We achieve a (randomized) polynomial-time approximation scheme (PTAS) for the Steiner Forest Problem in doubling metrics. Before our work, a PTAS is given only for the Euclidean plane in [FOCS 2008: Borradaile, Klein and Mathieu]. Our PTAS also shares similarities with the dynamic programming for sparse instances used in [STOC 2012: Bartal, Gottlieb and Krauthgamer] and [SODA 2016: Chan and Jiang]. However, extending previous approaches requires overcoming several non-trivial hurdles, and we make the following technical contributions. (1) We prove a technical lemma showing that Steiner points have to be "near" the terminals in an optimal Steiner tree. This enables us to define a heuristic to estimate the local behavior of the optimal solution, even though the Steiner points are unknown in advance. This lemma also generalizes previous results in the Euclidean plane, and may be of independent interest for related problems involving Steiner points. (2) We develop a novel algorithmic technique known as "adaptive cells" to overcome the difficulty of keeping track of multiple components in a solution. Our idea is based on but significantly different from the previously proposed "uniform cells" in the FOCS 2008 paper, whose techniques cannot be readily applied to doubling metrics.
我们实现了一个(随机的)多项式时间近似格式(PTAS)的斯坦纳森林问题的倍增指标。在我们的工作之前，只有在[fos 2008: Borradaile, Klein and Mathieu]中给出了欧几里得平面的PTAS。我们的PTAS也与[STOC 2012: Bartal, Gottlieb和Krauthgamer]和[SODA 2016: Chan和Jiang]中使用的稀疏实例的动态规划有相似之处。然而，扩展以前的方法需要克服几个重要的障碍，我们做出了以下技术贡献。(1)我们证明了一个技术引理，该引理表明Steiner点必须“靠近”最优Steiner树的终端。这使我们能够定义一个启发式来估计最优解的局部行为，即使斯坦纳点事先是未知的。这个引理也推广了以前在欧几里得平面上的结果，并且可能对涉及斯坦纳点的相关问题有独立的兴趣。(2)我们开发了一种新的算法技术，称为“自适应细胞”，以克服跟踪解决方案中多个组件的困难。我们的想法是基于FOCS 2008论文中先前提出的“均匀细胞”，但与之有很大不同，后者的技术不能轻易应用于倍增指标。
{"title":"A PTAS for the Steiner Forest Problem in Doubling Metrics","authors":"T-H. Hubert Chan, Shuguang Hu, S. Jiang","doi":"10.1109/FOCS.2016.91","DOIUrl":"https://doi.org/10.1109/FOCS.2016.91","url":null,"abstract":"We achieve a (randomized) polynomial-time approximation scheme (PTAS) for the Steiner Forest Problem in doubling metrics. Before our work, a PTAS is given only for the Euclidean plane in [FOCS 2008: Borradaile, Klein and Mathieu]. Our PTAS also shares similarities with the dynamic programming for sparse instances used in [STOC 2012: Bartal, Gottlieb and Krauthgamer] and [SODA 2016: Chan and Jiang]. However, extending previous approaches requires overcoming several non-trivial hurdles, and we make the following technical contributions. (1) We prove a technical lemma showing that Steiner points have to be \"near\" the terminals in an optimal Steiner tree. This enables us to define a heuristic to estimate the local behavior of the optimal solution, even though the Steiner points are unknown in advance. This lemma also generalizes previous results in the Euclidean plane, and may be of independent interest for related problems involving Steiner points. (2) We develop a novel algorithmic technique known as \"adaptive cells\" to overcome the difficulty of keeping track of multiple components in a solution. Our idea is based on but significantly different from the previously proposed \"uniform cells\" in the FOCS 2008 paper, whose techniques cannot be readily applied to doubling metrics.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125555564","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 present an Õ (m 7/10 U 1/7)-time algorithm for the maximum s-t flow problem (and the minimum s-t cut problem) in directed graphs with m arcs and largest integer capacity U. This matches the running time of the Õ (mU)10/7)- time algorithm of Madry [30] in the unit-capacity case, and improves over it, as well as over the Õ (m√n log U)-time algorithm of Lee and Sidford [25], whenever U is moderately large and the graph is sufficiently sparse. By well-known reductions, this also implies similar running time improvements for the maximum-cardinality bipartite b-matching problem. One of the advantages of our algorithm is that it is significantly simpler than the ones presented in [30] and [25]. In particular, these algorithms employ a sophisticated interior-point method framework, while our algorithm is cast directly in the classic augmenting path setting that almost all the combinatorial maximum flow algorithms use. At a high level, the presented algorithm takes a primal dual approach in which each iteration uses electrical flows computations both to find an augmenting s-t flow in the current residual graph and to update the dual solution. We show that by maintain certain careful coupling of these primal and dual solutions we are always guaranteed to make significant progress.
