We consider probabilistic circuits working over the real numbers and using arbitrary semialgebraic functions of bounded description complexity as gates. In particular, such circuits can use all arithmetic operations (+, −, ×, ÷), optimization operations (min and max), conditional branching (if-then-else), and many more. We show that probabilistic circuits using any of these operations as gates can be simulated by deterministic circuits with only about a quadratical blowup in size. A slightly larger blowup in circuit size is also shown when derandomizing approximating circuits. The algorithmic consequence, motivating the title, is that randomness cannot substantially speed up dynamic programming algorithms.
{"title":"Coin Flipping in Dynamic Programming Is Almost Useless","authors":"S. Jukna","doi":"10.1145/3397476","DOIUrl":"https://doi.org/10.1145/3397476","url":null,"abstract":"We consider probabilistic circuits working over the real numbers and using arbitrary semialgebraic functions of bounded description complexity as gates. In particular, such circuits can use all arithmetic operations (+, −, ×, ÷), optimization operations (min and max), conditional branching (if-then-else), and many more. We show that probabilistic circuits using any of these operations as gates can be simulated by deterministic circuits with only about a quadratical blowup in size. A slightly larger blowup in circuit size is also shown when derandomizing approximating circuits. The algorithmic consequence, motivating the title, is that randomness cannot substantially speed up dynamic programming algorithms.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129219923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Test","authors":"Cornetto","doi":"10.1145/3397480","DOIUrl":"https://doi.org/10.1145/3397480","url":null,"abstract":"","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"118 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132453721","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}
Unlike polynomial kernelization in general, for which many non-trivial results and methods exist, only few non-trival algorithms are known for polynomial-time sparsification. Furthermore, excepting problems on restricted inputs (such as graph problems on planar graphs), most such results rely upon encoding the instance as a system of bounded-degree polynomial equations. In particular, for satisfiability (SAT) problems with a fixed constraint language Γ, every previously known result is captured by this approach; for several such problems, this is known to be tight. In this work, we investigate the limits of this approach—in particular, does it really cover all cases of non-trivial polynomial-time sparsification? We generalize the method using tools from the algebraic approach to constraint satisfaction problems (CSP). Every constraint that can be modelled via a system of linear equations, over some finite field F, also admits a finite domain extension to a tractable CSP with a Maltsev polymorphism; using known algorithms for Maltsev languages, we can show that every problem of the latter type admits a “basis” of O(n) constraints, which implies a linear sparsification for the original problem. This generalization appears to be strict; other special cases include constraints modelled via group equations over some finite group G. For sparsifications of polynomial but super-linear size, we consider two extensions of this. Most directly, we can capture systems of bounded-degree polynomial equations in a “lift-and-project” manner, by finding Maltsev extensions for constraints over c-tuples of variables, for a basis with O(nc) constraints. Additionally, we may use extensions with k-edge polymorphisms instead of requiring a Maltsev polymorphism. We also investigate characterizations of when such extensions exist. We give an infinite sequence of partial polymorphisms φ1, φ2, …which characterizes whether a language Γ has a Maltsev extension (of possibly infinite domain). In the complementary direction of proving lower bounds on kernelizability, we prove that for any language not preserved by φ1, the corresponding SAT problem does not admit a kernel of size O(n2−ε) for any ε > 0 unless the polynomial hierarchy collapses.
