Pub Date : 2022-02-11DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.23
V. Arvind, Abhranil Chatterjee, P. Mukhopadhyay
Hrubev{s} and Wigderson (2015) initiated the complexity-theoretic study of noncommutative formulas with inverse gates. They introduced the Rational Identity Testing (RIT) problem which is to decide whether a noncommutative rational formula computes zero in the free skew field. In the white-box setting, deterministic polynomial-time algorithms are known for this problem following the works of Garg, Gurvits, Oliveira, and Wigderson (2016) and Ivanyos, Qiao, and Subrahmanyam (2018). A central open problem in this area is to design efficient deterministic black-box identity testing algorithm for rational formulas. In this paper, we solve this problem for the first nested inverse case. More precisely, we obtain a deterministic quasipolynomial-time black-box RIT algorithm for noncommutative rational formulas of inversion height two via a hitting set construction. Several new technical ideas are involved in the hitting set construction, including key concepts from matrix coefficient realization theory (Volv{c}iv{c}, 2018) and properties of cyclic division algebra (Lam, 2001). En route to the proof, an important step is to embed the hitting set of Forbes and Shpilka for noncommutative formulas (2013) inside a cyclic division algebra of small index.
Hrubev{s}和Wigderson(2015)开创了具有逆门的非交换公式的复杂性理论研究。他们引入了有理数恒等检验(RIT)问题，即判定非交换有理数公式在自由偏场中是否为零。在白盒环境中，确定性多项式时间算法在Garg, Gurvits, Oliveira, and Wigderson(2016)和Ivanyos, Qiao, and Subrahmanyam(2018)的工作之后被称为这个问题。该领域的一个核心开放问题是为有理公式设计高效的确定性黑盒恒等式检验算法。在本文中，我们解决了第一种嵌套逆情况下的这个问题。更精确地说，我们通过一个碰撞集构造得到了反演高度为2的非交换有理公式的确定性拟多项式时间黑箱RIT算法。撞击集的构造涉及了几个新的技术思想，包括矩阵系数实现理论(Volv{c}iv{c}， 2018)和循环除法代数性质(Lam, 2001)中的关键概念。在证明的过程中，一个重要的步骤是将Forbes和Shpilka的非交换公式(2013)的命中集嵌入到一个小指数的循环除法代数中。
{"title":"Black-box Identity Testing of Noncommutative Rational Formulas of Inversion Height Two in Deterministic Quasipolynomial-time","authors":"V. Arvind, Abhranil Chatterjee, P. Mukhopadhyay","doi":"10.4230/LIPIcs.APPROX/RANDOM.2022.23","DOIUrl":"https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.23","url":null,"abstract":"Hrubev{s} and Wigderson (2015) initiated the complexity-theoretic study of noncommutative formulas with inverse gates. They introduced the Rational Identity Testing (RIT) problem which is to decide whether a noncommutative rational formula computes zero in the free skew field. In the white-box setting, deterministic polynomial-time algorithms are known for this problem following the works of Garg, Gurvits, Oliveira, and Wigderson (2016) and Ivanyos, Qiao, and Subrahmanyam (2018). A central open problem in this area is to design efficient deterministic black-box identity testing algorithm for rational formulas. In this paper, we solve this problem for the first nested inverse case. More precisely, we obtain a deterministic quasipolynomial-time black-box RIT algorithm for noncommutative rational formulas of inversion height two via a hitting set construction. Several new technical ideas are involved in the hitting set construction, including key concepts from matrix coefficient realization theory (Volv{c}iv{c}, 2018) and properties of cyclic division algebra (Lam, 2001). En route to the proof, an important step is to embed the hitting set of Forbes and Shpilka for noncommutative formulas (2013) inside a cyclic division algebra of small index.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76623603","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 consider the Max-Cut problem, asking how much space is needed by a streaming algorithm in order to estimate the value of the maximum cut in a graph. This problem has been extensively studied over the last decade, and we now have a near-optimal lower bound for one-pass streaming algorithms, showing that they require linear space to guarantee a better-than-2 approximation [KKS15, KK19]. This result relies on a lower bound for the cycle-finding problem, showing that it is hard for a one-pass streaming algorithm to find a cycle in a union of matchings. The end-goal of our research is to prove a similar lower bound for multi-pass streaming algorithms that guarantee a better-than-2 approximation for Max-Cut, a highly challenging open problem. In this paper, we take a significant step in this direction, showing that even o(log n)-pass streaming algorithms need nΩ(1) space to solve the cycle-finding problem. Our proof is quite involved, dividing the cycles in the graph into “short” and “long” cycles, and using tailor-made lower bound techniques to handle each case. ∗UC Berkeley. †Princeton University. ‡Princeton University. §Microsoft Research. ¶Adobe Research. ‖Princeton University. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 161 (2022)
