Proceedings of the forty-sixth annual ACM symposium on Theory of computing最新文献

{"title":"Exponential improvement in precision for simulating sparse Hamiltonians","authors":"D. Berry, Andrew M. Childs, R. Cleve, Robin Kothari, R. Somma","doi":"10.1145/2591796.2591854","DOIUrl":"https://doi.org/10.1145/2591796.2591854","url":null,"abstract":"We provide a quantum algorithm for simulating the dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Specifically, we show that a d-sparse Hamiltonian H on n qubits can be simulated for time t with precision ε using O(τlog(τ/ε)/log log(τ/ε)) queries and O(τnlog2(τ/ε)/log log(τ/ε)) additional 2-qubit gates, where τ=d2||H||maxt. Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for time-varying Hamiltonians, the gate complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our algorithm is based on a significantly improved simulation of the continuous- and fractional-query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error. We also significantly simplify the analysis of this conversion, avoiding the need for a complex fault correction procedure. Our simplification relies on a new form of \"oblivious amplitude amplification\" that can be applied even though the reflection about the input state is unavailable. Finally, we prove new lower bounds showing that our algorithms are optimal as a function of the error.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134064593","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":"Formulas vs. circuits for small distance connectivity","authors":"Benjamin Rossman","doi":"10.1145/2591796.2591828","DOIUrl":"https://doi.org/10.1145/2591796.2591828","url":null,"abstract":"We give the first super-polynomial separation in the power of bounded-depth boolean formulas vs. circuits. Specifically, we consider the problem Distance k(n) Connectivity, which asks whether two specified nodes in a graph of size n are connected by a path of length at most k(n). This problem is solvable (by the recursive doubling technique) on circuits of depth O(log k) and size O(kn3). In contrast, we show that solving this problem on formulas of depth log n/(log log n)O(1) requires size nΩ(log k) for all k(n) ≤ log log n. As corollaries: (i) It follows that polynomial-size circuits for Distance k(n) Connectivity require depth Ω(log k) for all k(n) ≤ log log n. This matches the upper bound from recursive doubling and improves a previous Ω(log log k) lower bound of Beame, Impagliazzo and Pitassi [BIP98]. (ii) We get a tight lower bound of sΩ(d) on the size required to simulate size-s depth-d circuits by depth-d formulas for all s(n) = nO(1) and d(n) ≤ log log log n. No lower bound better than sΩ(1) was previously known for any d(n) ≮ O(1). Our proof technique is centered on a new notion of pathset complexity, which roughly speaking measures the minimum cost of constructing a set of (partial) paths in a universe of size n via the operations of union and relational join, subject to certain density constraints. Half of our proof shows that bounded-depth formulas solving Distance k(n) Connectivity imply upper bounds on pathset complexity. The other half is a combinatorial lower bound on pathset complexity.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124880733","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":"Super-polylogarithmic hypergraph coloring hardness via low-degree long codes","authors":"V. Guruswami, P. Harsha, J. Håstad, S. Srinivasan, G. Varma","doi":"10.1145/2591796.2591882","DOIUrl":"https://doi.org/10.1145/2591796.2591882","url":null,"abstract":"We prove improved inapproximability results for hypergraph coloring using the low-degree polynomial code (aka, the\"short code\" of Barak et. al. [FOCS 2012]) and the techniques proposed by Dinur and Guruswami [FOCS 2013] to incorporate this code for inapproximability results. In particular, we prove quasi-NP-hardness of the following problems on n-vertex hypergraphs: •Coloring a 2-colorable 8-uniform hypergraph with [EQUATION] colors. •Coloring a 4-colorable 4-uniform hypergraph with [EQUATION] colors. •Coloring a 3-colorable 3-uniform hypergraph with [EQUATION] colors. In each of these cases, the hardness results obtained are (at least) superpolynomially stronger (if not exponentially stronger as in the third case) than what was previously known for the respective cases. In fact, prior to this result, (log n)O(1) colors was the strongest quantitative bound on the number of colors ruled out by inapproximability results for O(1)-colorable hypergraphs. The fundamental bottleneck in obtaining coloring inapproximability results using the low-degree long code was a multipartite structural