{"title":"Sparsification of SAT and CSP Problems via Tractable Extensions","authors":"Victor Lagerkvist, Magnus Wahlström","doi":"10.1145/3389411","DOIUrl":"https://doi.org/10.1145/3389411","url":null,"abstract":"Unlike polynomial kernelization in general, for which many non-trivial results and methods exist, only few non-trival algorithms are known for polynomial-time sparsification. Furthermore, excepting problems on restricted inputs (such as graph problems on planar graphs), most such results rely upon encoding the instance as a system of bounded-degree polynomial equations. In particular, for satisfiability (SAT) problems with a fixed constraint language Γ, every previously known result is captured by this approach; for several such problems, this is known to be tight. In this work, we investigate the limits of this approach—in particular, does it really cover all cases of non-trivial polynomial-time sparsification? We generalize the method using tools from the algebraic approach to constraint satisfaction problems (CSP). Every constraint that can be modelled via a system of linear equations, over some finite field F, also admits a finite domain extension to a tractable CSP with a Maltsev polymorphism; using known algorithms for Maltsev languages, we can show that every problem of the latter type admits a “basis” of O(n) constraints, which implies a linear sparsification for the original problem. This generalization appears to be strict; other special cases include constraints modelled via group equations over some finite group G. For sparsifications of polynomial but super-linear size, we consider two extensions of this. Most directly, we can capture systems of bounded-degree polynomial equations in a “lift-and-project” manner, by finding Maltsev extensions for constraints over c-tuples of variables, for a basis with O(nc) constraints. Additionally, we may use extensions with k-edge polymorphisms instead of requiring a Maltsev polymorphism. We also investigate characterizations of when such extensions exist. We give an infinite sequence of partial polymorphisms φ1, φ2, …which characterizes whether a language Γ has a Maltsev extension (of possibly infinite domain). In the complementary direction of proving lower bounds on kernelizability, we prove that for any language not preserved by φ1, the corresponding SAT problem does not admit a kernel of size O(n2−ε) for any ε > 0 unless the polynomial hierarchy collapses.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129739387","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, D. Lokshtanov, Ivan Mihajlin, Saket Saurabh, M. Zehavi
We prove that the Hadwiger number of an n-vertex graph G (the maximum size of a clique minor in G) cannot be computed in time no(n), unless the Exponential Time Hypothesis (ETH) fails. This resolves a well-known open question in the area of exact exponential algorithms. The technique developed for resolving the Hadwiger number problem has a wider applicability. We use it to rule out the existence of no(n)-time algorithms (up to the ETH) for a large class of computational problems concerning edge contractions in graphs.
{"title":"Computation of Hadwiger Number and Related Contraction Problems","authors":"F. Fomin, D. Lokshtanov, Ivan Mihajlin, Saket Saurabh, M. Zehavi","doi":"10.1145/3448639","DOIUrl":"https://doi.org/10.1145/3448639","url":null,"abstract":"We prove that the Hadwiger number of an n-vertex graph G (the maximum size of a clique minor in G) cannot be computed in time no(n), unless the Exponential Time Hypothesis (ETH) fails. This resolves a well-known open question in the area of exact exponential algorithms. The technique developed for resolving the Hadwiger number problem has a wider applicability. We use it to rule out the existence of no(n)-time algorithms (up to the ETH) for a large class of computational problems concerning edge contractions in graphs.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130950295","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 spin system is a framework in which the vertices of a graph are assigned spins from a finite set. The interactions between neighbouring spins give rise to weights, so a spin assignment can also be viewed as a weighted graph homomorphism. The problem of approximating the partition function (the aggregate weight of spin assignments) or of sampling from the resulting probability distribution is typically intractable for general graphs. In this work, we consider arbitrary spin systems on bipartite expander Δ-regular graphs, including the canonical class of bipartite random Δ-regular graphs. We develop fast approximate sampling and counting algorithms for general spin systems whenever the degree and the spectral gap of the graph are sufficiently large. Roughly, this guarantees that the spin system is in the so-called low-temperature regime. Our approach generalises the techniques of Jenssen et al. and Chen et al. by showing that typical configurations on bipartite expanders correspond to “bicliques” of the spin system; then, using suitable polymer models, we show how to sample such configurations and approximate the partition function in Õ(n2) time, where n is the size of the graph.
