Christian Ikenmeyer, Balagopal Komarath, C. Lenzen, Vladimir Lysikov, A. Mokhov, Karteek Sreenivasaiah
The problem of constructing hazard-free Boolean circuits dates back to the 1940s and is an important problem in circuit design. Our main lower-bound result unconditionally shows the existence of functions whose circuit complexity is polynomially bounded while every hazard-free implementation is provably of exponential size. Previous lower bounds on the hazard-free complexity were only valid for depth 2 circuits. The same proof method yields that every subcubic implementation of Boolean matrix multiplication must have hazards. These results follow from a crucial structural insight: Hazard-free complexity is a natural generalization of monotone complexity to all (not necessarily monotone) Boolean functions. Thus, we can apply known monotone complexity lower bounds to find lower bounds on the hazard-free complexity. We also lift these methods from the monotone setting to prove exponential hazard-free complexity lower bounds for non-monotone functions. As our main upper-bound result we show how to efficiently convert a Boolean circuit into a bounded-bit hazard-free circuit with only a polynomially large blow-up in the number of gates. Previously, the best known method yielded exponentially large circuits in the worst case, so our algorithm gives an exponential improvement. As a side result we establish the NP-completeness of several hazard detection problems.
{"title":"On the complexity of hazard-free circuits","authors":"Christian Ikenmeyer, Balagopal Komarath, C. Lenzen, Vladimir Lysikov, A. Mokhov, Karteek Sreenivasaiah","doi":"10.1145/3188745.3188912","DOIUrl":"https://doi.org/10.1145/3188745.3188912","url":null,"abstract":"The problem of constructing hazard-free Boolean circuits dates back to the 1940s and is an important problem in circuit design. Our main lower-bound result unconditionally shows the existence of functions whose circuit complexity is polynomially bounded while every hazard-free implementation is provably of exponential size. Previous lower bounds on the hazard-free complexity were only valid for depth 2 circuits. The same proof method yields that every subcubic implementation of Boolean matrix multiplication must have hazards. These results follow from a crucial structural insight: Hazard-free complexity is a natural generalization of monotone complexity to all (not necessarily monotone) Boolean functions. Thus, we can apply known monotone complexity lower bounds to find lower bounds on the hazard-free complexity. We also lift these methods from the monotone setting to prove exponential hazard-free complexity lower bounds for non-monotone functions. As our main upper-bound result we show how to efficiently convert a Boolean circuit into a bounded-bit hazard-free circuit with only a polynomially large blow-up in the number of gates. Previously, the best known method yielded exponentially large circuits in the worst case, so our algorithm gives an exponential improvement. As a side result we establish the NP-completeness of several hazard detection problems.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83730360","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}
One of the simplest problems on directed graphs is that of identifying the set of vertices reachable from a designated source vertex. This problem can be solved easily sequentially by performing a graph search, but efficient parallel algorithms have eluded researchers for decades. For sparse high-diameter graphs in particular, there is no known work-efficient parallel algorithm with nontrivial parallelism. This amounts to one of the most fundamental open questions in parallel graph algorithms: Is there a parallel algorithm for digraph reachability with nearly linear work? This paper shows that the answer is yes. This paper presents a randomized parallel algorithm for digraph reachability and related problems with expected work Õ(m) and span Õ(n2/3), and hence parallelism Ω(m/n2/3) = Ω(n1/3), on any graph with n vertices and m arcs. This is the first parallel algorithm having both nearly linear work and strongly sublinear span, i.e., span Õ(n1−є) for any constant є>0. The algorithm can be extended to produce a directed spanning tree, determine whether the graph is acyclic, topologically sort the strongly connected components of the graph, or produce a directed ear decomposition, all with work Õ(m) and span Õ(n2/3). The main technical contribution is an efficient Monte Carlo algorithm that, through the addition of Õ(n) shortcuts, reduces the diameter of the graph to Õ(n2/3) with high probability. While both sequential and parallel algorithms are known with those combinatorial properties, even the sequential algorithms are not efficient, having sequential runtime Ω(mnΩ(1)). This paper presents a surprisingly simple sequential algorithm that achieves the stated diameter reduction and runs in Õ(m) time. Parallelizing that algorithm yields the main result, but doing so involves overcoming several other challenges.
