Dag-like communication protocols, a generalization of the classical tree-like communication protocols, are useful objects in the realm of proof complexity (most importantly for monotone feasible interpolation) and circuit complexity. We consider three kinds of protocols in this article (d is the degree of a protocol): — IEQ-d-dags: feasible sets of these protocols are described by inequality which means that the feasible sets are combinatorial triangles; these protocols are also called triangle-dags in the literature, — EQ-d-dags: feasible sets are described by equality, and — c-IEQ-d-dags: feasible sets are described by a conjunction of c inequalities.Garg, Göös, Kamath, and Sokolov (Theory of Computing, 2020) mentioned all these protocols, and they noted that EQ-d-dags are a special case of c-IEQ-d-dags. The exact relationship between these types of protocols is unclear. As our main contribution, we prove the following statement: EQ-2-dags can efficiently simulate c-IEQ-d-dags when c and d are constants. This implies that EQ-2-dags are at least as strong as IEQ-d-dags and that EQ-2-dags have the same strength as c-IEQ-d-dags for c ≥ 2 (because 2-IEQ-2-dags can trivially simulate EQ-2-dags). Hrubeš and Pudlák (Information Processing Letters, 2018) proved that IEQ-d-dags over the monotone Karchmer-Wigderson relation are equivalent to monotone real circuits which implies that we have exponential lower bounds for these protocols. Lower bounds for EQ-2-dags would directly imply lower bounds for the proof system R(LIN).
{"title":"On Protocols for Monotone Feasible Interpolation","authors":"Lukáš Folwarczný","doi":"10.1145/3583754","DOIUrl":"https://doi.org/10.1145/3583754","url":null,"abstract":"Dag-like communication protocols, a generalization of the classical tree-like communication protocols, are useful objects in the realm of proof complexity (most importantly for monotone feasible interpolation) and circuit complexity. We consider three kinds of protocols in this article (d is the degree of a protocol): — IEQ-d-dags: feasible sets of these protocols are described by inequality which means that the feasible sets are combinatorial triangles; these protocols are also called triangle-dags in the literature, — EQ-d-dags: feasible sets are described by equality, and — c-IEQ-d-dags: feasible sets are described by a conjunction of c inequalities.Garg, Göös, Kamath, and Sokolov (Theory of Computing, 2020) mentioned all these protocols, and they noted that EQ-d-dags are a special case of c-IEQ-d-dags. The exact relationship between these types of protocols is unclear. As our main contribution, we prove the following statement: EQ-2-dags can efficiently simulate c-IEQ-d-dags when c and d are constants. This implies that EQ-2-dags are at least as strong as IEQ-d-dags and that EQ-2-dags have the same strength as c-IEQ-d-dags for c ≥ 2 (because 2-IEQ-2-dags can trivially simulate EQ-2-dags). Hrubeš and Pudlák (Information Processing Letters, 2018) proved that IEQ-d-dags over the monotone Karchmer-Wigderson relation are equivalent to monotone real circuits which implies that we have exponential lower bounds for these protocols. Lower bounds for EQ-2-dags would directly imply lower bounds for the proof system R(LIN).","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46075631","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}
Akihiro Monde, Yukiko Yamauchi, S. Kijima, M. Yamashita
This article poses a question about a simple localization problem. The question is if an oblivious walker on a line segment can localize the midpoint of the line segment in a finite number of steps observing the direction (i.e., Left or Right) and the distance to the nearest end point. This problem arises from self-stabilizing location problems by autonomous mobile robots with limited visibility, which is an abstract model attracting a wide interest in distributed computing. Contrary to appearances, it is far from trivial whether this simple problem is solvable, and it is not settled yet. This article is concerned with three variants of the problem with a minimal relaxation and presents self-stabilizing algorithms for them. We also show an easy impossibility theorem for bilaterally symmetric algorithms.