自旋系统是一个框架，在这个框架中，图的顶点从一个有限集合中被分配自旋。相邻自旋之间的相互作用会产生权值，因此自旋分配也可以看作是加权图同态。对于一般图来说，逼近配分函数(自旋分配的总权重)或从结果概率分布中抽样的问题通常是难以解决的。在这项工作中，我们考虑了二部扩展Δ-regular图上的任意自旋系统，包括二部随机Δ-regular图的规范类。当图的度和谱隙足够大时，我们开发了一般自旋系统的快速近似采样和计数算法。粗略地说，这保证了自旋系统处于所谓的低温状态。我们的方法推广了Jenssen et al.和Chen et al.的技术，证明了二部展开器上的典型构型对应于自旋系统的“双线”;然后，使用合适的聚合物模型，我们展示了如何在Õ(n2)时间内对这种构型进行采样并近似配分函数，其中n是图的大小。
{"title":"Fast Algorithms for General Spin Systems on Bipartite Expanders","authors":"Andreas Galanis, L. A. Goldberg, James Stewart","doi":"10.1145/3470865","DOIUrl":"https://doi.org/10.1145/3470865","url":null,"abstract":"A spin system is a framework in which the vertices of a graph are assigned spins from a finite set. The interactions between neighbouring spins give rise to weights, so a spin assignment can also be viewed as a weighted graph homomorphism. The problem of approximating the partition function (the aggregate weight of spin assignments) or of sampling from the resulting probability distribution is typically intractable for general graphs. In this work, we consider arbitrary spin systems on bipartite expander Δ-regular graphs, including the canonical class of bipartite random Δ-regular graphs. We develop fast approximate sampling and counting algorithms for general spin systems whenever the degree and the spectral gap of the graph are sufficiently large. Roughly, this guarantees that the spin system is in the so-called low-temperature regime. Our approach generalises the techniques of Jenssen et al. and Chen et al. by showing that typical configurations on bipartite expanders correspond to “bicliques” of the spin system; then, using suitable polymer models, we show how to sample such configurations and approximate the partition function in Õ(n2) time, where n is the size of the graph.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"14 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132547387","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 ( x,yin lbrace 0,1rbrace ^n ) , Set Disjointness consists in deciding whether ( x_i=y_i=1 ) for some index ( i in [n] ) . We study the problem of computing this function in a distributed computing scenario in which the inputs ( x ) and ( y ) are given to the processors at the two extremities of a path of length ( d ) . Each vertex of the path has a quantum processor that can communicate with each of its neighbours by exchanging ( operatorname{O}(log n) ) qubits per round. We are interested in the number of rounds required for computing Set Disjointness with constant probability bounded away from ( 1/2 ) . We call this problem “Set Disjointness on a Line”. Set Disjointness on a Line was introduced by Le Gall and Magniez [14] for proving lower bounds on the quantum distributed complexity of computing the diameter of an arbitrary network in the CONGEST model. However, they were only able to provide a lower bound when the local memory used by the processors on the intermediate vertices of the path is severely limited. More precisely, their bound applies only when the local memory of each intermediate processor consists of ( operatorname{O}(log n) ) qubits. In this work, we prove an unconditional lower bound of ( widetilde{Omega }big (sqrt [3]{n d^2}+sqrt {n} , big) ) rounds for Set Disjointness on a Line with ( d + 1 ) processors. This is the first non-trivial lower bound when there is no restriction on the memory used by the processors. The result gives us a new lower bound of ( widetilde{Omega } big (sqrt [3]{ndelta ^2}+sqrt {n} , big) ) on the number of rounds required for computing the diameter ( delta ) of any ( n ) -node network with quantum messages of size ( operatorname{O}(log n) ) in the CONGEST model. We draw a connection between the distributed computing scenario above and a new model of query complexity. In this model, an algorithm computing a bi-variate function ( f ) (such as Set Disjointness) has access to the inputs ( x ) and ( y ) through two separate oracles ( {mathcal {O}}_x ) and ( {mathcal {O}}_y ) , respectively. The restriction is that the algorithm is required to alternately make ( d ) queries to ( {mathcal {O}}_x ) and ( d ) queries to ( {mathcal {O}}_y ) , with input-independent computation in between queries. The model reflects a “switching delay” of ( d ) queries between a “round” of queries to ( x ) and the following “round” of queries to ( y ) . The information-theoretic technique we use for deriving the round lower bound for Set Disjointness on a Line also applies to the number of rounds in this query model. We provide an algorithm for Set Disjointness in this query model with round complexity that matches the round lower bound stated above, up to a polylogarithmic factor. This presents a barrier for obtaining a better round lower bound for Set Disjointness on the Line. At the same time, it hints at the possibility of better communication protocols for the problem.
