A red-blue graph is a graph where every edge is colored either red or blue. The exact perfect matching problem asks for a perfect matching in a red-blue graph that has exactly a given number of red edges. We show that for complete and bipartite complete graphs, the exact perfect matching problem is logspace equivalent to the perfect matching problem. Hence, an efficient parallel algorithm for perfect matching would carry over to the exact perfect matching problem for this class of graphs. We also report some progress in extending the result to arbitrary graphs.
{"title":"Exact Perfect Matching in Complete Graphs","authors":"R. Gurjar, A. Korwar, J. Messner, T. Thierauf","doi":"10.1145/3041402","DOIUrl":"https://doi.org/10.1145/3041402","url":null,"abstract":"A red-blue graph is a graph where every edge is colored either red or blue. The exact perfect matching problem asks for a perfect matching in a red-blue graph that has exactly a given number of red edges. We show that for complete and bipartite complete graphs, the exact perfect matching problem is logspace equivalent to the perfect matching problem. Hence, an efficient parallel algorithm for perfect matching would carry over to the exact perfect matching problem for this class of graphs. We also report some progress in extending the result to arbitrary graphs.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128061684","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}
Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with a (Q ∪ {∞ })-valued objective function given as a sum of fixed-arity functions. In Boolean surjective VCSPs, variables take on labels from D = {0,1}, and an optimal assignment is required to use both labels from D. Examples include the classical global Min-Cut problem in graphs and the Minimum Distance problem studied in coding theory. We establish a dichotomy theorem and thus give a complete complexity classification of Boolean surjective VCSPs with respect to exact solvability. Our work generalises the dichotomy for {0, ∞}-valued constraint languages (corresponding to surjective decision CSPs) obtained by Creignou and Hébrard. For the maximisation problem of Q≥0-valued surjective VCSPs, we also establish a dichotomy theorem with respect to approximability. Unlike in the case of Boolean surjective (decision) CSPs, there appears a novel tractable class of languages that is trivial in the non-surjective setting. This newly discovered tractable class has an interesting mathematical structure related to downsets and upsets. Our main contribution is identifying this class and proving that it lies on the borderline of tractability. A crucial part of our proof is a polynomial-time algorithm for enumerating all near-optimal solutions to a generalised Min-Cut problem, which might be of independent interest.
{"title":"The Complexity of Boolean Surjective General-Valued CSPs","authors":"Peter Fulla, Hannes Uppman, Stanislav Živný","doi":"10.1145/3282429","DOIUrl":"https://doi.org/10.1145/3282429","url":null,"abstract":"Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with a (Q ∪ {∞ })-valued objective function given as a sum of fixed-arity functions. In Boolean surjective VCSPs, variables take on labels from D = {0,1}, and an optimal assignment is required to use both labels from D. Examples include the classical global Min-Cut problem in graphs and the Minimum Distance problem studied in coding theory. We establish a dichotomy theorem and thus give a complete complexity classification of Boolean surjective VCSPs with respect to exact solvability. Our work generalises the dichotomy for {0, ∞}-valued constraint languages (corresponding to surjective decision CSPs) obtained by Creignou and Hébrard. For the maximisation problem of Q≥0-valued surjective VCSPs, we also establish a dichotomy theorem with respect to approximability. Unlike in the case of Boolean surjective (decision) CSPs, there appears a novel tractable class of languages that is trivial in the non-surjective setting. This newly discovered tractable class has an interesting mathematical structure related to downsets and upsets. Our main contribution is identifying this class and proving that it lies on the borderline of tractability. A crucial part of our proof is a polynomial-time algorithm for enumerating all near-optimal solutions to a generalised Min-Cut problem, which might be of independent interest.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129190455","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}
For a Boolean function f: {0, 1}n → {0, 1}, let fˆ be the unique multilinear polynomial such that f(x) = fˆ(x) holds for every x ˆ {0, 1}n. We show that, assuming VP ≠ VNP, there exists a polynomial-time computable f such that fˆ requires superpolynomial arithmetic circuits. In fact, this f can be taken as a monotone 2-CNF, or a product of affine functions. This holds over any field. To prove the results in characteristic 2, we design new VNP-complete families in this characteristic. This includes the polynomial ECn counting edge covers in a graph and the polynomial mcliquen counting cliques in a graph with deleted perfect matching. They both correspond to polynomial-time decidable problems, a phenomenon previously encountered only in characteristic ≠ 2.