{"title":"Nearly work-efficient parallel algorithm for digraph reachability","authors":"Jeremy T. Fineman","doi":"10.1145/3188745.3188926","DOIUrl":"https://doi.org/10.1145/3188745.3188926","url":null,"abstract":"One of the simplest problems on directed graphs is that of identifying the set of vertices reachable from a designated source vertex. This problem can be solved easily sequentially by performing a graph search, but efficient parallel algorithms have eluded researchers for decades. For sparse high-diameter graphs in particular, there is no known work-efficient parallel algorithm with nontrivial parallelism. This amounts to one of the most fundamental open questions in parallel graph algorithms: Is there a parallel algorithm for digraph reachability with nearly linear work? This paper shows that the answer is yes. This paper presents a randomized parallel algorithm for digraph reachability and related problems with expected work Õ(m) and span Õ(n2/3), and hence parallelism Ω(m/n2/3) = Ω(n1/3), on any graph with n vertices and m arcs. This is the first parallel algorithm having both nearly linear work and strongly sublinear span, i.e., span Õ(n1−є) for any constant є>0. The algorithm can be extended to produce a directed spanning tree, determine whether the graph is acyclic, topologically sort the strongly connected components of the graph, or produce a directed ear decomposition, all with work Õ(m) and span Õ(n2/3). The main technical contribution is an efficient Monte Carlo algorithm that, through the addition of Õ(n) shortcuts, reduces the diameter of the graph to Õ(n2/3) with high probability. While both sequential and parallel algorithms are known with those combinatorial properties, even the sequential algorithms are not efficient, having sequential runtime Ω(mnΩ(1)). This paper presents a surprisingly simple sequential algorithm that achieves the stated diameter reduction and runs in Õ(m) time. Parallelizing that algorithm yields the main result, but doing so involves overcoming several other challenges.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82798122","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 develop a systematic approach, based on convex programming and real analysis, for obtaining upper bounds on the capacity of the binary deletion channel and, more generally, channels with i.i.d. insertions and deletions. Other than the classical deletion channel, we give a special attention to the Poisson-repeat channel introduced by Mitzenmacher and Drinea (IEEE Transactions on Information Theory, 2006). Our framework can be applied to obtain capacity upper bounds for any repetition distribution (the deletion and Poisson-repeat channels corresponding to the special cases of Bernoulli and Poisson distributions). Our techniques essentially reduce the task of proving capacity upper bounds to maximizing a univariate, real-valued, and often concave function over a bounded interval. The corresponding univariate function is carefully designed according to the underlying distribution of repetitions and the choices vary depending on the desired strength of the upper bounds as well as the desired simplicity of the function (e.g., being only efficiently computable versus having an explicit closed-form expression in terms of elementary, or common special, functions). Among our results, we show that the capacity of the binary deletion channel with deletion probability d is at most (1−d) logϕ for d ≥ 1/2, and, assuming the capacity function is convex, is at most 1−d log(4/ϕ) for d<1/2, where ϕ=(1+√5)/2 is the golden ratio. This is the first nontrivial capacity upper bound for any value of d outside the limiting case d → 0 that is fully explicit and proved without computer assistance. Furthermore, we derive the first set of capacity upper bounds for the Poisson-repeat channel. Our results uncover further striking connections between this channel and the deletion channel, and suggest, somewhat counter-intuitively, that the Poisson-repeat channel is actually analytically simpler than the deletion channel and may be of key importance to a complete understanding of the deletion channel. Finally, we derive several novel upper bounds on the capacity of the deletion channel. All upper bounds are maximums of efficiently computable, and concave, univariate real functions over a bounded domain. In turn, we upper bound these functions in terms of explicit elementary and standard special functions, whose maximums can be found even more efficiently (and sometimes, analytically, for example for d=1/2). Along the way, we develop several new techniques of potentially independent interest. For example, we develop systematic techniques to study channels with mean constraints over the reals. Furthermore, we motivate the study of novel probability distributions over non-negative integers, as well as novel special functions which could be of interest to mathematical analysis.