给定( x,yin lbrace 0,1rbrace ^n ), Set Disjointness决定是否( x_i=y_i=1 )适用于某些索引( i in [n] )。我们研究了在分布式计算场景中计算该函数的问题，其中输入( x )和( y )在长度为( d )的路径的两个极端处被给定给处理器。路径的每个顶点都有一个量子处理器，可以通过每轮交换( operatorname{O}(log n) )量子比特与每个相邻节点进行通信。我们感兴趣的是计算集脱节所需的轮数，具有恒定的概率，有界于( 1/2 )。我们称这个问题为“直线上的不相交集”。Le Gall和Magniez[14]引入了Set Disjointness on a Line，用于证明在CONGEST模型中计算任意网络直径的量子分布复杂度的下界。然而，只有当处理器在路径的中间顶点上使用的本地内存受到严重限制时，它们才能提供一个下界。更准确地说，它们的边界只适用于每个中间处理器的本地内存包含( operatorname{O}(log n) )量子位元的情况。在这项工作中，我们证明了具有( d + 1 )处理器的直线上集合不相交的一个( widetilde{Omega }big (sqrt [3]{n d^2}+sqrt {n} , big) )轮的无条件下界。当处理器使用的内存没有限制时，这是第一个非平凡的下界。结果给出了在CONGEST模型中计算任何具有大小为( operatorname{O}(log n) )的量子消息的( n )节点网络的直径( delta )所需的轮数的新下界( widetilde{Omega } big (sqrt [3]{ndelta ^2}+sqrt {n} , big) )。我们在上面的分布式计算场景和一个新的查询复杂性模型之间建立了联系。在这个模型中，计算双变量函数( f )(比如Set Disjointness)的算法可以分别通过两个独立的oracle ( {mathcal {O}}_x )和( {mathcal {O}}_y )访问输入( x )和( y )。限制是，该算法需要交替地对( {mathcal {O}}_x )进行( d )查询，对( {mathcal {O}}_y )进行( d )查询，在查询之间进行与输入无关的计算。该模型反映了( d )查询在到( x )的“一轮”查询和下一轮到( y )的“一轮”查询之间的“切换延迟”。我们用于推导直线上集合不相交的轮下界的信息理论技术也适用于该查询模型中的轮数。在这个查询模型中，我们提供了一种Set Disjointness算法，其复杂度与上面所述的round下界匹配，直到一个多对数因子。这为求直线上集合不相交的较好圆下界提供了一个障碍。同时，它暗示了解决这个问题的更好的通信协议的可能性。
{"title":"Quantum Distributed Complexity of Set Disjointness on a Line","authors":"F. Magniez, A. Nayak","doi":"10.1145/3512751","DOIUrl":"https://doi.org/10.1145/3512751","url":null,"abstract":"Given ( x,yin lbrace 0,1rbrace ^n ) , Set Disjointness consists in deciding whether ( x_i=y_i=1 ) for some index ( i in [n] ) . We study the problem of computing this function in a distributed computing scenario in which the inputs ( x ) and ( y ) are given to the processors at the two extremities of a path of length ( d ) . Each vertex of the path has a quantum processor that can communicate with each of its neighbours by exchanging ( operatorname{O}(log n) ) qubits per round. We are interested in the number of rounds required for computing Set Disjointness with constant probability bounded away from ( 1/2 ) . We call this problem “Set Disjointness on a Line”. Set Disjointness on a Line was introduced by Le Gall and Magniez [14] for proving lower bounds on the quantum distributed complexity of computing the diameter of an arbitrary network in the CONGEST model. However, they were only able to provide a lower bound when the local memory used by the processors on the intermediate vertices of the path is severely limited. More precisely, their bound applies only when the local memory of each intermediate processor consists of ( operatorname{O}(log n) ) qubits. In this work, we prove an unconditional lower bound of ( widetilde{Omega }big (sqrt [3]{n d^2}+sqrt {n} , big) ) rounds for Set Disjointness on a Line with ( d + 1 ) processors. This is the first non-trivial lower bound when there is no restriction on the memory used by the processors. The result gives us a new lower bound of ( widetilde{Omega } big (sqrt [3]{ndelta ^2}+sqrt {n} , big) ) on the number of rounds required for computing the diameter ( delta ) of any ( n ) -node network with quantum messages of size ( operatorname{O}(log n) ) in the CONGEST model. We draw a connection between the distributed computing scenario above and a new model of query complexity. In this model, an algorithm computing a bi-variate function ( f ) (such as Set Disjointness) has access to the inputs ( x ) and ( y ) through two separate oracles ( {mathcal {O}}_x ) and ( {mathcal {O}}_y ) , respectively. The restriction is that the algorithm is required to alternately make ( d ) queries to ( {mathcal {O}}_x ) and ( d ) queries to ( {mathcal {O}}_y ) , with input-independent computation in between queries. The model reflects a “switching delay” of ( d ) queries between a “round” of queries to ( x ) and the following “round” of queries to ( y ) . The information-theoretic technique we use for deriving the round lower bound for Set Disjointness on a Line also applies to the number of rounds in this query model. We provide an algorithm for Set Disjointness in this query model with round complexity that matches the round lower bound stated above, up to a polylogarithmic factor. This presents a barrier for obtaining a better round lower bound for Set Disjointness on the Line. At the same time, it hints at the possibility of better communication protocols for the problem.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130729772","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}