restriction in the PCP construction of Dinur-Guruswami. We are able to get around this restriction by simulating the multipartite structure implicitly by querying just one partition (albeit requiring 8 queries), which yields our result for 2-colorable 8-uniform hypergraphs. The result for 4-colorable 4-uniform hypergraphs is obtained via a \"query doubling\" method exploiting additional properties of the 8-query test. For 3-colorable 3-uniform hypergraphs, we exploit the ternary domain to design a test with an additive (as opposed to multiplicative) noise function, and analyze its efficacy in killing high weight Fourier coefficients via the pseudorandom properties of an associated quadratic form. The latter step involves extending the key algebraic ingredient of Dinur-Guruswami concerning testing binary Reed-Muller codes to the ternary alphabet.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125625749","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":"Efficient deterministic approximate counting for low-degree polynomial threshold functions","authors":"Anindya De, R. Servedio","doi":"10.1145/2591796.2591800","DOIUrl":"https://doi.org/10.1145/2591796.2591800","url":null,"abstract":"We give a deterministic algorithm for approximately counting satisfying assignments of a degree-d polynomial threshold function (PTF). Given a degree-d input polynomial p(x) over Rn and a parameter ε > 0, our algorithm approximates Pr [EQUATION] to within an additive ±ε in time Od,ε(1) · poly(nd). (Since it is NP-hard to determine whether the above probability is nonzero, any sort of efficient multiplicative approximation is almost certainly impossible even for randomized algorithms.) Note that the running time of our algorithm (as a function of nd, the number of coefficients of a degree-d PTF) is a fixed polynomial. The fastest previous algorithm for this problem [Kan12b], based on constructions of unconditional pseudorandom generators for degree-d PTFs, runs in time [EQUATION] for all c > 0. The key novel technical contributions of this work are • A new multivariate central limit theorem, proved using tools from Malliavin calculus and Stein's Method. This new CLT shows that any collection of Gaussian polynomials with small eigenvalues must have a joint distribution which is very close to a multidimensional Gaussian distribution. • A new decomposition of low-degree multilinear polynomials over Gaussian inputs. Roughly speaking we show that (up to some small error) any such polynomial can be decomposed into a bounded number of multilinear polynomials all of which have extremely small eigenvalues. We use these new ingredients to give a deterministic algorithm for a Gaussian-space version of the approximate counting problem, and then employ standard techniques for working with low-degree PTFs (invariance principles and regularity lemmas) to reduce the original approximate counting problem over the Boolean hypercube to the Gaussian version. As an application of our result, we give the first deterministic fixed-parameter tractable algorithm for the following moment approximation problem: given a degree-d polynomial p(x1,..., xn) over {--1, 1}n, a positive integer k and an error parameter ε, output a (1±ε)-multiplicatively accurate estimate to [EQUATION]. Our algorithm runs in time Od,ε,k(1) · poly(nd).","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"368 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124620967","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":"The limits of depth reduction for arithmetic formulas: it's all about the top fan-in","authors":"Mrinal Kumar, Shubhangi Saraf","doi":"10.1145/2591796.2591827","DOIUrl":"https://doi.org/10.1145/2591796.2591827","url":null,"abstract":"In recent years, a very exciting and promising method for proving lower bounds for arithmetic circuits has been proposed. This method combines the method of depth reduction developed in the works of Agrawal and Vinay[1], Koiran [11] and Tavenas [16], and the use of the shifted partial derivative complexity measure developed in the works of Kayal [9] and Gupta et al [5]. These results inspired a flurry of other beautiful results and strong lower bounds for various classes of arithmetic circuits, in particular a recent work of Kayal et al [10] showing superpolynomial lower bounds for regular arithmetic formulas via an improved depth reduction for these formulas. It was left as an intriguing question if these methods could prove superpolynomial lower bounds for general (homogeneous) arithmetic formulas, and if so this would indeed be a breakthrough in arithmetic circuit complexity. In this paper we study the power and