我们开发了一种基于凸规划和实数分析的系统方法，用于获得二进制删除信道的容量上界，更一般地说，具有iid插入和删除的信道。除了经典的删除信道，我们特别关注由Mitzenmacher和Drinea (IEEE Transactions on Information Theory, 2006)引入的泊松重复信道。我们的框架可用于获得任何重复分布(与伯努利分布和泊松分布的特殊情况相对应的删除和泊松-重复通道)的容量上界。我们的技术从本质上减少了证明容量上界的任务，使单变量、实值和通常在有界区间内凹函数最大化。相应的单变量函数是根据重复的潜在分布精心设计的，选择取决于上界的期望强度以及函数的期望简单性(例如，仅可有效计算，而不是根据初等函数或普通特殊函数具有显式的封闭形式表达式)。在我们的研究结果中，我们表明，对于d≥1/2，删除概率为d的二进制删除通道的容量最多为(1−d) logϕ，并且，假设容量函数是凸的，对于d<1/2，最大为1−d log(4/ϕ)，其中ϕ=(1+√5)/2是黄金比例。这是在极限情况d→0之外的任何d值的非平凡容量上界的第一个完全显式且无需计算机辅助证明的。进一步，我们导出了泊松重复信道的第一组容量上界。我们的研究结果进一步揭示了该通道和删除通道之间的惊人联系，并表明，在某种程度上与直觉相反，泊松重复通道实际上在分析上比删除通道更简单，并且可能对完全理解删除通道至关重要。最后，我们给出了删除信道容量的几个新的上界。所有上界都是有效可计算的、凹的、单变量实函数在有界域上的最大值。反过来，我们用显式初等函数和标准特殊函数为这些函数上界，它们的最大值可以更有效地找到(有时，解析地，例如d=1/2)。在此过程中，我们开发了几种潜在的独立兴趣的新技术。例如，我们开发了系统的技术来研究具有对实数的平均约束的信道。此外，我们鼓励研究非负整数上的新概率分布，以及可能对数学分析感兴趣的新特殊函数。
{"title":"Capacity upper bounds for deletion-type channels","authors":"Mahdi Cheraghchi","doi":"10.1145/3188745.3188768","DOIUrl":"https://doi.org/10.1145/3188745.3188768","url":null,"abstract":"We develop a systematic approach, based on convex programming and real analysis, for obtaining upper bounds on the capacity of the binary deletion channel and, more generally, channels with i.i.d. insertions and deletions. Other than the classical deletion channel, we give a special attention to the Poisson-repeat channel introduced by Mitzenmacher and Drinea (IEEE Transactions on Information Theory, 2006). Our framework can be applied to obtain capacity upper bounds for any repetition distribution (the deletion and Poisson-repeat channels corresponding to the special cases of Bernoulli and Poisson distributions). Our techniques essentially reduce the task of proving capacity upper bounds to maximizing a univariate, real-valued, and often concave function over a bounded interval. The corresponding univariate function is carefully designed according to the underlying distribution of repetitions and the choices vary depending on the desired strength of the upper bounds as well as the desired simplicity of the function (e.g., being only efficiently computable versus having an explicit closed-form expression in terms of elementary, or common special, functions). Among our results, we show that the capacity of the binary deletion channel with deletion probability d is at most (1−d) logϕ for d ≥ 1/2, and, assuming the capacity function is convex, is at most 1−d log(4/ϕ) for d<1/2, where ϕ=(1+√5)/2 is the golden ratio. This is the first nontrivial capacity upper bound for any value of d outside the limiting case d → 0 that is fully explicit and proved without computer assistance. Furthermore, we derive the first set of capacity upper bounds for the Poisson-repeat channel. Our results uncover further striking connections between this channel and the deletion channel, and suggest, somewhat counter-intuitively, that the Poisson-repeat channel is actually analytically simpler than the deletion channel and may be of key importance to a complete understanding of the deletion channel. Finally, we derive several novel upper bounds on the capacity of the deletion channel. All upper bounds are maximums of efficiently computable, and concave, univariate real functions over a bounded domain. In turn, we upper bound these functions in terms of explicit elementary and standard special functions, whose maximums can be found even more efficiently (and sometimes, analytically, for example for d=1/2). Along the way, we develop several new techniques of potentially independent interest. For example, we develop systematic techniques to study channels with mean constraints over the reals. Furthermore, we motivate the study of novel probability distributions over non-negative integers, as well as novel special functions which could be of interest to mathematical analysis.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"69 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84894013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we present a new iterative rounding framework for many clustering problems. Using this, we obtain an (α1 + є ≤ 7.081 + є)-approximation algorithm for k-median with outliers, greatly improving upon the large implicit constant approximation ratio of Chen. For k-means with outliers, we give an (α2+є ≤ 53.002 + є)-approximation, which is the first O(1)-approximation for this problem. The iterative algorithm framework is very versatile; we show how it can be used to give α1- and (α1 + є)-approximation algorithms for matroid and knapsack median problems respectively, improving upon the previous best approximations ratios of 8 due to Swamy and 17.46 due to Byrka et al. The natural LP relaxation for the k-median/k-means with outliers problem has an unbounded integrality gap. In spite of this negative result, our iterative rounding framework shows that we can round an LP solution to an almost-integral solution of small cost, in which we have at most two fractionally open facilities. Thus, the LP integrality gap arises due to the gap between almost-integral and fully-integral solutions. Then, using a pre-processing procedure, we show how to convert an almost-integral solution to a fully-integral solution losing only a constant-factor in the approximation ratio. By further using a sparsification technique, the additive factor loss incurred by the conversion can be reduced to any є > 0.