It is my honor to assume the role of editor-in-chief for Transactions on Computation Theory (ToCT). Thanks in large part to the passion and vision of the previous editor-in-chief, Venkat Guruswami, ToCT has grown significantly over the last several years, enhancing its role as a forum for cuttingedge research in computational complexity theory and related fields. On behalf of the community, I’d like to sincerely thank Venkat and the entire team of associate editors whose service to the journal has made this possible. At the same time, I’d also like to thank ACM’s journals manager Laura Lander and ToCT’s administrator Victoria White for their ongoing assistance with the journal and with the editor-in-chief transition. Chase Klingensmith has been added to the team in the role of information director, and I expect to invite a few new associate editors to the board in the near future. My main goals for ToCT in the coming years are threefold. First, I hope to raise its profile within the community, ensuring its status as leading destination for top research in complexity theory, broadly construed. A second goal is to maintain the quality of our reviews, while improving the response time to the authors, and decreasing the variance of time that papers remain in the system. Finally, I wish to continue to increase the number of high-quality results appearing in ToCT. To this end, the associate editors and I intend to play an active role in soliciting the most exciting new research in the field. For example, ToCT will bid to host the 2021 special issue for the Computational Complexity Conference, the flagship specialist conference in the area. Complexity theory is devoted to the foundational and mathematical underpinnings of computation, and as such it hosts some of the deepest results in theoretical computer science. Yet a continuing challenge in our field is the prevalence of research works whose final publication venue is conference proceedings. As a professional society journal, ToCT can play an invaluable archival role, and I hope to guide it in promoting the highest-quality scholarship in the years to come.
{"title":"Editorial from the New Editor-in-Chief","authors":"R. O'Donnell","doi":"10.1145/3381517","DOIUrl":"https://doi.org/10.1145/3381517","url":null,"abstract":"It is my honor to assume the role of editor-in-chief for Transactions on Computation Theory (ToCT). Thanks in large part to the passion and vision of the previous editor-in-chief, Venkat Guruswami, ToCT has grown significantly over the last several years, enhancing its role as a forum for cuttingedge research in computational complexity theory and related fields. On behalf of the community, I’d like to sincerely thank Venkat and the entire team of associate editors whose service to the journal has made this possible. At the same time, I’d also like to thank ACM’s journals manager Laura Lander and ToCT’s administrator Victoria White for their ongoing assistance with the journal and with the editor-in-chief transition. Chase Klingensmith has been added to the team in the role of information director, and I expect to invite a few new associate editors to the board in the near future. My main goals for ToCT in the coming years are threefold. First, I hope to raise its profile within the community, ensuring its status as leading destination for top research in complexity theory, broadly construed. A second goal is to maintain the quality of our reviews, while improving the response time to the authors, and decreasing the variance of time that papers remain in the system. Finally, I wish to continue to increase the number of high-quality results appearing in ToCT. To this end, the associate editors and I intend to play an active role in soliciting the most exciting new research in the field. For example, ToCT will bid to host the 2021 special issue for the Computational Complexity Conference, the flagship specialist conference in the area. Complexity theory is devoted to the foundational and mathematical underpinnings of computation, and as such it hosts some of the deepest results in theoretical computer science. Yet a continuing challenge in our field is the prevalence of research works whose final publication venue is conference proceedings. As a professional society journal, ToCT can play an invaluable archival role, and I hope to guide it in promoting the highest-quality scholarship in the years to come.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117225176","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 major problem in computational learning theory is whether the class of formulas in conjunctive normal form (CNF) is efficiently learnable. Although it is known that this class cannot be polynomially learned using either membership or equivalence queries alone, it is open whether the CNF class can be polynomially learned using both types of queries. One of the most important results concerning a restriction of the CNF class is that propositional Horn formulas are polynomial time learnable in Angluin’s exact learning model with membership and equivalence queries. In this work, we push this boundary and show that the class of multivalued dependency formulas (MVDF), which non-trivially extends propositional Horn, is polynomially learnable from interpretations. We then provide a notion of reduction between learning problems in Angluin’s model, showing that a transformation of the algorithm suffices to efficiently learn multivalued database dependencies from data relations. We also show via reductions that our main result extends well known previous results and allows us to find alternative solutions for them.
