients of the linear combination are varied. In this article, we derive better bounds using a mathematically simpler argument. We provide faster algorithms for enumerating these cuts. We give a lower bound showing our upper bounds have roughly the right degree. Our results also immediately generalize to parametric versions of other problems solved by the Contraction Algorithm, including approximate min-cuts, multi-way cuts, and a matroid optimization problem. We also give a first generalization to nonlinear parametric minimum cuts.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127714522","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}
Given a positive semidefinte matrix L whose columns and rows are indexed by a set U, and a partition matroid M=(U, I), we study the problem of selecting a basis B of M such that the determinant of the submatrix of L induced by the rows and columns in B is maximized. This problem appears in many areas including determinantal point processes in machine learning, experimental design, geographical placement problems, discrepancy theory and computational geometry to model subset selection problems that incorporate diversity. Our main result is to give a geometric concave program for the problem which approximates the optimum value within a factor of er+o(r), where r denotes the rank of the partition matroid M. We bound the integrality gap of the geometric concave program by giving a polynomial time randomized rounding algorithm. To analyze the rounding algorithm, we relate the solution of our algorithm as well the objective value of the relaxation to a certain stable polynomial. To prove the approximation guarantee, we utilize a general inequality about stable polynomials proved by Gurvits in the context of estimating the permanent of a doubly stochastic matrix.
{"title":"Maximizing determinants under partition constraints","authors":"Aleksandar Nikolov, Mohit Singh","doi":"10.1145/2897518.2897649","DOIUrl":"https://doi.org/10.1145/2897518.2897649","url":null,"abstract":"Given a positive semidefinte matrix L whose columns and rows are indexed by a set U, and a partition matroid M=(U, I), we study the problem of selecting a basis B of M such that the determinant of the submatrix of L induced by the rows and columns in B is maximized. This problem appears in many areas including determinantal point processes in machine learning, experimental design, geographical placement problems, discrepancy theory and computational geometry to model subset selection problems that incorporate diversity. Our main result is to give a geometric concave program for the problem which approximates the optimum value within a factor of er+o(r), where r denotes the rank of the partition matroid M. We bound the integrality gap of the geometric concave program by giving a polynomial time randomized rounding algorithm. To analyze the rounding algorithm, we relate the solution of our algorithm as well the objective value of the relaxation to a certain stable polynomial. To prove the approximation guarantee, we utilize a general inequality about stable polynomials proved by Gurvits in the context of estimating the permanent of a doubly stochastic matrix.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130194169","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}
Suppose that you have n truly random bits x=(x1,…,xn) and you wish to use them to generate m≫ n pseudorandom bits y=(y1,…, ym) using a local mapping, i.e., each yi should depend on at most d=O(1) bits of x. In the polynomial regime of m=ns, s>1, the only known solution, originates from (Goldreich, ECCC 2000), is based on Random Local Functions: Compute yi by applying some fixed (public) d-ary predicate P to a random (public) tuple of distinct inputs (xi1,…,xid). Our goal in this paper is to understand, for any value of s, how the pseudorandomness of the resulting sequence depends on the choice of the underlying predicate. We derive the following results: (1) We show that pseudorandomness against F2-linear adversaries (i.e., the distribution y has low-bias) is achieved if the predicate is (a) k=Ω(s)-resilience, i.e., uncorrelated with any k-subset of its inputs, and (b) has algebraic degree of Ω(s) even after fixing Ω(s) of its inputs. We also show that these requirements are necessary, and so they form a tight characterization (up to constants) of security against linear attacks. Our positive result shows that a d-local low-bias generator can have output length of nΩ(d), answering an open question of Mossel, Shpilka and Trevisan (FOCS, 2003). Our negative result shows that a candidate for pseudorandom generator proposed by the first author (computational complexity, 2015) and by O’Donnell and Witmer (CCC 2014) is insecure. We use similar techniques to refute a conjecture of Feldman, Perkins and Vempala (STOC 2015) regarding the hardness of planted constraint satisfaction problems. (2) Motivated by the cryptanalysis literature, we consider security against algebraic attacks. We provide the first theoretical treatment of such attacks by formalizing a general notion of algebraic inversion and distinguishing attacks based on the Polynomial Calculus proof system. We show that algebraic attacks succeed if and only if there exist a degree e=O(s) non-zero polynomial Q whose roots cover the roots of P or cover the roots of P’s complement. As a corollary, we obtain the first example of a predicate P for which the generated sequence y passes all linear tests but fails to pass some polynomial-time computable test, answering an open question posed by the first author (Question 4.9, computational complexity 2015).