{"title":"Constant approximation for k-median and k-means with outliers via iterative rounding","authors":"Ravishankar Krishnaswamy, Shi Li, Sai Sandeep","doi":"10.1145/3188745.3188882","DOIUrl":"https://doi.org/10.1145/3188745.3188882","url":null,"abstract":"In this paper, we present a new iterative rounding framework for many clustering problems. Using this, we obtain an (α1 + є ≤ 7.081 + є)-approximation algorithm for k-median with outliers, greatly improving upon the large implicit constant approximation ratio of Chen. For k-means with outliers, we give an (α2+є ≤ 53.002 + є)-approximation, which is the first O(1)-approximation for this problem. The iterative algorithm framework is very versatile; we show how it can be used to give α1- and (α1 + є)-approximation algorithms for matroid and knapsack median problems respectively, improving upon the previous best approximations ratios of 8 due to Swamy and 17.46 due to Byrka et al. The natural LP relaxation for the k-median/k-means with outliers problem has an unbounded integrality gap. In spite of this negative result, our iterative rounding framework shows that we can round an LP solution to an almost-integral solution of small cost, in which we have at most two fractionally open facilities. Thus, the LP integrality gap arises due to the gap between almost-integral and fully-integral solutions. Then, using a pre-processing procedure, we show how to convert an almost-integral solution to a fully-integral solution losing only a constant-factor in the approximation ratio. By further using a sparsification technique, the additive factor loss incurred by the conversion can be reduced to any є > 0.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"240 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75578578","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}
Sébastien Bubeck, Michael B. Cohen, Y. Lee, Yuanzhi Li
We consider the problem of linear regression where the ℓ2n norm loss (i.e., the usual least squares loss) is replaced by the ℓpn norm. We show how to solve such problems up to machine precision in Õp(n|1/2 − 1/p|) (dense) matrix-vector products and Õp(1) matrix inversions, or alternatively in Õp(n|1/2 − 1/p|) calls to a (sparse) linear system solver. This improves the state of the art for any p∉{1,2,+∞}. Furthermore we also propose a randomized algorithm solving such problems in input sparsity time, i.e., Õp(N + poly(d)) where N is the size of the input and d is the number of variables. Such a result was only known for p=2. Finally we prove that these results lie outside the scope of the Nesterov-Nemirovski’s theory of interior point methods by showing that any symmetric self-concordant barrier on the ℓpn unit ball has self-concordance parameter Ω(n).
{"title":"An homotopy method for lp regression provably beyond self-concordance and in input-sparsity time","authors":"Sébastien Bubeck, Michael B. Cohen, Y. Lee, Yuanzhi Li","doi":"10.1145/3188745.3188776","DOIUrl":"https://doi.org/10.1145/3188745.3188776","url":null,"abstract":"We consider the problem of linear regression where the ℓ2n norm loss (i.e., the usual least squares loss) is replaced by the ℓpn norm. We show how to solve such problems up to machine precision in Õp(n|1/2 − 1/p|) (dense) matrix-vector products and Õp(1) matrix inversions, or alternatively in Õp(n|1/2 − 1/p|) calls to a (sparse) linear system solver. This improves the state of the art for any p∉{1,2,+∞}. Furthermore we also propose a randomized algorithm solving such problems in input sparsity time, i.e., Õp(N + poly(d)) where N is the size of the input and d is the number of variables. Such a result was only known for p=2. Finally we prove that these results lie outside the scope of the Nesterov-Nemirovski’s theory of interior point methods by showing that any symmetric self-concordant barrier on the ℓpn unit ball has self-concordance parameter Ω(n).","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86017235","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}
Sébastien Bubeck, Michael B. Cohen, James R. Lee, Y. Lee, A. Madry
We present an O((logk)2)-competitive randomized algorithm for the k-server problem on hierarchically separated trees (HSTs). This is the first o(k)-competitive randomized algorithm for which the competitive ratio is independent of the size of the underlying HST. Our algorithm is designed in the framework of online mirror descent where the mirror map is a multiscale entropy. When combined with Bartal’s static HST embedding reduction, this leads to an O((logk)2 logn)-competitive algorithm on any n-point metric space. We give a new dynamic HST embedding that yields an O((logk)3 logΔ)-competitive algorithm on any metric space where the ratio of the largest to smallest non-zero distance is at most Δ.