{"title":"Towards Multi-Pass Streaming Lower Bounds for Optimal Approximation of Max-Cut","authors":"Lijie Chen, Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena, Zhao Song, Huacheng Yu","doi":"10.1137/1.9781611977554.ch35","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch35","url":null,"abstract":"We consider the Max-Cut problem, asking how much space is needed by a streaming algorithm in order to estimate the value of the maximum cut in a graph. This problem has been extensively studied over the last decade, and we now have a near-optimal lower bound for one-pass streaming algorithms, showing that they require linear space to guarantee a better-than-2 approximation [KKS15, KK19]. This result relies on a lower bound for the cycle-finding problem, showing that it is hard for a one-pass streaming algorithm to find a cycle in a union of matchings. The end-goal of our research is to prove a similar lower bound for multi-pass streaming algorithms that guarantee a better-than-2 approximation for Max-Cut, a highly challenging open problem. In this paper, we take a significant step in this direction, showing that even o(log n)-pass streaming algorithms need nΩ(1) space to solve the cycle-finding problem. Our proof is quite involved, dividing the cycles in the graph into “short” and “long” cycles, and using tailor-made lower bound techniques to handle each case. ∗UC Berkeley. †Princeton University. ‡Princeton University. §Microsoft Research. ¶Adobe Research. ‖Princeton University. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 161 (2022)","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"276 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79160115","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}
Pub Date : 2022-01-01DOI: 10.1007/978-3-031-15802-5_6
Marshall Ball, D. Dachman-Soled, J. Loss
{"title":"(Nondeterministic) Hardness vs. Non-Malleability","authors":"Marshall Ball, D. Dachman-Soled, J. Loss","doi":"10.1007/978-3-031-15802-5_6","DOIUrl":"https://doi.org/10.1007/978-3-031-15802-5_6","url":null,"abstract":"","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"100 1","pages":"148-177"},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79326813","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}
Pub Date : 2022-01-01DOI: 10.1137/1.9781611977073.23
M. Henzinger, Andrea Lincoln, B. Saha
Statistics of small subgraph counts such as triangles, four-cycles, and s - t paths of short lengths reveal important structural properties of the underlying graph. These problems have been widely studied in social network analysis. In most relevant applications, the graphs are not only massive but also change dynamically over time. Most of these problems become hard in the dynamic setting when considering the worst case. In this paper, we ask whether the question of small subgraph counting over dynamic graphs is hard also in the average case. We consider the simplest possible average case model where the updates follow an Erd˝os-R´enyi graph: each update selects a pair of vertices ( u , v ) uniformly at random and flips the existence of the edge ( u , v ) . We develop new lower bounds and matching algorithms in this model for counting four-cycles, counting triangles through a specified point s , or a random queried point, and st paths of length 3, 4 and 5. Our results indicate while computing st paths of length 3, and 4 are easy in the average case with O ( 1 ) update time (note that they are hard in the worst case), it becomes hard when considering st paths of length 5. We introduce new techniques which allow us to get average-case hardness for these graph problems from the worst-case hardness of the Online Matrix vector problem (OMv). Our techniques rely on recent advances in fine-grained average-case complexity. Our techniques advance this literature, giving the ability to prove new lower bounds on average-case dynamic algorithms.