我们提出了一种Õ (m 7/10 U 1/7)时间算法，用于求解具有m条曲线和最大整数容量U的有向图中的最大s-t流问题(和最小s-t切问题)。这与Madry[30]在单位容量情况下的Õ (mU)10/7)-时间算法的运行时间相匹配，并且在U中等大且图足够稀疏的情况下，改进了Madry[30]的Õ (mU)10/7)-时间算法以及Lee和Sidford[25]的Õ (m√n log U)时间算法。通过众所周知的缩减，这也意味着对于最大基数二部b匹配问题的类似运行时间改进。我们的算法的优点之一是它比[30]和[25]中提出的算法简单得多。特别是，这些算法采用了一个复杂的内点法框架，而我们的算法直接投射在几乎所有组合最大流量算法使用的经典增强路径设置中。在高层次上，所提出的算法采用原始对偶方法，其中每次迭代都使用电流计算来在当前残差图中找到增大的s-t流并更新对偶解。我们证明，通过保持这些原解和对偶解的一定的小心耦合，我们总是保证取得重大进展。
{"title":"Computing Maximum Flow with Augmenting Electrical Flows","authors":"A. Madry","doi":"10.1109/FOCS.2016.70","DOIUrl":"https://doi.org/10.1109/FOCS.2016.70","url":null,"abstract":"We present an Õ (m 7/10 U 1/7)-time algorithm for the maximum s-t flow problem (and the minimum s-t cut problem) in directed graphs with m arcs and largest integer capacity U. This matches the running time of the Õ (mU)10/7)- time algorithm of Madry [30] in the unit-capacity case, and improves over it, as well as over the Õ (m√n log U)-time algorithm of Lee and Sidford [25], whenever U is moderately large and the graph is sufficiently sparse. By well-known reductions, this also implies similar running time improvements for the maximum-cardinality bipartite b-matching problem. One of the advantages of our algorithm is that it is significantly simpler than the ones presented in [30] and [25]. In particular, these algorithms employ a sophisticated interior-point method framework, while our algorithm is cast directly in the classic augmenting path setting that almost all the combinatorial maximum flow algorithms use. At a high level, the presented algorithm takes a primal dual approach in which each iteration uses electrical flows computations both to find an augmenting s-t flow in the current residual graph and to update the dual solution. We show that by maintain certain careful coupling of these primal and dual solutions we are always guaranteed to make significant progress.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125011546","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 design new polynomials for representing threshold functions in three different regimes: probabilistic polynomials of low degree, which need far less randomness than previous constructions, polynomial threshold functions (PTFs) with "nice" threshold behavior and degree almost as low as the probabilistic polynomials, and a new notion of probabilistic PTFs where we combine the above techniques to achieve even lower degree with similar "nice" threshold behavior. Utilizing these polynomial constructions, we design faster algorithms for a variety of problems: · Offline Hamming Nearest (and Furthest) Neighbors: Given n red and n blue points in d-dimensional Hamming space for d = c log n, we can find an (exact) nearest (or furthest) blue neighbor for every red point in randomized time n2-1/O(√clog2/3 c) or deterministic time n2-1/O(c log2 c). These improve on a randomized n2-1/O(c log2 c) bound by Alman and Williams (FOCS'15), and also lead to faster MAX-SAT algorithms for sparse CNFs. · Offline Approximate Nearest (and Furthest) Neighbors: Given n red and n blue points in d-dimensional ℓ1 or Euclidean space, we can find a (1+ε)-approximate nearest (or furthest) blue neighbor for each red point in randomized time near dn+n2-Ω(ε1/3/log(1/ε)). This improves on an algorithm by Valiant (FOCS'12) with randomized time near dn+n2-Ω(√ε), which in turn improves previous methods based on locality-sensitive hashing. · SAT Algorithms and Lower Bounds for Circuits With Linear Threshold Functions: We give a satisfiability algorithm for AC0[m] o LTF LTF circuits with a subquadratic number of LTF gates on the bottom layer, and a subexponential number of gates on the other layers, that runs in deterministic 2n-nε time. This strictly generalizes a SAT algorithm for ACC0 oLTF circuits of subexponential size by Williams (STOC'14) and also implies new circuit lower bounds for threshold circuits, improving a recent gate lower bound of Kane and Williams (STOC'16). We also give a randomized 2n-nε-time SAT algorithm for subexponential-size MAJ o AC0 oLTF o AC0 oLTF circuits, where the top MAJ gate and middle LTF gates have O(n6/5-δ) fan-in.