{"title":"k-server via multiscale entropic regularization","authors":"Sébastien Bubeck, Michael B. Cohen, James R. Lee, Y. Lee, A. Madry","doi":"10.1145/3188745.3188798","DOIUrl":"https://doi.org/10.1145/3188745.3188798","url":null,"abstract":"We present an O((logk)2)-competitive randomized algorithm for the k-server problem on hierarchically separated trees (HSTs). This is the first o(k)-competitive randomized algorithm for which the competitive ratio is independent of the size of the underlying HST. Our algorithm is designed in the framework of online mirror descent where the mirror map is a multiscale entropy. When combined with Bartal’s static HST embedding reduction, this leads to an O((logk)2 logn)-competitive algorithm on any n-point metric space. We give a new dynamic HST embedding that yields an O((logk)3 logΔ)-competitive algorithm on any metric space where the ratio of the largest to smallest non-zero distance is at most Δ.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"80 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90685912","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 the problem of *shadow tomography*: given an unknown D-dimensional quantum mixed state ρ, as well as known two-outcome measurements E1,…,EM, estimate the probability that Ei accepts ρ, to within additive error ε, for each of the M measurements. How many copies of ρ are needed to achieve this, with high probability? Surprisingly, we give a procedure that solves the problem by measuring only O( ε−5·log4 M·logD) copies. This means, for example, that we can learn the behavior of an arbitrary n-qubit state, on *all* accepting/rejecting circuits of some fixed polynomial size, by measuring only nO( 1) copies of the state. This resolves an open problem of the author, which arose from his work on private-key quantum money schemes, but which also has applications to quantum copy-protected software, quantum advice, and quantum one-way communication. Recently, building on this work, Brandão et al. have given a different approach to shadow tomography using semidefinite programming, which achieves a savings in computation time.
{"title":"Shadow tomography of quantum states","authors":"S. Aaronson","doi":"10.1145/3188745.3188802","DOIUrl":"https://doi.org/10.1145/3188745.3188802","url":null,"abstract":"We introduce the problem of *shadow tomography*: given an unknown D-dimensional quantum mixed state ρ, as well as known two-outcome measurements E1,…,EM, estimate the probability that Ei accepts ρ, to within additive error ε, for each of the M measurements. How many copies of ρ are needed to achieve this, with high probability? Surprisingly, we give a procedure that solves the problem by measuring only O( ε−5·log4 M·logD) copies. This means, for example, that we can learn the behavior of an arbitrary n-qubit state, on *all* accepting/rejecting circuits of some fixed polynomial size, by measuring only nO( 1) copies of the state. This resolves an open problem of the author, which arose from his work on private-key quantum money schemes, but which also has applications to quantum copy-protected software, quantum advice, and quantum one-way communication. Recently, building on this work, Brandão et al. have given a different approach to shadow tomography using semidefinite programming, which achieves a savings in computation time.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"491 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77808562","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}
Stable matching is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal paper by Gale and Shapley. In this paper, we provide a new upper bound on f(n), the maximum number of stable matchings that a stable matching instance with n men and n women can have. It has been a long-standing open problem to understand the asymptotic behavior of f(n) as n→∞, first posed by Donald Knuth in the 1970s. Until now the best lower bound was approximately 2.28n, and the best upper bound was 2nlogn− O(n). In this paper, we show that for all n, f(n) ≤ cn for some universal constant c. This matches the lower bound up to the base of the exponent. Our proof is based on a reduction to counting the number of downsets of a family of posets that we call “mixing”. The latter might be of independent interest.