{"title":"On Hardness of Multilinearization and VNP-Completeness in Characteristic 2","authors":"P. Hrubes","doi":"10.1145/2940323","DOIUrl":"https://doi.org/10.1145/2940323","url":null,"abstract":"For a Boolean function f: {0, 1}n → {0, 1}, let fˆ be the unique multilinear polynomial such that f(x) = fˆ(x) holds for every x ˆ {0, 1}n. We show that, assuming VP ≠ VNP, there exists a polynomial-time computable f such that fˆ requires superpolynomial arithmetic circuits. In fact, this f can be taken as a monotone 2-CNF, or a product of affine functions. This holds over any field. To prove the results in characteristic 2, we design new VNP-complete families in this characteristic. This includes the polynomial ECn counting edge covers in a graph and the polynomial mcliquen counting cliques in a graph with deleted perfect matching. They both correspond to polynomial-time decidable problems, a phenomenon previously encountered only in characteristic ≠ 2.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"264 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116687398","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}
Algorithmic fractal dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions of intersections and Cartesian products of fractals in Euclidean spaces. This approach shows that two prominent, fundamental results about the dimension of Borel or analytic sets also hold for arbitrary sets.
{"title":"Fractal Intersections and Products via Algorithmic Dimension","authors":"Neil Lutz","doi":"10.1145/3460948","DOIUrl":"https://doi.org/10.1145/3460948","url":null,"abstract":"Algorithmic fractal dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions of intersections and Cartesian products of fractals in Euclidean spaces. This approach shows that two prominent, fundamental results about the dimension of Borel or analytic sets also hold for arbitrary sets.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132931149","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}
P. Beame, Nathan Grosshans, P. McKenzie, L. Segoufin
A formulation of Nečiporuk’s lower bound method slightly more inclusive than the usual complexity-measure-specific formulation is presented. Using this general formulation, limitations to lower bounds achievable by the method are obtained for several computation models, such as branching programs and Boolean formulas having access to a sublinear number of nondeterministic bits. In particular, it is shown that any lower bound achievable by the method of Nečiporuk for the size of nondeterministic and parity branching programs is at most O(n3/2/logn).
{"title":"Nondeterminism and An Abstract Formulation of Nečiporuk’s Lower Bound Method","authors":"P. Beame, Nathan Grosshans, P. McKenzie, L. Segoufin","doi":"10.1145/3013516","DOIUrl":"https://doi.org/10.1145/3013516","url":null,"abstract":"A formulation of Nečiporuk’s lower bound method slightly more inclusive than the usual complexity-measure-specific formulation is presented. Using this general formulation, limitations to lower bounds achievable by the method are obtained for several computation models, such as branching programs and Boolean formulas having access to a sublinear number of nondeterministic bits. In particular, it is shown that any lower bound achievable by the method of Nečiporuk for the size of nondeterministic and parity branching programs is at most O(n3/2/logn).","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130534729","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 investigate the sensitivity complexity of hypergraph properties. We present a k-uniform hypergraph property with sensitivity complexity O(n(⌈k/3⌉) for any k≥3, where n is the number of vertices. Moreover, we can do better when k≡1 (mod 3) by presenting a k-uniform hypergraph property with sensitivity O(n⌈k/3⌉-1/2). This result disproves a conjecture of Babai, which conjectures that the sensitivity complexity of k-uniform hypergraph properties is at least Ω (nk/2). We also investigate the sensitivity complexity of other symmetric functions and show that for many classes of transitive Boolean functions the minimum achievable sensitivity complexity can be O(N1/3), where N is the number of variables.