{"title":"Can a Skywalker Localize the Midpoint of a Rope?","authors":"Akihiro Monde, Yukiko Yamauchi, S. Kijima, M. Yamashita","doi":"10.1145/3460954","DOIUrl":"https://doi.org/10.1145/3460954","url":null,"abstract":"This article poses a question about a simple localization problem. The question is if an oblivious walker on a line segment can localize the midpoint of the line segment in a finite number of steps observing the direction (i.e., Left or Right) and the distance to the nearest end point. This problem arises from self-stabilizing location problems by autonomous mobile robots with limited visibility, which is an abstract model attracting a wide interest in distributed computing. Contrary to appearances, it is far from trivial whether this simple problem is solvable, and it is not settled yet. This article is concerned with three variants of the problem with a minimal relaxation and presents self-stabilizing algorithms for them. We also show an easy impossibility theorem for bilaterally symmetric algorithms.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64042163","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}
Recent developments in approximate counting have made startling progress in developing fast algorithmic methods for approximating the number of solutions to constraint satisfaction problems (CSPs) with large arities, using connections to the Lovász Local Lemma. Nevertheless, the boundaries of these methods for CSPs with non-Boolean domain are not well-understood. Our goal in this article is to fill in this gap and obtain strong inapproximability results by studying the prototypical problem in this class of CSPs, hypergraph colourings. More precisely, we focus on the problem of approximately counting q-colourings on K-uniform hypergraphs with bounded degree Δ. An efficient algorithm exists if ({{Delta lesssim frac{q^{K/3-1}}{4^KK^2}}}) [Jain et al. 25; He et al. 23]. Somewhat surprisingly however, a hardness bound is not known even for the easier problem of finding colourings. For the counting problem, the situation is even less clear and there is no evidence of the right constant controlling the growth of the exponent in terms of K. To this end, we first establish that for general q computational hardness for finding a colouring on simple/linear hypergraphs occurs at Δ ≳ KqK, almost matching the algorithm from the Lovász Local Lemma. Our second and main contribution is to obtain a far more refined bound for the counting problem that goes well beyond the hardness of finding a colouring and which we conjecture is asymptotically tight (up to constant factors). We show in particular that for all even q ≥ 4 it is NP-hard to approximate the number of colourings when Δ ≳ qK/2. Our approach is based on considering an auxiliary weighted binary CSP model on graphs, which is obtained by “halving” the K-ary hypergraph constraints. This allows us to utilise reduction techniques available for the graph case, which hinge upon understanding the behaviour of random regular bipartite graphs that serve as gadgets in the reduction. The major challenge in our setting is to analyse the induced matrix norm of the interaction matrix of the new CSP which captures the most likely solutions of the system. In contrast to previous analyses in the literature, the auxiliary CSP demonstrates both symmetry and asymmetry, making the analysis of the optimisation problem severely more complicated and demanding the combination of delicate perturbation arguments and careful asymptotic estimates.