{"title":"A simply exponential upper bound on the maximum number of stable matchings","authors":"Anna R. Karlin, S. Gharan, Robbie Weber","doi":"10.1145/3188745.3188848","DOIUrl":"https://doi.org/10.1145/3188745.3188848","url":null,"abstract":"Stable matching is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal paper by Gale and Shapley. In this paper, we provide a new upper bound on f(n), the maximum number of stable matchings that a stable matching instance with n men and n women can have. It has been a long-standing open problem to understand the asymptotic behavior of f(n) as n→∞, first posed by Donald Knuth in the 1970s. Until now the best lower bound was approximately 2.28n, and the best upper bound was 2nlogn− O(n). In this paper, we show that for all n, f(n) ≤ cn for some universal constant c. This matches the lower bound up to the base of the exponent. Our proof is based on a reduction to counting the number of downsets of a family of posets that we call “mixing”. The latter might be of independent interest.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"111 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85815321","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}
Ivona Bezáková, Andreas Galanis, L. A. Goldberg, Daniel Stefankovic
We study the complexity of approximating the value of the independent set polynomial ZG(λ) of a graph G with maximum degree Δ when the activity λ is a complex number. When λ is real, the complexity picture is well-understood, and is captured by two real-valued thresholds λ* and λc, which depend on Δ and satisfy 0<λ*<λc. It is known that if λ is a real number in the interval (−λ*,λc) then there is an FPTAS for approximating ZG(λ) on graphs G with maximum degree at most Δ. On the other hand, if λ is a real number outside of the (closed) interval, then approximation is NP-hard. The key to establishing this picture was the interpretation of the thresholds λ* and λc on the Δ-regular tree. The ”occupation ratio” of a Δ-regular tree T is the contribution to ZT(λ) from independent sets containing the root of the tree, divided by ZT(λ) itself. This occupation ratio converges to a limit, as the height of the tree grows, if and only if λ∈ [−λ*,λc]. Unsurprisingly, the case where λ is complex is more challenging. It is known that there is an FPTAS when λ is a complex number with norm at most λ* and also when λ is in a small strip surrounding the real interval [0,λc). However, neither of these results is believed to fully capture the truth about when approximation is possible. Peters and Regts identified the values of λ for which the occupation ratio of the Δ-regular tree converges. These values carve a cardioid-shaped region ΛΔ in the complex plane, whose boundary includes the critical points −λ* and λc. Motivated by the picture in the real case, they asked whether ΛΔ marks the true approximability threshold for general complex values λ. Our main result shows that for every λ outside of ΛΔ, the problem of approximating ZG(λ) on graphs G with maximum degree at most Δ is indeed NP-hard. In fact, when λ is outside of ΛΔ and is not a positive real number, we give the stronger result that approximating ZG(λ) is actually #P-hard. Further, on the negative real axis, when λ<−λ*, we show that it is #P-hard to even decide whether ZG(λ)>0, resolving in the affirmative a conjecture of Harvey, Srivastava and Vondrak. Our proof techniques are based around tools from complex analysis — specifically the study of iterative multivariate rational maps.
{"title":"Inapproximability of the independent set polynomial in the complex plane","authors":"Ivona Bezáková, Andreas Galanis, L. A. Goldberg, Daniel Stefankovic","doi":"10.1145/3188745.3188788","DOIUrl":"https://doi.org/10.1145/3188745.3188788","url":null,"abstract":"We study the complexity of approximating the value of the independent set polynomial ZG(λ) of a graph G with maximum degree Δ when the activity λ is a complex number. When λ is real, the complexity picture is well-understood, and is captured by two real-valued thresholds λ* and λc, which depend on Δ and satisfy 0<λ*<λc. It is known that if λ is a real number in the interval (−λ*,λc) then there is an FPTAS for approximating ZG(λ) on graphs G with maximum degree at most Δ. On the other hand, if λ is a real number outside of the (closed) interval, then approximation is NP-hard. The key to establishing this picture was the interpretation of the thresholds λ* and λc on the Δ-regular tree. The ”occupation ratio” of a Δ-regular tree T is the contribution to ZT(λ) from independent sets containing the root of the tree, divided by ZT(λ) itself. This occupation ratio converges to a limit, as the height of the tree grows, if and only if λ∈ [−λ*,λc]. Unsurprisingly, the case where λ is complex is more challenging. It is known that there is an FPTAS when λ is a complex number with norm at most λ* and also when λ is in a small strip surrounding the real interval [0,λc). However, neither of these results is believed to fully capture the truth about when approximation is possible. Peters and Regts identified the values of λ for which the occupation ratio of the Δ-regular tree converges. These values carve a cardioid-shaped region ΛΔ in the complex plane, whose boundary includes the critical points −λ* and λc. Motivated by the picture in the real case, they asked whether ΛΔ marks the true approximability threshold for general complex values λ. Our main result shows that for every λ outside of ΛΔ, the problem of approximating ZG(λ) on graphs G with maximum degree at most Δ is indeed NP-hard. In fact, when λ is outside of ΛΔ and is not a positive real number, we give the stronger result that approximating ZG(λ) is actually #P-hard. Further, on the negative real axis, when λ<−λ*, we show that it is #P-hard to even decide whether ZG(λ)>0, resolving in the affirmative a conjecture of Harvey, Srivastava and Vondrak. Our proof techniques are based around tools from complex analysis — specifically the study of iterative multivariate rational maps.