{"title":"On the Sensitivity Complexity of k-Uniform Hypergraph Properties","authors":"Qian Li, Xiaoming Sun","doi":"10.1145/3448643","DOIUrl":"https://doi.org/10.1145/3448643","url":null,"abstract":"In this article, we investigate the sensitivity complexity of hypergraph properties. We present a k-uniform hypergraph property with sensitivity complexity O(n(⌈k/3⌉) for any k≥3, where n is the number of vertices. Moreover, we can do better when k≡1 (mod 3) by presenting a k-uniform hypergraph property with sensitivity O(n⌈k/3⌉-1/2). This result disproves a conjecture of Babai, which conjectures that the sensitivity complexity of k-uniform hypergraph properties is at least Ω (nk/2). We also investigate the sensitivity complexity of other symmetric functions and show that for many classes of transitive Boolean functions the minimum achievable sensitivity complexity can be O(N1/3), where N is the number of variables.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"135 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114663349","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}
Eric Blais, C. Canonne, T. Eden, Amit Levi, D. Ron
A function f:{ −1,1}n → { −1,1} is a k-junta if it depends on at most k of its variables. We consider the problem of tolerant testing of k-juntas, where the testing algorithm must accept any function that is ε-close to some k-junta and reject any function that is ε′-far from every k′-junta for some ε′ = O(ε) and k′ = O(k). Our first result is an algorithm that solves this problem with query complexity polynomial in k and 1/ε. This result is obtained via a new polynomial-time approximation algorithm for submodular function minimization (SFM) under large cardinality constraints, which holds even when only given an approximate oracle access to the function. Our second result considers the case where k′ = k. We show how to obtain a smooth tradeoff between the amount of tolerance and the query complexity in this setting. Specifically, we design an algorithm that, given ρ ∈ (0,1), accepts any function that is ε ρ/16-close to some k-junta and rejects any function that is ε-far from every k-junta. The query complexity of the algorithm is O (k log k/ε ρ (1-ρ)k. Finally, we show how to apply the second result to the problem of tolerant isomorphism testing between two unknown Boolean functions f and g. We give an algorithm for this problem whose query complexity only depends on the (unknown) smallest k such that either f or g is close to being a k-junta.
{"title":"Tolerant Junta Testing and the Connection to Submodular Optimization and Function Isomorphism","authors":"Eric Blais, C. Canonne, T. Eden, Amit Levi, D. Ron","doi":"10.1145/3337789","DOIUrl":"https://doi.org/10.1145/3337789","url":null,"abstract":"A function f:{ −1,1}n → { −1,1} is a k-junta if it depends on at most k of its variables. We consider the problem of tolerant testing of k-juntas, where the testing algorithm must accept any function that is ε-close to some k-junta and reject any function that is ε′-far from every k′-junta for some ε′ = O(ε) and k′ = O(k). Our first result is an algorithm that solves this problem with query complexity polynomial in k and 1/ε. This result is obtained via a new polynomial-time approximation algorithm for submodular function minimization (SFM) under large cardinality constraints, which holds even when only given an approximate oracle access to the function. Our second result considers the case where k′ = k. We show how to obtain a smooth tradeoff between the amount of tolerance and the query complexity in this setting. Specifically, we design an algorithm that, given ρ ∈ (0,1), accepts any function that is ε ρ/16-close to some k-junta and rejects any function that is ε-far from every k-junta. The query complexity of the algorithm is O (k log k/ε ρ (1-ρ)k. Finally, we show how to apply the second result to the problem of tolerant isomorphism testing between two unknown Boolean functions f and g. We give an algorithm for this problem whose query complexity only depends on the (unknown) smallest k such that either f or g is close to being a k-junta.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123572266","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}
Marthe Bonamy, Lukasz Kowalik, Michal Pilipczuk, Arkadiusz Socala, Marcin Wrochna
In the multicoloring problem, also known as (a:b)-coloring or b-fold coloring, we are given a graph G and a set of a colors, and the task is to assign a subset of b colors to each vertex of G so that adjacent vertices receive disjoint color subsets. This natural generalization of the classic coloring problem (the b=1 case) is equivalent to finding a homomorphism to the Kneser graph KGa,b and gives relaxations approaching the fractional chromatic number. We study the complexity of determining whether a graph has an (a:b)-coloring. Our main result is that this problem does not admit an algorithm with runtime f(b)ċ 2o(log b)ċ n for any computable f(b) unless the Exponential Time Hypothesis (ETH) fails. A (b+1)nċ poly(n)-time algorithm due to Nederlof [33] shows that this is tight. A direct corollary of our result is that the graph homomorphism problem does not admit a 2O(n+h) algorithm unless the ETH fails even if the target graph is required to be a Kneser graph. This refines the understanding given by the recent lower bound of Cygan et al. [9]. The crucial ingredient in our hardness reduction is the usage of detecting matrices of Lindström [28], which is a combinatorial tool that, to the best of our knowledge, has not yet been used for proving complexity lower bounds. As a side result, we prove that the runtime of the algorithms of Abasi et al. [1] and of Gabizon et al. [14] for the r-monomial detection problem are optimal under the ETH.
