Maximally recoverable codes are codes designed for distributed storage which combine quick recovery from single node failure and optimal recovery from catastrophic failure. Gopalan et al [SODA 2017] studied the alphabet size needed for such codes in grid topologies and gave a combinatorial characterization for it.Consider a labeling of the edges of the complete bipartite graph K_{n,n} with labels coming from F_2^d, that satisfies the following condition: for any simple cycle, the sum of the labels over its edges is nonzero. The minimal d where this is possible controls the alphabet size needed for maximally recoverable codes in n × n grid topologies.Prior to the current work, it was known that d is between log(n)^2 and n log n. We improve both bounds and show that d is linear in n. The upper bound is a recursive construction which beats the random construction. The lower bound follows by first relating the problem to the independence number of the Birkhoff polytope graph, and then providing tight bounds for it using the representation theory of the symmetric group.
{"title":"The Independence Number of the Birkhoff Polytope Graph, and Applications to Maximally Recoverable Codes","authors":"D. Kane, Shachar Lovett, Sankeerth Rao","doi":"10.1109/FOCS.2017.31","DOIUrl":"https://doi.org/10.1109/FOCS.2017.31","url":null,"abstract":"Maximally recoverable codes are codes designed for distributed storage which combine quick recovery from single node failure and optimal recovery from catastrophic failure. Gopalan et al [SODA 2017] studied the alphabet size needed for such codes in grid topologies and gave a combinatorial characterization for it.Consider a labeling of the edges of the complete bipartite graph K_{n,n} with labels coming from F_2^d, that satisfies the following condition: for any simple cycle, the sum of the labels over its edges is nonzero. The minimal d where this is possible controls the alphabet size needed for maximally recoverable codes in n × n grid topologies.Prior to the current work, it was known that d is between log(n)^2 and n log n. We improve both bounds and show that d is linear in n. The upper bound is a recursive construction which beats the random construction. The lower bound follows by first relating the problem to the independence number of the Birkhoff polytope graph, and then providing tight bounds for it using the representation theory of the symmetric group.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129309682","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 give a nearly linear-time randomized approximation scheme for the Held-Karp bound [22] for Metric-TSP. Formally, given an undirected edge-weighted graph G = (V,E) on m edges and ε 0, the algorithm outputs in O(m log^4 n/ε^2) time, with high probability, a (1 + ε)-approximation to the Held-Karp bound on the Metric-TSP instance induced by the shortest path metric on G. The algorithm can also be used to output a corresponding solution to the Subtour Elimination LP. We substantially improve upon the O(m^2 log^2(m)/ε^2) running time achieved previously by Garg and Khandekar.
{"title":"Approximating the Held-Karp Bound for Metric TSP in Nearly-Linear Time","authors":"C. Chekuri, Kent Quanrud","doi":"10.1109/FOCS.2017.78","DOIUrl":"https://doi.org/10.1109/FOCS.2017.78","url":null,"abstract":"We give a nearly linear-time randomized approximation scheme for the Held-Karp bound [22] for Metric-TSP. Formally, given an undirected edge-weighted graph G = (V,E) on m edges and ε 0, the algorithm outputs in O(m log^4 n/ε^2) time, with high probability, a (1 + ε)-approximation to the Held-Karp bound on the Metric-TSP instance induced by the shortest path metric on G. The algorithm can also be used to output a corresponding solution to the Subtour Elimination LP. We substantially improve upon the O(m^2 log^2(m)/ε^2) running time achieved previously by Garg and Khandekar.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115038854","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}
Vincent Cohen-Addad, Søren Dahlgaard, Christian Wulff-Nilsen
For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an O(n 5/3)-space distance oracle which answers exact distance queries in O(log n) time for n-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space (i.e., space O(n 2- ) for some constant 0) either required query time polynomial in n or could only answer approximate distance queries.Furthermore, we show how to trade-off time and space: for any S ≥ n 3/2, we show how to obtain an S-space distance 5/2 oracle that answers queries in time O(S n 3/2 log n). This is a polynomial improvement over the previous planar distance oracles with o(n 1/4) query time.
{"title":"Fast and Compact Exact Distance Oracle for Planar Graphs","authors":"Vincent Cohen-Addad, Søren Dahlgaard, Christian Wulff-Nilsen","doi":"10.1109/FOCS.2017.93","DOIUrl":"https://doi.org/10.1109/FOCS.2017.93","url":null,"abstract":"For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an O(n 5/3)-space distance oracle which answers exact distance queries in O(log n) time for n-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space (i.e., space O(n 2- ) for some constant 0) either required query time polynomial in n or could only answer approximate distance queries.Furthermore, we show how to trade-off time and space: for any S ≥ n 3/2, we show how to obtain an S-space distance 5/2 oracle that answers queries in time O(S n 3/2 log n). This is a polynomial improvement over the previous planar distance oracles with o(n 1/4) query time.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117266851","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 classic k-median and k-means clustering objectives in the beyond-worst-case scenario. We consider three well-studied notions of structured data that aim at characterizing real-world inputs:• Distribution Stability (introduced by Awasthi, Blum, and Sheffet, FOCS 2010)• Spectral Separability (introduced by Kumar and Kannan, FOCS 2010)• Perturbation Resilience (introduced by Bilu and Linial, ICS 2010)We prove structural results showing that inputs satisfying at least one of the conditions are inherently local. Namely, for any such input, any local optimum is close both in term of structure and in term of objective value to the global optima.As a corollary we obtain that the widely-used Local Search algorithm has strong performance guarantees for both the tasks of recovering the underlying optimal clustering and obtaining a clustering of small cost. This is a significant step toward understanding the success of local search heuristics in clustering applications.