我们设计了新的多项式来表示三种不同的阈值函数:低阶的概率多项式，它比以前的结构需要更少的随机性;具有“良好”阈值行为和程度几乎与概率多项式一样低的多项式阈值函数(ptf);以及一个新的概率ptf概念，我们结合上述技术来实现更低的度和类似的“良好”阈值行为。利用这些多项式结构，我们为各种问题设计了更快的算法:·离线汉明最近(和最远)邻居:给定d维Hamming空间中n个红点和n个蓝点，d = clog n，我们可以在随机时间n2-1/O(√clog2/3 c)或确定性时间n2-1/O(c log2c)内为每个红点找到一个(精确的)最近(或最远)的蓝色邻居。这些改进了由Alman和Williams (FOCS'15)约束的随机n2-1/O(c log2c)，并且还导致更快的MAX-SAT算法用于稀疏CNFs。·离线近似最近(和最远)邻居:在d维或欧几里德空间中给定n个红点和n个蓝点，我们可以在dn+n2-Ω(ε1/3/log(1/ε))附近的随机时间内为每个红点找到一个(1+ε)-近似最近(或最远)的蓝邻居。该算法改进了Valiant (FOCS'12)在dn+n2-Ω(√ε)附近随机化时间的算法，进而改进了先前基于位置敏感哈希的方法。·具有线性阈值函数的电路的SAT算法和下界:我们给出了AC0[m] o LTF LTF电路的可满足性算法，其底层具有次二次LTF门数，其他层具有次指数门数，在确定性的2n-nε时间内运行。这严格推广了Williams (STOC'14)针对亚指数大小的ACC0 oLTF电路的SAT算法，并且还暗示了阈值电路的新电路下界，改进了Kane和Williams (STOC'16)最近提出的门下界。我们还给出了一种随机2n-nε时间SAT算法，用于亚指数大小的MAJ o AC0 oLTF o AC0 oLTF电路，其中顶部MAJ门和中间LTF门具有o (n6/5-δ)扇入。
{"title":"Polynomial Representations of Threshold Functions and Algorithmic Applications","authors":"Josh Alman, Timothy M. Chan, Ryan Williams","doi":"10.1109/FOCS.2016.57","DOIUrl":"https://doi.org/10.1109/FOCS.2016.57","url":null,"abstract":"We design new polynomials for representing threshold functions in three different regimes: probabilistic polynomials of low degree, which need far less randomness than previous constructions, polynomial threshold functions (PTFs) with \"nice\" threshold behavior and degree almost as low as the probabilistic polynomials, and a new notion of probabilistic PTFs where we combine the above techniques to achieve even lower degree with similar \"nice\" threshold behavior. Utilizing these polynomial constructions, we design faster algorithms for a variety of problems: · Offline Hamming Nearest (and Furthest) Neighbors: Given n red and n blue points in d-dimensional Hamming space for d = c log n, we can find an (exact) nearest (or furthest) blue neighbor for every red point in randomized time n<sup>2-1</sup>/O(√clog<sup>2/3</sup> c) or deterministic time n<sup>2-1/O(c log2 c)</sup>. These improve on a randomized n<sup>2-1/O(c log2 c)</sup> bound by Alman and Williams (FOCS'15), and also lead to faster MAX-SAT algorithms for sparse CNFs. · Offline Approximate Nearest (and Furthest) Neighbors: Given n red and n blue points in d-dimensional ℓ<sub>1</sub> or Euclidean space, we can find a (1+ε)-approximate nearest (or furthest) blue neighbor for each red point in randomized time near dn+n<sup>2-Ω(ε1/3/log(1/ε))</sup>. This improves on an algorithm by Valiant (FOCS'12) with randomized time near dn+n<sup>2-Ω(√ε)</sup>, which in turn improves previous methods based on locality-sensitive hashing. · SAT Algorithms and Lower Bounds for Circuits With Linear Threshold Functions: We give a satisfiability algorithm for AC<sup>0</sup>[m] o LTF LTF circuits with a subquadratic number of LTF gates on the bottom layer, and a subexponential number of gates on the other layers, that runs in deterministic 2<sup>n-n</sup><sup>ε</sup> time. This strictly generalizes a SAT algorithm for ACC<sup>0</sup> oLTF circuits of subexponential size by Williams (STOC'14) and also implies new circuit lower bounds for threshold circuits, improving a recent gate lower bound of Kane and Williams (STOC'16). We also give a randomized 2<sup>n-n</sup><sup>ε</sup>-time SAT algorithm for subexponential-size MAJ o AC<sub>0</sub> oLTF o AC<sub>0</sub> oLTF circuits, where the top MAJ gate and middle LTF gates have O(n<sup>6/5-δ</sup>) fan-in.