{"title":"Tight Lower Bounds for the Complexity of Multicoloring","authors":"Marthe Bonamy, Lukasz Kowalik, Michal Pilipczuk, Arkadiusz Socala, Marcin Wrochna","doi":"10.1145/3313906","DOIUrl":"https://doi.org/10.1145/3313906","url":null,"abstract":"In the multicoloring problem, also known as (a:b)-coloring or b-fold coloring, we are given a graph G and a set of a colors, and the task is to assign a subset of b colors to each vertex of G so that adjacent vertices receive disjoint color subsets. This natural generalization of the classic coloring problem (the b=1 case) is equivalent to finding a homomorphism to the Kneser graph KGa,b and gives relaxations approaching the fractional chromatic number. We study the complexity of determining whether a graph has an (a:b)-coloring. Our main result is that this problem does not admit an algorithm with runtime f(b)ċ 2o(log b)ċ n for any computable f(b) unless the Exponential Time Hypothesis (ETH) fails. A (b+1)nċ poly(n)-time algorithm due to Nederlof [33] shows that this is tight. A direct corollary of our result is that the graph homomorphism problem does not admit a 2O(n+h) algorithm unless the ETH fails even if the target graph is required to be a Kneser graph. This refines the understanding given by the recent lower bound of Cygan et al. [9]. The crucial ingredient in our hardness reduction is the usage of detecting matrices of Lindström [28], which is a combinatorial tool that, to the best of our knowledge, has not yet been used for proving complexity lower bounds. As a side result, we prove that the runtime of the algorithms of Abasi et al. [1] and of Gabizon et al. [14] for the r-monomial detection problem are optimal under the ETH.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124878891","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. Gál, Jing-Tang Jang, N. Limaye, M. Mahajan, Karteek Sreenivasaiah
SubsetSum is a well-known NP-complete problem: given t ∈ Z+ and a set S of m positive integers, output YES if and only if there is a subset S′⊆S such that the sum of all numbers in S′ equals t. The problem and its search and optimization versions are known to be solvable in pseudopolynomial time in general. We develop a one-pass deterministic streaming algorithm that uses space O(log t / ε) and decides if some subset of the input stream adds up to a value in the range {(1 ± ϵ)t}. Using this algorithm, we design space-efficient fully polynomial-time approximation schemes (FPTAS) solving the search and optimization versions of SubsetSum. Our algorithms run in O(1 / ϵ m2) time and O(1 / ϵ) space on unit-cost RAMs, where 1 + ϵ is the approximation factor. This implies constant space quadratic time FPTAS on unit-cost RAMs when ϵ is a constant. Previous FPTAS used space linear in m. In addition, we show that on certain inputs, when a solution is located within a short prefix of the input sequence, our algorithms may run in sublinear time. We apply our techniques to the problem of finding balanced separators, and we extend our results to some other variants of the more general knapsack problem. When the input numbers are encoded in unary, the decision version has been known to be in log space. We give streaming space lower and upper bounds for unary SubsetSum (USS). If the input length is N when the numbers are encoded in unary, we show that randomized s-pass streaming algorithms for exact SubsetSum need space Ω (√N/s) and give a simple deterministic two-pass streaming algorithm using O(√N log N) space. Finally, we formulate an encoding under which USS is monotone and show that the exact and approximate versions in this formulation have monotone O(log2t) depth Boolean circuits. We also show that any circuit using ε-approximator gates for SubsetSum under this encoding needs Ω(n/logn) gates to compute the disjointness function.