{"title":"Algebraic attacks against random local functions and their countermeasures","authors":"B. Applebaum, Shachar Lovett","doi":"10.1145/2897518.2897554","DOIUrl":"https://doi.org/10.1145/2897518.2897554","url":null,"abstract":"Suppose that you have n truly random bits x=(x1,…,xn) and you wish to use them to generate m≫ n pseudorandom bits y=(y1,…, ym) using a local mapping, i.e., each yi should depend on at most d=O(1) bits of x. In the polynomial regime of m=ns, s>1, the only known solution, originates from (Goldreich, ECCC 2000), is based on Random Local Functions: Compute yi by applying some fixed (public) d-ary predicate P to a random (public) tuple of distinct inputs (xi1,…,xid). Our goal in this paper is to understand, for any value of s, how the pseudorandomness of the resulting sequence depends on the choice of the underlying predicate. We derive the following results: (1) We show that pseudorandomness against F2-linear adversaries (i.e., the distribution y has low-bias) is achieved if the predicate is (a) k=Ω(s)-resilience, i.e., uncorrelated with any k-subset of its inputs, and (b) has algebraic degree of Ω(s) even after fixing Ω(s) of its inputs. We also show that these requirements are necessary, and so they form a tight characterization (up to constants) of security against linear attacks. Our positive result shows that a d-local low-bias generator can have output length of nΩ(d), answering an open question of Mossel, Shpilka and Trevisan (FOCS, 2003). Our negative result shows that a candidate for pseudorandom generator proposed by the first author (computational complexity, 2015) and by O’Donnell and Witmer (CCC 2014) is insecure. We use similar techniques to refute a conjecture of Feldman, Perkins and Vempala (STOC 2015) regarding the hardness of planted constraint satisfaction problems. (2) Motivated by the cryptanalysis literature, we consider security against algebraic attacks. We provide the first theoretical treatment of such attacks by formalizing a general notion of algebraic inversion and distinguishing attacks based on the Polynomial Calculus proof system. We show that algebraic attacks succeed if and only if there exist a degree e=O(s) non-zero polynomial Q whose roots cover the roots of P or cover the roots of P’s complement. As a corollary, we obtain the first example of a predicate P for which the generated sequence y passes all linear tests but fails to pass some polynomial-time computable test, answering an open question posed by the first author (Question 4.9, computational complexity 2015).","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133870350","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}
T. Pitassi, Benjamin Rossman, R. Servedio, Li-Yang Tan
We show that any polynomial-size Frege refutation of a certain linear-size unsatisfiable 3-CNF formula over n variables must have depth Ω(√logn). This is an exponential improvement over the previous best results (Pitassi et al. 1993, Krajíček et al. 1995, Ben-Sasson 2002) which give Ω(loglogn) lower bounds. The 3-CNF formulas which we use to establish this result are Tseitin contradictions on 3-regular expander graphs. In more detail, our main result is a proof that for every d, any depth-d Frege refutation of the Tseitin contradiction over these n-node graphs must have size nΩ((logn)/d2). A key ingredient of our approach is a new switching lemma for a carefully designed random restriction process over these expanders. These random restrictions reduce a Tseitin instance on a 3-regular n-node expander to a Tseitin instance on a random subgraph which is a topological embedding of a 3-regular n′-node expander, for some n′ which is not too much less than n. Our result involves Ω(√logn) iterative applications of this type of random restriction.