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125161501","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 work, we present a new algorithm for maximizing a non-monotone submodular function subject to a general constraint. Our algorithm finds an approximate fractional solution for maximizing the multilinear extension of the function over a down-closed polytope. The approximation guarantee is 0.372 and it is the first improvement over the 1/e approximation achieved by the unified Continuous Greedy algorithm [Feldman et al., FOCS 2011].
在这项工作中，我们提出了一种新的算法来最大化受一般约束的非单调子模函数。我们的算法找到了函数在下闭多面体上的多线性扩展最大化的近似分数解。近似保证为0.372，这是对统一连续贪婪算法实现的1/e近似的第一次改进[Feldman et al.， FOCS 2011]。
{"title":"Constrained Submodular Maximization: Beyond 1/e","authors":"Alina Ene, Huy L. Nguyen","doi":"10.1109/FOCS.2016.34","DOIUrl":"https://doi.org/10.1109/FOCS.2016.34","url":null,"abstract":"In this work, we present a new algorithm for maximizing a non-monotone submodular function subject to a general constraint. Our algorithm finds an approximate fractional solution for maximizing the multilinear extension of the function over a down-closed polytope. The approximation guarantee is 0.372 and it is the first improvement over the 1/e approximation achieved by the unified Continuous Greedy algorithm [Feldman et al., FOCS 2011].","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122523085","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 finite relational structure A, let CSP(A) denote the CSP instances whose constraint relations are taken from A. The resulting family of problems CSP(A) has been considered heavily in a variety of computational contexts. In this article, we consider this family from the perspective of property testing: given an instance of a CSP and query access to an assignment, one wants to decide whether the assignment satisfies the instance, or is far from so doing. While previous work on this scenario studied concrete templates or restricted classes of structures, this article presents comprehensive classification theorems. Our first contribution is a dichotomy theorem completely characterizing the structures A such that CSP(A) is constant-query testable: (i) If A has a majority polymorphism and a Maltsev polymorphism, then CSP(A) is constant-query testable with one-sided error. (ii) Else, testing CSP(A) requires a super-constant number of queries. Let ∃CSP(A) denote the extension of CSP(A) to instances which may include existentially quantified variables. Our second contribution is to classify all structures A in terms of the number of queries needed to test assignments to instances of ∃CSP(A), with one-sided error. More specifically, we show the following trichotomy (i) If A has a majority polymorphism and a Maltsev polymorphism, then ∃CSP(A) is constant-query testable with one-sided error. (ii) Else, if A has a (k + 1)-ary near-unanimity polymorphism for some k ≥ 2, and no Maltsev polymorphism then ∃CSP(A) is not constant-query testable (even with two-sided error) but is sublinear-query testable with one-sided error. (iii) Else, testing ∃CSP(A) with one-sided error requires a linear number of queries.