{"title":"Exact Learning","authors":"Montserrat Hermo, A. Ozaki","doi":"10.1145/3369930","DOIUrl":"https://doi.org/10.1145/3369930","url":null,"abstract":"A major problem in computational learning theory is whether the class of formulas in conjunctive normal form (CNF) is efficiently learnable. Although it is known that this class cannot be polynomially learned using either membership or equivalence queries alone, it is open whether the CNF class can be polynomially learned using both types of queries. One of the most important results concerning a restriction of the CNF class is that propositional Horn formulas are polynomial time learnable in Angluin’s exact learning model with membership and equivalence queries. In this work, we push this boundary and show that the class of multivalued dependency formulas (MVDF), which non-trivially extends propositional Horn, is polynomially learnable from interpretations. We then provide a notion of reduction between learning problems in Angluin’s model, showing that a transformation of the algorithm suffices to efficiently learn multivalued database dependencies from data relations. We also show via reductions that our main result extends well known previous results and allows us to find alternative solutions for them.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114345655","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}
H. Fernau, F. Manea, Robert Mercas, Markus L. Schmid
A pattern ɑ (i.e., a string of variables and terminals) matches a word w, if w can be obtained by uniformly replacing the variables of ɑ by terminal words. The respective matching problem, i.e., deciding whether or not a given pattern matches a given word, is generally NP-complete, but can be solved in polynomial-time for restricted classes of patterns. We present efficient algorithms for the matching problem with respect to patterns with a bounded number of repeated variables and patterns with a structural restriction on the order of variables. Furthermore, we show that it is NP-complete to decide, for a given number k and a word w, whether w can be factorised into k distinct factors. As an immediate consequence of this hardness result, the injective version (i.e., different variables are replaced by different words) of the matching problem is NP-complete even for very restricted classes of patterns.
{"title":"Pattern Matching with Variables","authors":"H. Fernau, F. Manea, Robert Mercas, Markus L. Schmid","doi":"10.1145/3369935","DOIUrl":"https://doi.org/10.1145/3369935","url":null,"abstract":"A pattern ɑ (i.e., a string of variables and terminals) matches a word w, if w can be obtained by uniformly replacing the variables of ɑ by terminal words. The respective matching problem, i.e., deciding whether or not a given pattern matches a given word, is generally NP-complete, but can be solved in polynomial-time for restricted classes of patterns. We present efficient algorithms for the matching problem with respect to patterns with a bounded number of repeated variables and patterns with a structural restriction on the order of variables. Furthermore, we show that it is NP-complete to decide, for a given number k and a word w, whether w can be factorised into k distinct factors. As an immediate consequence of this hardness result, the injective version (i.e., different variables are replaced by different words) of the matching problem is NP-complete even for very restricted classes of patterns.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131895071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that over the field of complex numbers, every homogeneous polynomial of degree d can be approximated (in the border complexity sense) by a depth-3 arithmetic circuit of top fan-in at most 2. This is quite surprising, since there exist homogeneous polynomials P on n variables of degree 2, such that any depth-3 arithmetic circuit computing P must have top fan-in at least Ω (n). As an application, we get a new tradeoff between the top fan-in and formal degree in an approximate analog of the celebrated depth reduction result of Gupta, Kamath, Kayal, and Saptharishi [7, 10]. Formally, we show that if a degree d homogeneous polynomial P can be computed by an arithmetic circuit of size s ≥ d, then for every t ≤ d, P is in the border of a depth-3 circuit of top fan-in sO(t) and formal degree sO(d/t). To the best of our knowledge, the upper bound on the top fan-in in the original proof of Reference [7] is always at least sΩ (√d), regardless of the formal degree.
{"title":"On the Power of Border of Depth-3 Arithmetic Circuits","authors":"Mrinal Kumar","doi":"10.1145/3371506","DOIUrl":"https://doi.org/10.1145/3371506","url":null,"abstract":"We show that over the field of complex numbers, every homogeneous polynomial of degree d can be approximated (in the border complexity sense) by a depth-3 arithmetic circuit of top fan-in at most 2. This is quite surprising, since there exist homogeneous polynomials P on n variables of degree 2, such that any depth-3 arithmetic circuit computing P must have top fan-in at least Ω (n). As an application, we get a new tradeoff between the top fan-in and formal degree in an approximate analog of the celebrated depth reduction result of Gupta, Kamath, Kayal, and Saptharishi [7, 10]. Formally, we show that if a degree d homogeneous polynomial P can be computed by an arithmetic circuit of size s ≥ d, then for every t ≤ d, P is in the border of a depth-3 circuit of top fan-in sO(t) and formal degree sO(d/t). To the best of our knowledge, the upper bound on the top fan-in in the original proof of Reference [7] is always at least sΩ (√d), regardless of the formal degree.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126008373","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：