我们证明了对n个变量上的某个线性大小的不可满足3-CNF公式的任何多项式大小的Frege反驳必须具有深度Ω(√logn)。与之前的最佳结果(Pitassi et al. 1993, Krajíček et al. 1995, Ben-Sasson 2002)相比，这是一个指数级的改进，后者给出了Ω(对数)下界。我们用来建立这一结果的3-CNF公式是3正则展开图上的tseittin矛盾。更详细地说，我们的主要结果是证明对于每一个d，在这些n节点图上对tseittin矛盾的任何深度d Frege反驳必须具有nΩ((logn)/d2)的大小。我们方法的一个关键组成部分是一个新的开关引理，用于在这些展开式上精心设计的随机限制过程。这些随机限制将一个3-正则n节点展开器上的tseittin实例减少到一个随机子图上的tseittin实例，该子图是一个3-正则n '节点展开器的拓扑嵌入，对于一些n '并不比n小太多。我们的结果涉及这种类型的随机限制的Ω(√logn)迭代应用。
{"title":"Poly-logarithmic Frege depth lower bounds via an expander switching lemma","authors":"T. Pitassi, Benjamin Rossman, R. Servedio, Li-Yang Tan","doi":"10.1145/2897518.2897637","DOIUrl":"https://doi.org/10.1145/2897518.2897637","url":null,"abstract":"We show that any polynomial-size Frege refutation of a certain linear-size unsatisfiable 3-CNF formula over n variables must have depth Ω(√logn). This is an exponential improvement over the previous best results (Pitassi et al. 1993, Krajíček et al. 1995, Ben-Sasson 2002) which give Ω(loglogn) lower bounds. The 3-CNF formulas which we use to establish this result are Tseitin contradictions on 3-regular expander graphs. In more detail, our main result is a proof that for every d, any depth-d Frege refutation of the Tseitin contradiction over these n-node graphs must have size nΩ((logn)/d2). A key ingredient of our approach is a new switching lemma for a carefully designed random restriction process over these expanders. These random restrictions reduce a Tseitin instance on a 3-regular n-node expander to a Tseitin instance on a random subgraph which is a topological embedding of a 3-regular n′-node expander, for some n′ which is not too much less than n. Our result involves Ω(√logn) iterative applications of this type of random restriction.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130972586","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}
Let r be an integer. Let us call a polynomial f as a multi-r-ic polynomial if the degree of f with respect to any variable is at most r (this generalizes the notion of multilinear polynomials). We investigate arithmetic circuits in which the output is syntactically forced to be a multi-r-ic polynomial and refer to these as multi-r-ic circuits. Specifically, first define the formal degree of a node a with respect to a variable x inductively as follows. For a leaf it is 1 if a is labelled with x and zero otherwise; for an internal node labelled with * (respectively +) it is the sum of (respectively the maximum of) the formal degrees of the children with respect to x. We call an arithmetic circuit as a multi-r-ic circuit if the formal degree of the output node with respect to any variable is at most r. We prove lower bounds for various subclasses of multi-r-ic circuits.
{"title":"On the size of homogeneous and of depth four formulas with low individual degree","authors":"N. Kayal, Chandan Saha, Sébastien Tavenas","doi":"10.1145/2897518.2897550","DOIUrl":"https://doi.org/10.1145/2897518.2897550","url":null,"abstract":"Let r be an integer. Let us call a polynomial f as a multi-r-ic polynomial if the degree of f with respect to any variable is at most r (this generalizes the notion of multilinear polynomials). We investigate arithmetic circuits in which the output is syntactically forced to be a multi-r-ic polynomial and refer to these as multi-r-ic circuits. Specifically, first define the formal degree of a node a with respect to a variable x inductively as follows. For a leaf it is 1 if a is labelled with x and zero otherwise; for an internal node labelled with * (respectively +) it is the sum of (respectively the maximum of) the formal degrees of the children with respect to x. We call an arithmetic circuit as a multi-r-ic circuit if the formal degree of the output node with respect to any variable is at most r. We prove lower bounds for various subclasses of multi-r-ic circuits.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127673470","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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