{"title":"Inapproximability of Counting Hypergraph Colourings","authors":"Andreas Galanis, Heng Guo, Jiaheng Wang","doi":"10.1145/3558554","DOIUrl":"https://doi.org/10.1145/3558554","url":null,"abstract":"Recent developments in approximate counting have made startling progress in developing fast algorithmic methods for approximating the number of solutions to constraint satisfaction problems (CSPs) with large arities, using connections to the Lovász Local Lemma. Nevertheless, the boundaries of these methods for CSPs with non-Boolean domain are not well-understood. Our goal in this article is to fill in this gap and obtain strong inapproximability results by studying the prototypical problem in this class of CSPs, hypergraph colourings. More precisely, we focus on the problem of approximately counting q-colourings on K-uniform hypergraphs with bounded degree Δ. An efficient algorithm exists if ({{Delta lesssim frac{q^{K/3-1}}{4^KK^2}}}) [Jain et al. 25; He et al. 23]. Somewhat surprisingly however, a hardness bound is not known even for the easier problem of finding colourings. For the counting problem, the situation is even less clear and there is no evidence of the right constant controlling the growth of the exponent in terms of K. To this end, we first establish that for general q computational hardness for finding a colouring on simple/linear hypergraphs occurs at Δ ≳ KqK, almost matching the algorithm from the Lovász Local Lemma. Our second and main contribution is to obtain a far more refined bound for the counting problem that goes well beyond the hardness of finding a colouring and which we conjecture is asymptotically tight (up to constant factors). We show in particular that for all even q ≥ 4 it is NP-hard to approximate the number of colourings when Δ ≳ qK/2. Our approach is based on considering an auxiliary weighted binary CSP model on graphs, which is obtained by “halving” the K-ary hypergraph constraints. This allows us to utilise reduction techniques available for the graph case, which hinge upon understanding the behaviour of random regular bipartite graphs that serve as gadgets in the reduction. The major challenge in our setting is to analyse the induced matrix norm of the interaction matrix of the new CSP which captures the most likely solutions of the system. In contrast to previous analyses in the literature, the auxiliary CSP demonstrates both symmetry and asymmetry, making the analysis of the optimisation problem severely more complicated and demanding the combination of delicate perturbation arguments and careful asymptotic estimates.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49631299","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}
Sayan Bandyapadhyay, F. Fomin, P. Golovach, Kirill Simonov
We develop new algorithmic methods with provable guarantees for feature selection in regard to categorical data clustering. While feature selection is one of the most common approaches to reduce dimensionality in practice, most of the known feature selection methods are heuristics. We study the following mathematical model. We assume that there are some inadvertent (or undesirable) features of the input data that unnecessarily increase the cost of clustering. Consequently, we want to select a subset of the original features from the data such that there is a small-cost clustering on the selected features. More precisely, for given integers ℓ (the number of irrelevant features) and k (the number of clusters), budget B, and a set of n categorical data points (represented by m-dimensional vectors whose elements belong to a finite set of values Σ), we want to select m − ℓ relevant features such that the cost of any optimal k-clustering on these features does not exceed B. Here the cost of a cluster is the sum of Hamming distances (ℓ0-distances) between the selected features of the elements of the cluster and its center. The clustering cost is the total sum of the costs of the clusters. We use the framework of parameterized complexity to identify how the complexity of the problem depends on parameters k, B, and |Σ|. Our main result is an algorithm that solves the Feature Selection problem in time f(k, B, |Σ|) · mg(k, |Σ|) · n2 for some functions f and g. In other words, the problem is fixed-parameter tractable parameterized by B when |Σ| and k are constants. Our algorithm for Feature Selection is based on a solution to a more general problem, Constrained Clustering with Outliers. In this problem, we want to delete a certain number of outliers such that the remaining points could be clustered around centers satisfying specific constraints. One interesting fact about Constrained Clustering with Outliers is that besides Feature Selection, it encompasses many other fundamental problems regarding categorical data such as Robust Clustering, and Binary and Boolean Low-rank Matrix Approximation with Outliers. Thus as a byproduct of our theorem, we obtain algorithms for all these problems. We also complement our algorithmic findings with complexity lower bounds.