{"title":"Testing Assignments to Constraint Satisfaction Problems","authors":"Hubie Chen, M. Valeriote, Yuichi Yoshida","doi":"10.1109/FOCS.2016.63","DOIUrl":"https://doi.org/10.1109/FOCS.2016.63","url":null,"abstract":"For a finite relational structure A, let CSP(A) denote the CSP instances whose constraint relations are taken from A. The resulting family of problems CSP(A) has been considered heavily in a variety of computational contexts. In this article, we consider this family from the perspective of property testing: given an instance of a CSP and query access to an assignment, one wants to decide whether the assignment satisfies the instance, or is far from so doing. While previous work on this scenario studied concrete templates or restricted classes of structures, this article presents comprehensive classification theorems. Our first contribution is a dichotomy theorem completely characterizing the structures A such that CSP(A) is constant-query testable: (i) If A has a majority polymorphism and a Maltsev polymorphism, then CSP(A) is constant-query testable with one-sided error. (ii) Else, testing CSP(A) requires a super-constant number of queries. Let ∃CSP(A) denote the extension of CSP(A) to instances which may include existentially quantified variables. Our second contribution is to classify all structures A in terms of the number of queries needed to test assignments to instances of ∃CSP(A), with one-sided error. More specifically, we show the following trichotomy (i) If A has a majority polymorphism and a Maltsev polymorphism, then ∃CSP(A) is constant-query testable with one-sided error. (ii) Else, if A has a (k + 1)-ary near-unanimity polymorphism for some k ≥ 2, and no Maltsev polymorphism then ∃CSP(A) is not constant-query testable (even with two-sided error) but is sublinear-query testable with one-sided error. (iii) Else, testing ∃CSP(A) with one-sided error requires a linear number of queries.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122055928","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}
As a mathematical model of tile-based self-assembling systems, Winfree's abstract Tile Assembly Model (aTAM) has proven to be a remarkable platform for studying and understanding the behaviors and powers of self-assembling systems. Furthermore, as it is capable of Turing universal computation, the aTAM allows algorithmic self-assembly, in which the components can be designed so that the rules governing their behaviors force them to inherently execute prescribed algorithms as they combine. This power has yielded a wide variety of theoretical results in the aTAM utilizing algorithmic self-assembly to design systems capable of performing complex computations and forming extremely intricate structures. Adding to the completeness of the model, in FOCS 2012 the aTAM was shown to also be intrinsically universal, which means that there exists one single tile set such that for any arbitrary input aTAM system, that tile set can be configured into a "seed" structure which will then cause self-assembly using that tile set to simulate the input system, capturing its full dynamics modulo only a scale factor. However, the "universal simulator" of that result makes use of nondeterminism in terms of the tiles placed in several key locations when different assembly sequences are followed. This nondeterminism remains even when the simulator is simulating a system which is directed, meaning that it has exactly one unique terminal assembly and for any given location, no matter which assembly sequence is followed, the same tile type is always placed there. The question which then arose was whether or not that nondeterminism is fundamentally required, and if any universal simulator must in fact utilize more nondeterminism than directed systems when simulating them. In this paper, we answer that question in the affirmative: the class of directed systems in the aTAM is not intrinsically universal, meaning there is no universal simulator for directed systems which itself is always directed. This result provides a powerful insight into the role of nondeterminism in self-assembly, which is itself a fundamentally nondeterministic process occurring via unguided local interactions. Furthermore, to achieve this result we leverage powerful results of computational complexity hierarchies, including tight bounds on both best and worst-case complexities of decidable languages, to tailor design systems with precisely controllable space resources available to computations embedded within them. We also develop novel techniques for designing systems containing subsystems with disjoint, mutually exclusive computational powers. The main result will be important in the development of future simulation systems, and the supporting design techniques and lemmas will provide powerful tools for the development of future aTAM systems as well as proofs of their computational abilities.