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84796726","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 well-known fact in the field of lossless text compression is that high-order entropy is a weak model when the input contains long repetitions. Motivated by this fact, decades of research have generated myriads of so-called dictionary compressors: algorithms able to reduce the text’s size by exploiting its repetitiveness. Lempel-Ziv 77 is one of the most successful and well-known tools of this kind, followed by straight-line programs, run-length Burrows-Wheeler transform, macro schemes, collage systems, and the compact directed acyclic word graph. In this paper, we show that these techniques are different solutions to the same, elegant, combinatorial problem: to find a small set of positions capturing all distinct text’s substrings. We call such a set a string attractor. We first show reductions between dictionary compressors and string attractors. This gives the approximation ratios of dictionary compressors with respect to the smallest string attractor and allows us to uncover new asymptotic relations between the output sizes of different dictionary compressors. We then show that the k-attractor problem — deciding whether a text has a size-t set of positions capturing all substrings of length at most k — is NP-complete for k≥ 3. This, in particular, includes the full string attractor problem. We provide several approximation techniques for the smallest k-attractor, show that the problem is APX-complete for constant k, and give strong inapproximability results. To conclude, we provide matching lower and upper bounds for the random access problem on string attractors. The upper bound is proved by showing a data structure supporting queries in optimal time. Our data structure is universal: by our reductions to string attractors, it supports random access on any dictionary-compression scheme. In particular, it matches the lower bound also on LZ77, straight-line programs, collage systems, and macro schemes, and therefore essentially closes (at once) the random access problem for all these compressors.
{"title":"At the roots of dictionary compression: string attractors","authors":"Dominik Kempa, N. Prezza","doi":"10.1145/3188745.3188814","DOIUrl":"https://doi.org/10.1145/3188745.3188814","url":null,"abstract":"A well-known fact in the field of lossless text compression is that high-order entropy is a weak model when the input contains long repetitions. Motivated by this fact, decades of research have generated myriads of so-called dictionary compressors: algorithms able to reduce the text’s size by exploiting its repetitiveness. Lempel-Ziv 77 is one of the most successful and well-known tools of this kind, followed by straight-line programs, run-length Burrows-Wheeler transform, macro schemes, collage systems, and the compact directed acyclic word graph. In this paper, we show that these techniques are different solutions to the same, elegant, combinatorial problem: to find a small set of positions capturing all distinct text’s substrings. We call such a set a string attractor. We first show reductions between dictionary compressors and string attractors. This gives the approximation ratios of dictionary compressors with respect to the smallest string attractor and allows us to uncover new asymptotic relations between the output sizes of different dictionary compressors. We then show that the k-attractor problem — deciding whether a text has a size-t set of positions capturing all substrings of length at most k — is NP-complete for k≥ 3. This, in particular, includes the full string attractor problem. We provide several approximation techniques for the smallest k-attractor, show that the problem is APX-complete for constant k, and give strong inapproximability results. To conclude, we provide matching lower and upper bounds for the random access problem on string attractors. The upper bound is proved by showing a data structure supporting queries in optimal time. Our data structure is universal: by our reductions to string attractors, it supports random access on any dictionary-compression scheme. In particular, it matches the lower bound also on LZ77, straight-line programs, collage systems, and macro schemes, and therefore essentially closes (at once) the random access problem for all these compressors.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83927484","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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