{"title":"Parameterized Complexity of Feature Selection for Categorical Data Clustering","authors":"Sayan Bandyapadhyay, F. Fomin, P. Golovach, Kirill Simonov","doi":"10.1145/3604797","DOIUrl":"https://doi.org/10.1145/3604797","url":null,"abstract":"We develop new algorithmic methods with provable guarantees for feature selection in regard to categorical data clustering. While feature selection is one of the most common approaches to reduce dimensionality in practice, most of the known feature selection methods are heuristics. We study the following mathematical model. We assume that there are some inadvertent (or undesirable) features of the input data that unnecessarily increase the cost of clustering. Consequently, we want to select a subset of the original features from the data such that there is a small-cost clustering on the selected features. More precisely, for given integers ℓ (the number of irrelevant features) and k (the number of clusters), budget B, and a set of n categorical data points (represented by m-dimensional vectors whose elements belong to a finite set of values Σ), we want to select m − ℓ relevant features such that the cost of any optimal k-clustering on these features does not exceed B. Here the cost of a cluster is the sum of Hamming distances (ℓ0-distances) between the selected features of the elements of the cluster and its center. The clustering cost is the total sum of the costs of the clusters. We use the framework of parameterized complexity to identify how the complexity of the problem depends on parameters k, B, and |Σ|. Our main result is an algorithm that solves the Feature Selection problem in time f(k, B, |Σ|) · mg(k, |Σ|) · n2 for some functions f and g. In other words, the problem is fixed-parameter tractable parameterized by B when |Σ| and k are constants. Our algorithm for Feature Selection is based on a solution to a more general problem, Constrained Clustering with Outliers. In this problem, we want to delete a certain number of outliers such that the remaining points could be clustered around centers satisfying specific constraints. One interesting fact about Constrained Clustering with Outliers is that besides Feature Selection, it encompasses many other fundamental problems regarding categorical data such as Robust Clustering, and Binary and Boolean Low-rank Matrix Approximation with Outliers. Thus as a byproduct of our theorem, we obtain algorithms for all these problems. We also complement our algorithmic findings with complexity lower bounds.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43512867","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 a recent article, Kim and Kopparty (2017) gave a deterministic algorithm for the unique decoding problem for polynomials of bounded total degree over a general grid S1 Sm. We show that their alg...
{"title":"Decoding Variants of Reed-Muller Codes over Finite Grids","authors":"SrinivasanSrikanth, TripathiUtkarsh, VenkiteshS.","doi":"10.1145/3417754","DOIUrl":"https://doi.org/10.1145/3417754","url":null,"abstract":"In a recent article, Kim and Kopparty (2017) gave a deterministic algorithm for the unique decoding problem for polynomials of bounded total degree over a general grid S1 Sm. We show that their alg...","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3417754","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64033791","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}
Following Newman (2010), we initiate a study of testing properties of graphs that are presented as subgraphs of a fixed (or an explicitly given) graph. The tester is given free access to a base gra...
{"title":"The Subgraph Testing Model","authors":"GoldreichOded, RonDana","doi":"10.1145/3428675","DOIUrl":"https://doi.org/10.1145/3428675","url":null,"abstract":"Following Newman (2010), we initiate a study of testing properties of graphs that are presented as subgraphs of a fixed (or an explicitly given) graph. The tester is given free access to a base gra...","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78098130","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}
Consider the following secret-sharing problem: A file s should be distributed between n servers such that (d-1)-subsets cannot recover the file, (d+1)-subsets can recover the file, and d-subsets sh...
{"title":"On the Power of Amortization in Secret Sharing: d -Uniform Secret Sharing and CDS with Constant Information Rate","authors":"ApplebaumBenny, ArkisBarak","doi":"10.1145/3417756","DOIUrl":"https://doi.org/10.1145/3417756","url":null,"abstract":"Consider the following secret-sharing problem: A file s should be distributed between n servers such that (d-1)-subsets cannot recover the file, (d+1)-subsets can recover the file, and d-subsets sh...","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76957129","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}
F. Fomin, P. Golovach, Giannos Stamoulis, D. Thilikos
In general, a graph modification problem is defined by a graph modification operation ⊠ and a target graph property 𝒫. Typically, the modification operation ⊠ may be vertex deletion, edge deletion, edge contraction, or edge addition and the question is, given a graph G and an integer k, whether it is possible to transform G to a graph in 𝒫 after applying the operation ⊠ k times on G. This problem has been extensively studied for particular instantiations of ⊠ and 𝒫. In this article, we consider the general property 𝒫𝛗 of being planar and, additionally, being a model of some First-Order Logic (FOL) sentence 𝛗 (an FOL-sentence). We call the corresponding meta-problem Graph ⊠-Modification to Planarity and 𝛗 and prove the following algorithmic meta-theorem: there exists a function f : ℕ2 → ℕ such that, for every ⊠ and every FOL-sentence 𝛗, the Graph ⊠-Modification to Planarity and 𝛗 is solvable in f(k,|𝛗|)⋅ n2 time. The proof constitutes a hybrid of two different classic techniques in graph algorithms. The first is the irrelevant vertex technique that is typically used in the context of Graph Minors and deals with properties such as planarity or surface-embeddability (that are not FOL-expressible) and the second is the use of Gaifman’s locality theorem that is the theoretical base for the meta-algorithmic study of FOL-expressible problems.