{"title":"Universal Simulation of Directed Systems in the Abstract Tile Assembly Model Requires Undirectedness","authors":"Jacob Hendricks, Matthew J. Patitz, T. Rogers","doi":"10.1109/FOCS.2016.90","DOIUrl":"https://doi.org/10.1109/FOCS.2016.90","url":null,"abstract":"As a mathematical model of tile-based self-assembling systems, Winfree's abstract Tile Assembly Model (aTAM) has proven to be a remarkable platform for studying and understanding the behaviors and powers of self-assembling systems. Furthermore, as it is capable of Turing universal computation, the aTAM allows algorithmic self-assembly, in which the components can be designed so that the rules governing their behaviors force them to inherently execute prescribed algorithms as they combine. This power has yielded a wide variety of theoretical results in the aTAM utilizing algorithmic self-assembly to design systems capable of performing complex computations and forming extremely intricate structures. Adding to the completeness of the model, in FOCS 2012 the aTAM was shown to also be intrinsically universal, which means that there exists one single tile set such that for any arbitrary input aTAM system, that tile set can be configured into a \"seed\" structure which will then cause self-assembly using that tile set to simulate the input system, capturing its full dynamics modulo only a scale factor. However, the \"universal simulator\" of that result makes use of nondeterminism in terms of the tiles placed in several key locations when different assembly sequences are followed. This nondeterminism remains even when the simulator is simulating a system which is directed, meaning that it has exactly one unique terminal assembly and for any given location, no matter which assembly sequence is followed, the same tile type is always placed there. The question which then arose was whether or not that nondeterminism is fundamentally required, and if any universal simulator must in fact utilize more nondeterminism than directed systems when simulating them. In this paper, we answer that question in the affirmative: the class of directed systems in the aTAM is not intrinsically universal, meaning there is no universal simulator for directed systems which itself is always directed. This result provides a powerful insight into the role of nondeterminism in self-assembly, which is itself a fundamentally nondeterministic process occurring via unguided local interactions. Furthermore, to achieve this result we leverage powerful results of computational complexity hierarchies, including tight bounds on both best and worst-case complexities of decidable languages, to tailor design systems with precisely controllable space resources available to computations embedded within them. We also develop novel techniques for designing systems containing subsystems with disjoint, mutually exclusive computational powers. The main result will be important in the development of future simulation systems, and the supporting design techniques and lemmas will provide powerful tools for the development of future aTAM systems as well as proofs of their computational abilities.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128507906","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}
Michael B. Cohen, Jonathan A. Kelner, John Peebles, Richard Peng, Aaron Sidford, Adrian Vladu
In this paper, we provide faster algorithms for computing various fundamental quantities associated with random walks on a directed graph, including the stationary distribution, personalized PageRank vectors, hitting times, and escape probabilities. In particular, on a directed graph with n vertices and m edges, we show how to compute each quantity in time Õ(m3/4n + mn2/3), where the Õ notation suppresses polylog factors in n, the desired accuracy, and the appropriate condition number (i.e. the mixing time or restart probability). Our result improves upon the previous fastest running times for these problems; previous results either invoke a general purpose linear system solver on a n × n matrix with m nonzero entries, or depend polynomially on the desired error or natural condition number associated with the problem (i.e. the mixing time or restart probability). For sparse graphs, we obtain a running time of Õ(n7/4), breaking the O(n2) barrier of the best running time one could hope to achieve using fast matrix multiplication. We achieve our result by providing a similar running time improvement for solving directed Laplacian systems, a natural directed or asymmetric analog of the well studied symmetric or undirected Laplacian systems. We show how to solve such systems in time Õ(m3/4n + mn2/3), and efficiently reduce a broad range of problems to solving Õ(1) directed Laplacian systems on Eulerian graphs. We hope these results and our analysis open the door for further study into directed spectral graph theory.