小子图计数的统计，如三角形、四圈和短长度的s - t路径，揭示了底层图的重要结构特性。这些问题在社会网络分析中得到了广泛的研究。在大多数相关的应用程序中，图不仅非常庞大，而且还会随时间动态变化。当考虑到最坏的情况时，大多数这些问题在动态环境中变得困难。在本文中，我们讨论了在平均情况下动态图上的小子图计数问题是否也很困难。我们考虑最简单的可能的平均情况模型，其中更新遵循Erd ' os ' - r ' enyi图:每次更新随机均匀地选择一对顶点(u, v)并翻转边缘(u, v)的存在性。在该模型中，我们开发了新的下界和匹配算法，用于计数四个循环，计数经过指定点s或随机查询点的三角形，以及长度为3,4,5的st路径。我们的结果表明，虽然计算长度为3和4的st条路径在O(1)更新时间的平均情况下很容易(注意，在最坏情况下它们很难)，但考虑长度为5的st条路径时就变得困难了。我们引入了新的技术，使我们能够从在线矩阵向量问题(OMv)的最坏情况硬度得到这些图问题的平均情况硬度。我们的技术依赖于细粒度平均情况复杂性的最新进展。我们的技术推进了这一文献，提供了证明平均情况下动态算法的新下界的能力。
{"title":"The Complexity of Average-Case Dynamic Subgraph Counting","authors":"M. Henzinger, Andrea Lincoln, B. Saha","doi":"10.1137/1.9781611977073.23","DOIUrl":"https://doi.org/10.1137/1.9781611977073.23","url":null,"abstract":"Statistics of small subgraph counts such as triangles, four-cycles, and s - t paths of short lengths reveal important structural properties of the underlying graph. These problems have been widely studied in social network analysis. In most relevant applications, the graphs are not only massive but also change dynamically over time. Most of these problems become hard in the dynamic setting when considering the worst case. In this paper, we ask whether the question of small subgraph counting over dynamic graphs is hard also in the average case. We consider the simplest possible average case model where the updates follow an Erd˝os-R´enyi graph: each update selects a pair of vertices ( u , v ) uniformly at random and flips the existence of the edge ( u , v ) . We develop new lower bounds and matching algorithms in this model for counting four-cycles, counting triangles through a specified point s , or a random queried point, and st paths of length 3, 4 and 5. Our results indicate while computing st paths of length 3, and 4 are easy in the average case with O ( 1 ) update time (note that they are hard in the worst case), it becomes hard when considering st paths of length 5. We introduce new techniques which allow us to get average-case hardness for these graph problems from the worst-case hardness of the Online Matrix vector problem (OMv). Our techniques rely on recent advances in fine-grained average-case complexity. Our techniques advance this literature, giving the ability to prove new lower bounds on average-case dynamic algorithms.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76789411","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}
Pub Date : 2022-01-01DOI: 10.4230/LIPIcs.ITCS.2023.46
K. Efremenko, Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena
Much of today’s communication is carried out over large wireless systems with different input-output behaviors. In this work, we compare the power of central abstractions of wireless communication through the general notion of boolean symmetric f -channels : In every round of the f -channel, each of its n parties decides to either broadcast or not, and the channel outputs f ( m ), where m is the number of broadcasting parties. Our first result is that the well studied beeping channel , where f is the threshold-1 function, is not stronger than any other f -channel. To this end, we design a protocol over the f -channel and prove that any protocol that simulates it over the beeping channel blows up the round complexity by a factor of Ω(log n ). Our lower bound technique may be of independent interest, as it essentially generalizes the popular fooling set technique by exploiting a “local” relaxation of combinatorial rectangles. Curiously, while this result shows the limitations of a noiseless channel, namely, the beeping channel, we are able to use it to show the limitations of the noisy version of many other channels. This includes the extensively studied single-hop radio network model with collisions-as-silence (CAS), which is equivalent to the f -channel with f ( m ) = 1 iff m = 1. In particular, our second and main result, obtained from the first, shows that converting CAS protocols to noise resilient ones may incur a large performance overhead, i.e., no constant rate interactive code exists. To this end, we design a CAS protocol and prove that any protocol that simulates it over the noisy CAS model with correlated stochastic noise, blows up the round complexity by a factor of Ω(log n ). We mention that the Ω(log n ) overhead in both our results is tight.