{"title":"An Algorithmic Meta-Theorem for Graph Modification to Planarity and FOL","authors":"F. Fomin, P. Golovach, Giannos Stamoulis, D. Thilikos","doi":"10.1145/3571278","DOIUrl":"https://doi.org/10.1145/3571278","url":null,"abstract":"In general, a graph modification problem is defined by a graph modification operation ⊠ and a target graph property 𝒫. Typically, the modification operation ⊠ may be vertex deletion, edge deletion, edge contraction, or edge addition and the question is, given a graph G and an integer k, whether it is possible to transform G to a graph in 𝒫 after applying the operation ⊠ k times on G. This problem has been extensively studied for particular instantiations of ⊠ and 𝒫. In this article, we consider the general property 𝒫𝛗 of being planar and, additionally, being a model of some First-Order Logic (FOL) sentence 𝛗 (an FOL-sentence). We call the corresponding meta-problem Graph ⊠-Modification to Planarity and 𝛗 and prove the following algorithmic meta-theorem: there exists a function f : ℕ2 → ℕ such that, for every ⊠ and every FOL-sentence 𝛗, the Graph ⊠-Modification to Planarity and 𝛗 is solvable in f(k,|𝛗|)⋅ n2 time. The proof constitutes a hybrid of two different classic techniques in graph algorithms. The first is the irrelevant vertex technique that is typically used in the context of Graph Minors and deals with properties such as planarity or surface-embeddability (that are not FOL-expressible) and the second is the use of Gaifman’s locality theorem that is the theoretical base for the meta-algorithmic study of FOL-expressible problems.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45963136","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 article, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which the polynomial computed at every node has a bound on the individual degree of r≥ 1 with respect to all its variables (referred to as multi-r-ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better bounds as the value of r increases. Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degree r computing an explicit multilinear polynomial on nO(1) variables and degree d must have size at least (n/r1.1)Ω(√d/r). This bound, however, deteriorates as the value of r increases. It is a natural question to ask if we can prove a bound that does not deteriorate as the value of r increases, or a bound that holds for a larger regime of r. In this article, we prove a lower bound that does not deteriorate with increasing values of r, albeit for a specific instance of d = d(n) but for a wider range of r. Formally, for all large enough integers n and a small constant η, we show that there exists an explicit polynomial on nO(1) variables and degree Θ (log2 n) such that any depth four circuit of bounded individual degree r ≤ nη must have size at least exp(Ω(log2 n)). This improvement is obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017).