{"title":"Faster Algorithms for Computing the Stationary Distribution, Simulating Random Walks, and More","authors":"Michael B. Cohen, Jonathan A. Kelner, John Peebles, Richard Peng, Aaron Sidford, Adrian Vladu","doi":"10.1109/FOCS.2016.69","DOIUrl":"https://doi.org/10.1109/FOCS.2016.69","url":null,"abstract":"In this paper, we provide faster algorithms for computing various fundamental quantities associated with random walks on a directed graph, including the stationary distribution, personalized PageRank vectors, hitting times, and escape probabilities. In particular, on a directed graph with n vertices and m edges, we show how to compute each quantity in time Õ(m3/4n + mn2/3), where the Õ notation suppresses polylog factors in n, the desired accuracy, and the appropriate condition number (i.e. the mixing time or restart probability). Our result improves upon the previous fastest running times for these problems; previous results either invoke a general purpose linear system solver on a n × n matrix with m nonzero entries, or depend polynomially on the desired error or natural condition number associated with the problem (i.e. the mixing time or restart probability). For sparse graphs, we obtain a running time of Õ(n7/4), breaking the O(n2) barrier of the best running time one could hope to achieve using fast matrix multiplication. We achieve our result by providing a similar running time improvement for solving directed Laplacian systems, a natural directed or asymmetric analog of the well studied symmetric or undirected Laplacian systems. We show how to solve such systems in time Õ(m3/4n + mn2/3), and efficiently reduce a broad range of problems to solving Õ(1) directed Laplacian systems on Eulerian graphs. We hope these results and our analysis open the door for further study into directed spectral graph theory.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127357657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the Maximum Independent Set of Rectangles (MISR) problem: given a set of n axis-parallel rectangles, find a largest-cardinality subset of the rectangles, such that no two of them overlap. MISR is a basic geometric optimization problem with many applications, that has been studied extensively. Until recently, the best approximation algorithm for it achieved an O(log log n)-approximation factor. In a recent breakthrough, Adamaszek and Wiese provided a quasi-polynomial time approximation scheme: a (1-ε)-approximation algorithm with running time nO(poly(log n)/ε). Despite this result, obtaining a PTAS or even a polynomial-time constant-factor approximation remains a challenging open problem. In this paper we make progress towards this goal by providing an algorithm for MISR that achieves a (1 - ε)-approximation in time nO(poly(log logn/ε)). We introduce several new technical ideas, that we hope will lead to further progress on this and related problems.
{"title":"On Approximating Maximum Independent Set of Rectangles","authors":"Julia Chuzhoy, Alina Ene","doi":"10.1109/FOCS.2016.92","DOIUrl":"https://doi.org/10.1109/FOCS.2016.92","url":null,"abstract":"We study the Maximum Independent Set of Rectangles (MISR) problem: given a set of n axis-parallel rectangles, find a largest-cardinality subset of the rectangles, such that no two of them overlap. MISR is a basic geometric optimization problem with many applications, that has been studied extensively. Until recently, the best approximation algorithm for it achieved an O(log log n)-approximation factor. In a recent breakthrough, Adamaszek and Wiese provided a quasi-polynomial time approximation scheme: a (1-ε)-approximation algorithm with running time nO(poly(log n)/ε). Despite this result, obtaining a PTAS or even a polynomial-time constant-factor approximation remains a challenging open problem. In this paper we make progress towards this goal by providing an algorithm for MISR that achieves a (1 - ε)-approximation in time nO(poly(log logn/ε)). We introduce several new technical ideas, that we hope will lead to further progress on this and related problems.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"171 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133600610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that in the document exchange problem, where Alice holds x ϵ {0, 1}n and Bob holds y ϵ {0, 1}n, Alice can send Bob a message of size O(K(log2 K + log n)) bits such that Bob can recover x using the message and his input y if the edit distance between x and y is no more than K, and output "error" otherwise. Both the encoding and decoding can be done in time Õ(n + poly(K)). This result significantly improves on the previous communication bounds under polynomial encoding/decoding time. We also show that in the referee model, where Alice and Bob hold x and y respectively, they can compute sketches of x and y of sizes poly(K log n) bits (the encoding), and send to the referee, who can then compute the edit distance between x and y together with all the edit operations if the edit distance is no more than K, and output "error" otherwise (the decoding). To the best of our knowledge, this is the first result for sketching edit distance using poly(K log n) bits. Moreover, the encoding phase of our sketching algorithm can be performed by scanning the input string in one pass. Thus our sketching algorithm also implies the first streaming algorithm for computing edit distance and all the edits exactly using poly(K log n) bits of space.