{"title":"Noisy Radio Network Lower Bounds via Noiseless Beeping Lower Bounds","authors":"K. Efremenko, Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena","doi":"10.4230/LIPIcs.ITCS.2023.46","DOIUrl":"https://doi.org/10.4230/LIPIcs.ITCS.2023.46","url":null,"abstract":"Much of today’s communication is carried out over large wireless systems with different input-output behaviors. In this work, we compare the power of central abstractions of wireless communication through the general notion of boolean symmetric f -channels : In every round of the f -channel, each of its n parties decides to either broadcast or not, and the channel outputs f ( m ), where m is the number of broadcasting parties. Our first result is that the well studied beeping channel , where f is the threshold-1 function, is not stronger than any other f -channel. To this end, we design a protocol over the f -channel and prove that any protocol that simulates it over the beeping channel blows up the round complexity by a factor of Ω(log n ). Our lower bound technique may be of independent interest, as it essentially generalizes the popular fooling set technique by exploiting a “local” relaxation of combinatorial rectangles. Curiously, while this result shows the limitations of a noiseless channel, namely, the beeping channel, we are able to use it to show the limitations of the noisy version of many other channels. This includes the extensively studied single-hop radio network model with collisions-as-silence (CAS), which is equivalent to the f -channel with f ( m ) = 1 iff m = 1. In particular, our second and main result, obtained from the first, shows that converting CAS protocols to noise resilient ones may incur a large performance overhead, i.e., no constant rate interactive code exists. To this end, we design a CAS protocol and prove that any protocol that simulates it over the noisy CAS model with correlated stochastic noise, blows up the round complexity by a factor of Ω(log n ). We mention that the Ω(log n ) overhead in both our results is tight.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89187776","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}
Pub Date : 2022-01-01DOI: 10.4230/LIPIcs.STACS.2023.16
N. Bshouty
We give the first polynomial-time non-adaptive proper learning algorithm of Boolean sparse multivariate polynomial under the uniform distribution. Our algorithm, for s -sparse polynomial over n variables, makes q = ( s/ϵ ) γ ( s,ϵ ) log n queries where 2 . 66 ≤ γ ( s, ϵ ) ≤ 6 . 922 and runs in ˜ O ( n ) · poly ( s, 1 /ϵ ) time. We also show that for any ϵ = 1 /s O (1) any non-adaptive learning algorithm must make at least ( s/ϵ ) Ω(1) log n queries. Therefore, the query complexity of our algorithm is also polynomial in the optimal query complexity and optimal in n .
{"title":"Non-Adaptive Proper Learning Polynomials","authors":"N. Bshouty","doi":"10.4230/LIPIcs.STACS.2023.16","DOIUrl":"https://doi.org/10.4230/LIPIcs.STACS.2023.16","url":null,"abstract":"We give the first polynomial-time non-adaptive proper learning algorithm of Boolean sparse multivariate polynomial under the uniform distribution. Our algorithm, for s -sparse polynomial over n variables, makes q = ( s/ϵ ) γ ( s,ϵ ) log n queries where 2 . 66 ≤ γ ( s, ϵ ) ≤ 6 . 922 and runs in ˜ O ( n ) · poly ( s, 1 /ϵ ) time. We also show that for any ϵ = 1 /s O (1) any non-adaptive learning algorithm must make at least ( s/ϵ ) Ω(1) log n queries. Therefore, the query complexity of our algorithm is also polynomial in the optimal query complexity and optimal in n .","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90624721","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}
Pub Date : 2022-01-01DOI: 10.4230/LIPIcs.FSTTCS.2022.10
Pranav Bisht, Ilya Volkovich
In a recent result of Bhargava, Saraf and Volkovich [FOCS’18; JACM’20], the first factor sparsity bound for constant individual degree polynomials was shown. In particular, it was shown that any factor of a polynomial with at most s terms and individual degree bounded by d can itself have at most s O ( d 2 log n ) terms. It is conjectured, though, that the “true” sparsity bound should be polynomial (i.e. s poly( d ) ). In this paper we provide supporting evidence for this conjecture by presenting polynomial-time algorithms for several problems that would be implied by a polynomial-size sparsity bound. In particular, we give efficient (deterministic) algorithms for identity testing of Σ [2] ΠΣΠ [ ind - deg d ] circuits and testing if a sparse polynomial is an exact power. Hence, our algorithms rely on different techniques.