在本文中，我们感兴趣的是理解使用深度四电路计算多线性多项式的复杂性，其中在每个节点计算的多项式相对于其所有变量(称为多r-ic电路)的单个度r≥1有一个界。本研究的目标是通过证明r值增加时更好的边界，在证明一般深度四电路计算多线性多项式的超多项式下界方面取得进展。最近，Kayal, Saha和Tavenas (Theory of Computing, 2018)表明，在nO(1)个变量和d次上计算显式多线性多项式的任何有限个体度r的深度四算术电路必须具有至少(n/r1.1)Ω(√d/r)的大小。然而，随着r值的增加，这个界限会变差。一个自然的问题是，我们是否能证明一个不随r值的增加而恶化的边界，或者一个适用于更大范围r的边界。在本文中，我们证明了一个不随r值的增加而恶化的下界，尽管是针对d = d(n)的特定实例，但适用于更广泛的r范围。形式上，对于所有足够大的整数n和一个小的常数η，我们证明了在nO(1)个变量和阶次Θ (log2 n)上存在一个显式多项式，使得任何有界的单个阶次r≤nη的深度四回路必须至少具有exp(Ω(log2 n))的大小。这种改进是通过适当地采用Kayal等人的复杂性度量来获得的(Theory of Computing, 2018)。这种对度量的适应受到Kayal等人使用的复杂性度量的启发(SIAM J. Computing, 2017)。
{"title":"On Computing Multilinear Polynomials Using Multi-r-ic Depth Four Circuits","authors":"S. Chillara","doi":"10.1145/3460952","DOIUrl":"https://doi.org/10.1145/3460952","url":null,"abstract":"In this article, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which the polynomial computed at every node has a bound on the individual degree of r≥ 1 with respect to all its variables (referred to as multi-r-ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better bounds as the value of r increases. Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degree r computing an explicit multilinear polynomial on nO(1) variables and degree d must have size at least (n/r1.1)Ω(√d/r). This bound, however, deteriorates as the value of r increases. It is a natural question to ask if we can prove a bound that does not deteriorate as the value of r increases, or a bound that holds for a larger regime of r. In this article, we prove a lower bound that does not deteriorate with increasing values of r, albeit for a specific instance of d = d(n) but for a wider range of r. Formally, for all large enough integers n and a small constant η, we show that there exists an explicit polynomial on nO(1) variables and degree Θ (log2 n) such that any depth four circuit of bounded individual degree r ≤ nη must have size at least exp(Ω(log2 n)). This improvement is obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017).","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3460952","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64042116","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 problem of finding K collision pairs in a random function f : [N] → [N] by using a quantum computer. We prove that the number of queries to the function in the quantum random oracle model must increase significantly when the size of the available memory is limited. Namely, we demonstrate that any algorithm using S qubits of memory must perform a number T of queries that satisfies the tradeoff T3 S ≥ Ω (K3 N). Classically, the same question has only been settled recently by Dinur [22], who showed that the Parallel Collision Search algorithm of van Oorschot and Wiener [36] achieves the optimal time–space tradeoff of T2 S = Θ (K2 N). Our result limits the extent to which quantum computing may decrease this tradeoff. Our method is based on a novel application of Zhandry’s recording query technique [42] for proving lower bounds in the exponentially small success probability regime. As a second application, we give a simpler proof of the time–space tradeoff T2 S ≥ Ω (N3) for sorting N numbers on a quantum computer, which was first obtained by Klauck, Špalek, and de Wolf [30].
{"title":"Quantum Time–Space Tradeoff for Finding Multiple Collision Pairs","authors":"Yassine Hamoudi, F. Magniez","doi":"10.1145/3589986","DOIUrl":"https://doi.org/10.1145/3589986","url":null,"abstract":"We study the problem of finding K collision pairs in a random function f : [N] → [N] by using a quantum computer. We prove that the number of queries to the function in the quantum random oracle model must increase significantly when the size of the available memory is limited. Namely, we demonstrate that any algorithm using S qubits of memory must perform a number T of queries that satisfies the tradeoff T3 S ≥ Ω (K3 N). Classically, the same question has only been settled recently by Dinur [22], who showed that the Parallel Collision Search algorithm of van Oorschot and Wiener [36] achieves the optimal time–space tradeoff of T2 S = Θ (K2 N). Our result limits the extent to which quantum computing may decrease this tradeoff. Our method is based on a novel application of Zhandry’s recording query technique [42] for proving lower bounds in the exponentially small success probability regime. As a second application, we give a simpler proof of the time–space tradeoff T2 S ≥ Ω (N3) for sorting N numbers on a quantum computer, which was first obtained by Klauck, Špalek, and de Wolf [30].","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49280626","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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