limitations of depth reduction and shifted partial derivatives for arithmetic formulas. We do it via studying the class of depth 4 homogeneous arithmetic circuits. We show: (1) the first superpolynomial lower bounds for the class of homogeneous depth 4 circuits with top fan-in o(log n). The core of our result is to show improved depth reduction for these circuits. This class of circuits has received much attention for the problem of polynomial identity testing. We give the first nontrivial lower bounds for these circuits for any top fan-in ≥ 2. (2) We show that improved depth reduction is not possible when the top fan-in is Ω(log n). In particular this shows that the depth reduction procedure of Koiran and Tavenas [11, 16] cannot be improved even for homogeneous formulas, thus strengthening the results of Fournier et al [3] who showed that depth reduction is tight for circuits, and answering some of the main open questions of [10, 3]. Our results in particular suggest that the method of improved depth reduction and shifted partial derivatives may not be powerful enough to prove superpolynomial lower bounds for (even homogeneous) arithmetic formulas.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115743172","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":"Approximation algorithms for regret-bounded vehicle routing and applications to distance-constrained vehicle routing","authors":"Zachary Friggstad, Chaitanya Swamy","doi":"10.1145/2591796.2591840","DOIUrl":"https://doi.org/10.1145/2591796.2591840","url":null,"abstract":"We consider vehicle-routing problems (VRPs) that incorporate the notion of regret of a client, which is a measure of the waiting time of a client relative to its shortest-path distance from the depot. Formally, we consider both the additive and multiplicative versions of, what we call, the regret-bounded vehicle routing problem (RVRP). In these problems, we are given an undirected complete graph G = ({r} ∪ V,E) on n nodes with a distinguished root (depot) node r, edge costs {cuv} that form a metric, and a regret bound R. Given a path P rooted at r and a node v ∈ P, let cP(v) be the distance from r to v along P. The goal is to find the fewest number of paths rooted at r that cover all the nodes so that for every node v covered by (say) path P: (i) its additive regret cP(v) -- crv, with respect to P is at most R in additive-RVRP; or (ii) its multiplicative regret, cP(v)/crv, with respect to P is at most R in multiplicative-RVRP. Our main result is the first constant-factor approximation algorithm for additive-RVRP. This is a substantial improvement over the previous-best O(log n)-approximation. Additive-RVRP turns out be a rather central vehicle-routing problem, whose study reveals insights into a variety of other regret-related problems as well as the classical distance-constrained VRP (DVRP), enabling us to obtain guarantees for these various problems by leveraging our algorithm for additive-RVRP and the underlying techniques. We obtain approximation ratios of [EQUATION] for multiplicative-RVRP, and [EQUATION] for DVRP with distance bound D via reductions to additive-RVRP; the latter improves upon the previous-best approximation for DVRP. A noteworthy aspect of our results is that they are obtained by devising rounding techniques for a natural configuration-style LP. This furthers our understanding of LP-relaxations for VRPs and enriches the toolkit of techniques that have been utilized for configuration LPs.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115687757","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":"Economic efficiency requires interaction","authors":"Shahar Dobzinski, N. Nisan, Sigal Oren","doi":"10.1145/2591796.2591815","DOIUrl":"https://doi.org/10.1145/2591796.2591815","url":null,"abstract":"We study the necessity of interaction between individuals for obtaining approximately efficient economic allocations. We view this as a formalization of Hayek's classic point of view that focuses on the information transfer advantages that markets have relative to centralized planning. We study two settings: combinatorial auctions with unit demand bidders (bipartite matching) and combinatorial auctions with subadditive bidders. In both settings we prove that non-interactive protocols require exponentially larger communication costs than interactive ones, even those that only use a modest amount of interaction.