{"title":"On Solving Sparse Polynomial Factorization Related Problems","authors":"Pranav Bisht, Ilya Volkovich","doi":"10.4230/LIPIcs.FSTTCS.2022.10","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSTTCS.2022.10","url":null,"abstract":"In a recent result of Bhargava, Saraf and Volkovich [FOCS’18; JACM’20], the first factor sparsity bound for constant individual degree polynomials was shown. In particular, it was shown that any factor of a polynomial with at most s terms and individual degree bounded by d can itself have at most s O ( d 2 log n ) terms. It is conjectured, though, that the “true” sparsity bound should be polynomial (i.e. s poly( d ) ). In this paper we provide supporting evidence for this conjecture by presenting polynomial-time algorithms for several problems that would be implied by a polynomial-size sparsity bound. In particular, we give efficient (deterministic) algorithms for identity testing of Σ [2] ΠΣΠ [ ind - deg d ] circuits and testing if a sparse polynomial is an exact power. Hence, our algorithms rely on different techniques.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"PC-20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84835735","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}
Pub Date : 2022-01-01DOI: 10.4230/LIPIcs.ITCS.2023.59
O. Goldreich, G. Rothblum, Tal Skverer
Interactive proofs of proximity (IPPs) offer ultra-fast approximate verification of assertions regarding their input, where ultra-fast means that only a small portion of the input is read and approximate verification is analogous to the notion of approximate decision that underlies property testing. Specifically, in an IPP, the prover can make the verifier accept each input in the property, but cannot fool the verifier into accepting an input that is far from the property (except for with small probability). The verifier in an IPP system engages in two very different types of activities: interacting with an untrusted prover, and querying its input. The definition allows for arbitrary coordination between these two activities, but keeping them separate is both conceptually interesting and necessary for important applications such as addressing temporal considerations (i.e., at what time is each of the services available) and facilitating the construction of zero-knowledge schemes. In this work we embark on a systematic study of IPPs with proof-oblivious queries, where the queries should not be affected by the interaction with the prover. We assign the query and interaction activities to separate modules, and consider different limitations on their coordination. The most strict limitation requires these activities to be totally isolated from one another; they just feed their views to a separate deciding module. We show that such systems can be efficiently emulated by standard testers. Going to the other extreme, we only disallow information to flow from the interacting module to the querying module, but allow free information flow in the other direction. We show that extremely efficient one-round (i.e., two-message) systems of such type can be used to verify properties that are extremely hard to test (without the help of a prover). That is, the complexity of verifying can be polylogarithmic in the complexity of testing. This stands in contrast the MAPs (viewed as 1 / 2-round systems) in which proof-oblivious queries are as limited as our isolated model. Our focus is on an intermediate model that allows shared randomness between the querying and interacting modules but no information flow between them. In this case we show that 1-round systems are efficiently emulated by standard testers but 3 / 2-round systems of extremely low complexity exist for properties that are extremely hard to test. One additional result about this model is that it can efficiently emulate any IPP for any property of low-degree polynomials.
{"title":"On Interactive Proofs of Proximity with Proof-Oblivious Queries","authors":"O. Goldreich, G. Rothblum, Tal Skverer","doi":"10.4230/LIPIcs.ITCS.2023.59","DOIUrl":"https://doi.org/10.4230/LIPIcs.ITCS.2023.59","url":null,"abstract":"Interactive proofs of proximity (IPPs) offer ultra-fast approximate verification of assertions regarding their input, where ultra-fast means that only a small portion of the input is read and approximate verification is analogous to the notion of approximate decision that underlies property testing. Specifically, in an IPP, the prover can make the verifier accept each input in the property, but cannot fool the verifier into accepting an input that is far from the property (except for with small probability). The verifier in an IPP system engages in two very different types of activities: interacting with an untrusted prover, and querying its input. The definition allows for arbitrary coordination between these two activities, but keeping them separate is both conceptually interesting and necessary for important applications such as addressing temporal considerations (i.e., at what time is each of the services available) and facilitating the construction of