{"title":"Edit Distance: Sketching, Streaming, and Document Exchange","authors":"D. Belazzougui, Qin Zhang","doi":"10.1109/FOCS.2016.15","DOIUrl":"https://doi.org/10.1109/FOCS.2016.15","url":null,"abstract":"We show that in the document exchange problem, where Alice holds x ϵ {0, 1}n and Bob holds y ϵ {0, 1}n, Alice can send Bob a message of size O(K(log2 K + log n)) bits such that Bob can recover x using the message and his input y if the edit distance between x and y is no more than K, and output \"error\" otherwise. Both the encoding and decoding can be done in time Õ(n + poly(K)). This result significantly improves on the previous communication bounds under polynomial encoding/decoding time. We also show that in the referee model, where Alice and Bob hold x and y respectively, they can compute sketches of x and y of sizes poly(K log n) bits (the encoding), and send to the referee, who can then compute the edit distance between x and y together with all the edit operations if the edit distance is no more than K, and output \"error\" otherwise (the decoding). To the best of our knowledge, this is the first result for sketching edit distance using poly(K log n) bits. Moreover, the encoding phase of our sketching algorithm can be performed by scanning the input string in one pass. Thus our sketching algorithm also implies the first streaming algorithm for computing edit distance and all the edits exactly using poly(K log n) bits of space.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"162 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128163159","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 present decidability results for a sub-class of "non-interactive" simulation problems, a well-studied class of problems in information theory. A non-interactive simulation problem is specified by two distributions P(x, y) and Q(u, v): The goal is to determine if two players, Alice and Bob, that observe sequences Xn and Yn respectively where {(Xi, Yi)}ni = 1 are drawn i.i.d. from P(x, y) can generate pairs U and V respectively (without communicating with each other) with a joint distribution that is arbitrarily close in total variation to Q(u, v). Even when P and Q are extremely simple: e.g., P is uniform on the triples (0, 0), (0,1), (1,0) and Q is a "doubly symmetric binary source", i.e., U and V are uniform ± 1 variables with correlation say 0.49, it is open if P can simulate Q. In this work, we show that whenever P is a distribution on a finite domain and Q is a 2 × 2 distribution, then the non-interactive simulation problem is decidable: specifically, given δ > 0 the algorithm runs in time bounded by some function of P and δ and either gives a non-interactive simulation protocol that is δ-close to Q or asserts that no protocol gets O(δ)-close to Q. The main challenge to such a result is determining explicit (computable) convergence bounds on the number n of samples that need to be drawn from P(x, y) to get δ-close to Q. We invoke contemporary results from the analysis of Boolean functions such as the invariance principle and a regularity lemma to obtain such explicit bounds.
{"title":"Decidability of Non-interactive Simulation of Joint Distributions","authors":"Badih Ghazi, Pritish Kamath, M. Sudan","doi":"10.1109/FOCS.2016.65","DOIUrl":"https://doi.org/10.1109/FOCS.2016.65","url":null,"abstract":"We present decidability results for a sub-class of \"non-interactive\" simulation problems, a well-studied class of problems in information theory. A non-interactive simulation problem is specified by two distributions P(x, y) and Q(u, v): The goal is to determine if two players, Alice and Bob, that observe sequences Xn and Yn respectively where {(Xi, Yi)}ni = 1 are drawn i.i.d. from P(x, y) can generate pairs U and V respectively (without communicating with each other) with a joint distribution that is arbitrarily close in total variation to Q(u, v). Even when P and Q are extremely simple: e.g., P is uniform on the triples (0, 0), (0,1), (1,0) and Q is a \"doubly symmetric binary source\", i.e., U and V are uniform ± 1 variables with correlation say 0.49, it is open if P can simulate Q. In this work, we show that whenever P is a distribution on a finite domain and Q is a 2 × 2 distribution, then the non-interactive simulation problem is decidable: specifically, given δ > 0 the algorithm runs in time bounded by some function of P and δ and either gives a non-interactive simulation protocol that is δ-close to Q or asserts that no protocol gets O(δ)-close to Q. The main challenge to such a result is determining explicit (computable) convergence bounds on the number n of samples that need to be drawn from P(x, y) to get δ-close to Q. We invoke contemporary results from the analysis of Boolean functions such as the invariance principle and a regularity lemma to obtain such explicit bounds.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"34 48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132863010","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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