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131915373","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":"Deciding first-order properties of nowhere dense graphs","authors":"Martin Grohe, S. Kreutzer, S. Siebertz","doi":"10.1145/2591796.2591851","DOIUrl":"https://doi.org/10.1145/2591796.2591851","url":null,"abstract":"Nowhere dense graph classes, introduced by Nešetřil and Ossona de Mendez [30], form a large variety of classes of \"sparse graphs\" including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion. We show that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes. At least for graph classes closed under taking subgraphs, this result is optimal: it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, then C must be nowhere dense (under a reasonable complexity theoretic assumption). As a by-product, we give an algorithmic construction of sparse neighbourhood covers for nowhere dense graphs. This extends and improves previous constructions of neighbourhood covers for graph classes with excluded minors. At the same time, our construction is considerably simpler than those. Our proofs are based on a new game-theoretic characterisation of nowhere dense graphs that allows for a recursive version of locality-based algorithms on these classes. On the logical side, we prove a \"rank-preserving\" version of Gaifman's locality theorem.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128169024","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":"Smoothed analysis of tensor decompositions","authors":"Aditya Bhaskara, M. Charikar, Ankur Moitra, Aravindan Vijayaraghavan","doi":"10.1145/2591796.2591881","DOIUrl":"https://doi.org/10.1145/2591796.2591881","url":null,"abstract":"Low rank decomposition of tensors is a powerful tool for learning generative models. The uniqueness results that hold for tensors give them a significant advantage over matrices. However, tensors pose serious algorithmic challenges; in particular, much of the matrix algebra toolkit fails to generalize to tensors. Efficient decomposition in the overcomplete case (where rank exceeds dimension) is particularly challenging. We introduce a smoothed analysis model for studying these questions and develop an efficient algorithm for tensor decomposition in the highly overcomplete case (rank polynomial in the dimension). In this setting, we show that our algorithm is robust to inverse polynomial error -- a crucial property for applications in learning since we are only allowed a polynomial number of samples. While algorithms are known for exact tensor decomposition in some overcomplete settings, our main contribution is in analyzing their stability in the framework of smoothed analysis. Our main technical contribution is to show that tensor products of perturbed vectors are linearly independent in a robust sense (i.e. the associated matrix has singular values that are at least an inverse polynomial). This key result paves the way for applying tensor methods to learning problems in the smoothed setting. In particular, we use it to obtain results for learning multi-view models and mixtures of axis-aligned Gaussians where there are many more \"components\" than dimensions. The assumption here is that the model is not adversarially chosen, which we formalize by thinking of the model parameters as being perturbed. We believe this an appealing way to analyze realistic instances of learning problems, since this framework allows us to overcome many of the usual limitations of using tensor methods.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126844699","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":"Community detection thresholds and the weak Ramanujan property","authors":"L. Massoulié","doi":"10.1145/2591796.2591857","DOIUrl":"https://doi.org/10.1145/2591796.2591857","url":null,"abstract":"Decelle et al. [1] conjectured the existence of a sharp threshold on model parameters for community detection in sparse random graphs drawn from the stochastic block model. Mossel, Neeman and Sly [2] established the negative part of the conjecture, proving impossibility of non-trivial reconstruction below the threshold. In this work we solve the positive part of the conjecture. To that end we introduce a modified adjacency matrix B which counts self-avoiding paths of a given length ℓ between pairs of nodes. We then prove that for logarithmic length ℓ, the leading eigenvectors of this modified matrix provide a non-trivial reconstruction of the underlying structure, thereby settling the conjecture. A key step in the proof consists in establishing a weak Ramanujan property of the constructed matrix B. Namely, the spectrum of B consists in two leading eigenvalues ρ(B), λ2 and n -- 2 eigenvalues of a lower order O(nε √ρ(B) for all ε 0, ρ(B) denoting B's spectral radius.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124122565","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}