zero-knowledge schemes. In this work we embark on a systematic study of IPPs with proof-oblivious queries, where the queries should not be affected by the interaction with the prover. We assign the query and interaction activities to separate modules, and consider different limitations on their coordination. The most strict limitation requires these activities to be totally isolated from one another; they just feed their views to a separate deciding module. We show that such systems can be efficiently emulated by standard testers. Going to the other extreme, we only disallow information to flow from the interacting module to the querying module, but allow free information flow in the other direction. We show that extremely efficient one-round (i.e., two-message) systems of such type can be used to verify properties that are extremely hard to test (without the help of a prover). That is, the complexity of verifying can be polylogarithmic in the complexity of testing. This stands in contrast the MAPs (viewed as 1 / 2-round systems) in which proof-oblivious queries are as limited as our isolated model. Our focus is on an intermediate model that allows shared randomness between the querying and interacting modules but no information flow between them. In this case we show that 1-round systems are efficiently emulated by standard testers but 3 / 2-round systems of extremely low complexity exist for properties that are extremely hard to test. One additional result about this model is that it can efficiently emulate any IPP for any property of low-degree polynomials.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"20 1","pages":"59:1-59:16"},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84076491","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}
Pub Date : 2022-01-01DOI: 10.4230/LIPIcs.CCC.2023.8
Gil Cohen, Itay Cohen
Dinitz, Schapira, and Valadarsky [DSV17] introduced the intriguing notion of expanding expanders – a family of expander graphs with the property that every two consecutive graphs in the family differ only on a small number of edges. Such a family allows one to add and remove vertices with only few edge updates, making them useful in dynamic settings such as for datacenter network topologies and for the design of distributed algorithms for self-healing expanders. [DSV17] constructed explicit expanding-expanders based on the Bilu-Linial construction of spectral expanders [BL06]. The construction of expanding expanders, however, ends up being of edge expanders, thus, an open problem raised by [DSV17] is to construct spectral expanding expanders (SEE). In this work, we resolve this question by constructing SEE with spectral expansion which, like [BL06], is optimal up to a poly-logarithmic factor, and the number of edge updates is optimal up to a constant. We further give a simple proof for the existence of SEE that are close to Ramanujan up to a small additive term. As in [DSV17], our construction is based on interpolating between a graph and its lift. However, to establish spectral expansion, we carefully weigh the interpolated graphs, dubbed partial lifts, in a way that enables us to conduct a delicate analysis of their spectrum. In particular, at a crucial point in the analysis, we consider the eigenvectors structure of the partial lifts.
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Pub Date : 2022-01-01DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.11
Dan Karliner, A. Ta-Shma
Multiplicity codes are a generalization of Reed-Muller codes which include derivatives as well as the values of low degree polynomials, evaluated in every point in F mp . Similarly to Reed-Muller codes, multiplicity codes have a local nature that allows for local correction and local testing. Recently, [6] showed that the plane test , which tests the degree of the codeword on a random plane, is a good local tester for small enough degrees . In this work we simplify and extend the analysis of local testing for multiplicity codes, giving a more general and tight analysis. In particular, we show that multiplicity codes MRM p ( m, d, s ) over prime fields with arbitrary d are locally testable by an appropriate k -flat test , which tests the degree of the codeword on a random k -dimensional affine subspace. The relationship between the degree parameter d and the required dimension k is shown to be nearly optimal, and improves on [6] in the case of planes. Our analysis relies on a generalization of the technique of canonincal monomials introduced in [5]. Generalizing canonical monomials to the multiplicity case requires substantially different proofs which exploit the algebraic structure of multiplicity codes.
多重码是Reed-Muller码的一种推广，它包括导数和低次多项式的值，在fmp中的每个点上进行评估。与Reed-Muller码类似，多重码具有局部性质，允许局部校正和局部测试。最近，[6]证明了平面测试，即测试码字在随机平面上的度，是一个足够小度的很好的局部测试方法。在本文中，我们简化和扩展了多重码的局部测试分析，给出了一个更一般和严格的分析。特别地，我们证明了具有任意d的素域上的多重码MRM p (m, d, s)可以用一个适当的k -平坦检验来局部检验，该检验用于检验码字在随机k维仿射子空间上的度。度参数d与所需维度k之间的关系接近最优，并且在平面的情况下在[6]上得到改善。我们的分析依赖于b[5]中引入的标准单项式技术的推广。将正则单项式推广到多重情况需要大量不同的证明，这些证明利用了多重码的代数结构。
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