SubsetSum是一个著名的np完全问题:给定t∈Z+，有一个m个正整数的集合S，当且仅当存在一个子集S’≥t且S’中所有数的和等于t时，输出YES。该问题及其搜索优化版本已知在一般伪多项式时间内可解。我们开发了一种单遍确定性流算法，它使用空间O(log t / ε)，并决定输入流的某个子集是否加起来等于{(1±λ)t}范围内的值。利用该算法，我们设计了空间高效的全多项式时间近似方案(FPTAS)来解决SubsetSum的搜索和优化版本。我们的算法在单位成本ram上运行在O(1 /御柱m2)时间和O(1 /御柱)空间中，其中1 +御柱是近似因子。这意味着当λ为常数时，单位成本ram上的常数空间二次时间FPTAS。以前的FPTAS在m中使用空间线性。此外，我们表明，在某些输入上，当一个解位于输入序列的短前缀内时，我们的算法可能在亚线性时间内运行。我们将我们的技术应用于寻找平衡分隔符的问题，并将我们的结果扩展到更一般的背包问题的一些其他变体。当输入数字以一元编码时，决策版本已知在日志空间中。我们给出一元SubsetSum (USS)的流空间下界和上界。如果输入长度为N，当数字以一元编码时，我们证明了精确SubsetSum的随机s-pass流算法需要Ω(√N/s)空间，并给出了一个简单的确定性两通道流算法，使用O(√N log N)空间。最后，我们给出了USS为单调的编码，并证明了该编码的精确和近似版本具有单调的O(log2t)深度布尔电路。我们还证明，在这种编码下，任何对SubsetSum使用ε-近似门的电路都需要Ω(n/logn)门来计算不相交函数。
{"title":"Space-Efficient Approximations for Subset Sum","authors":"A. Gál, Jing-Tang Jang, N. Limaye, M. Mahajan, Karteek Sreenivasaiah","doi":"10.1145/2894843","DOIUrl":"https://doi.org/10.1145/2894843","url":null,"abstract":"SubsetSum is a well-known NP-complete problem: given t ∈ Z+ and a set S of m positive integers, output YES if and only if there is a subset S′⊆S such that the sum of all numbers in S′ equals t. The problem and its search and optimization versions are known to be solvable in pseudopolynomial time in general. We develop a one-pass deterministic streaming algorithm that uses space O(log t / ε) and decides if some subset of the input stream adds up to a value in the range {(1 ± ϵ)t}. Using this algorithm, we design space-efficient fully polynomial-time approximation schemes (FPTAS) solving the search and optimization versions of SubsetSum. Our algorithms run in O(1 / ϵ m2) time and O(1 / ϵ) space on unit-cost RAMs, where 1 + ϵ is the approximation factor. This implies constant space quadratic time FPTAS on unit-cost RAMs when ϵ is a constant. Previous FPTAS used space linear in m. In addition, we show that on certain inputs, when a solution is located within a short prefix of the input sequence, our algorithms may run in sublinear time. We apply our techniques to the problem of finding balanced separators, and we extend our results to some other variants of the more general knapsack problem. When the input numbers are encoded in unary, the decision version has been known to be in log space. We give streaming space lower and upper bounds for unary SubsetSum (USS). If the input length is N when the numbers are encoded in unary, we show that randomized s-pass streaming algorithms for exact SubsetSum need space Ω (√N/s) and give a simple deterministic two-pass streaming algorithm using O(√N log N) space. Finally, we formulate an encoding under which USS is monotone and show that the exact and approximate versions in this formulation have monotone O(log2t) depth Boolean circuits. We also show that any circuit using ε-approximator gates for SubsetSum under this encoding needs Ω(n/logn) gates to compute the disjointness function.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132289049","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 introduce a new information-theoretic measure, which we call Public Information Complexity (PIC), as a tool for the study of multi-party computation protocols, and of quantities such as their communication complexity, or the amount of randomness they require in the context of information-theoretic private computations. We are able to use this measure directly in the natural asynchronous message-passing peer-to-peer model and show a number of interesting properties and applications of our new notion: The Public Information Complexity is a lower bound on the Communication Complexity and an upper bound on the Information Complexity; the difference between the Public Information Complexity and the Information Complexity provides a lower bound on the amount of randomness used in a protocol; any communication protocol can be compressed to its Public Information Cost; and an explicit calculation of the zero-error Public Information Complexity of the k-party, n-bit Parity function, where a player outputs the bitwise parity of the inputs. The latter result also establishes that the amount of randomness needed by a private protocol that computes this function is Ω (n).
{"title":"Multi-Party Protocols, Information Complexity and Privacy","authors":"Iordanis Kerenidis, A. Rosén, Florent Urrutia","doi":"10.1145/3313230","DOIUrl":"https://doi.org/10.1145/3313230","url":null,"abstract":"We introduce a new information-theoretic measure, which we call Public Information Complexity (PIC), as a tool for the study of multi-party computation protocols, and of quantities such as their communication complexity, or the amount of randomness they require in the context of information-theoretic private computations. We are able to use this measure directly in the natural asynchronous message-passing peer-to-peer model and show a number of interesting properties and applications of our new notion: The Public Information Complexity is a lower bound on the Communication Complexity and an upper bound on the Information Complexity; the difference between the Public Information Complexity and the Information Complexity provides a lower bound on the amount of randomness used in a protocol; any communication protocol can be compressed to its Public Information Cost; and an explicit calculation of the zero-error Public Information Complexity of the k-party, n-bit Parity function, where a player outputs the bitwise parity of the inputs. The latter result also establishes that the amount of randomness needed by a private protocol that computes this function is Ω (n).","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134512081","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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