我们研究了超越最坏情况下的经典k中值和k均值聚类目标。我们考虑了三个经过充分研究的结构化数据概念，旨在表征现实世界的输入:•分配稳定性(Awasthi, Blum, and Sheffet, FOCS 2010引入)•光谱可分性(由Kumar和Kannan引入，FOCS 2010)•扰动弹性(由Bilu和Linial引入，ICS 2010)我们证明了结构结果，表明满足至少一个条件的输入本质上是局部的。也就是说，对于任何这样的输入，任何局部最优在结构和目标值上都接近全局最优。结果表明，广泛使用的局部搜索算法对于恢复底层最优聚类和获得小代价聚类都有很强的性能保证。这是理解本地搜索启发式在集群应用程序中的成功的重要一步。
{"title":"On the Local Structure of Stable Clustering Instances","authors":"Vincent Cohen-Addad, Chris Schwiegelshohn","doi":"10.1109/FOCS.2017.14","DOIUrl":"https://doi.org/10.1109/FOCS.2017.14","url":null,"abstract":"We study the classic k-median and k-means clustering objectives in the beyond-worst-case scenario. We consider three well-studied notions of structured data that aim at characterizing real-world inputs:• Distribution Stability (introduced by Awasthi, Blum, and Sheffet, FOCS 2010)• Spectral Separability (introduced by Kumar and Kannan, FOCS 2010)• Perturbation Resilience (introduced by Bilu and Linial, ICS 2010)We prove structural results showing that inputs satisfying at least one of the conditions are inherently local. Namely, for any such input, any local optimum is close both in term of structure and in term of objective value to the global optima.As a corollary we obtain that the widely-used Local Search algorithm has strong performance guarantees for both the tasks of recovering the underlying optimal clustering and obtaining a clustering of small cost. This is a significant step toward understanding the success of local search heuristics in clustering applications.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"261 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133937716","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}
More than 25 years ago, inspired by applications in computer graphics, Chazelle etal (FOCS 1991) studied the following question: Is it possible to cut any set of n lines or other objects in Reals^3 into a subquadratic number of fragments such that the resulting fragments admit a depth order? They managed to prove an O(n^{9/4}) bound on the number of fragments, but only for the very special case of bipartite weavings of lines. Since then only little progress was made, until a recent breakthrough by Aronov and Sharir (STOC 2016) who showed that O(n^{3/2}polylog n) fragments suffice for any set of lines. In a follow-up paper Aronov, Miller and Sharir (SODA 2017) proved an O(n^{3/2+≥}) bound for triangles, but their method uses high-degree algebraic arcs to perform the cuts. Hence, the resulting pieces have curved boundaries. Moreover, their method uses polynomial partitions, for which currently no algorithm is known. Thus the most natural version of the problem is still wide open: Is it possible to cut any collection of n disjoint triangles in Reals^3 into a subquadratic number of triangular fragments that admit a depth order? And if so, can we compute the cuts efficiently?We answer this question by presenting an algorithm that cuts any set of n disjoint triangles in Reals^3 into O(n^{7/4}polylog n) triangular fragments that admit a depth order. The running time of our algorithm is O(n^{3.69}). We also prove a refined bound that depends on the number, K, of intersections between the projections of the triangle edges onto the xy-plane: we show that O(n^{1+≥} + n^{1/4} K^{3/4}polylog n) fragments suffice to obtain a depth order. This result extends to xy-monotone surface patches bounded by a constant number of bounded-degree algebraic arcs in general position, constituting the first subquadratic bound for surface patches. Finally, as a byproduct of our approach we obtain a faster algorithm to cut a set of lines into O(n^{3/2}polylog n) fragments that admit a depth order. Our algorithm for lines runs in O(n^{5.38}) time, while the previous algorithm uses O(n^{8.77}) time.
25年前，受到计算机图形学应用的启发，Chazelle etal (FOCS 1991)研究了以下问题:是否有可能将任何一组n条线或其他物体在 real ^3中切割成次二次数的碎片，从而使产生的碎片承认深度顺序?他们成功地证明了碎片数量上的O(n^{9/4})界，但只适用于非常特殊的线段二部编织的情况。从那时起，只有很少的进展，直到最近由Aronov和Sharir (STOC 2016)取得突破，他们表明O(n^{3/2}polylog n)片段足以满足任何一组线。在随后的一篇论文中，Aronov, Miller和Sharir (SODA 2017)证明了三角形的O(n^{3/2+≥})界，但他们的方法使用高次代数弧来执行切割。因此，产生的碎片具有弯曲的边界。此外，他们的方法使用多项式划分，目前还没有已知的算法。因此，这个问题的最自然的版本仍然是开放的:是否有可能将 real ^3中n个不相交三角形的任何集合切割成允许深度顺序的次二次三角形碎片?如果是这样，我们能有效地计算出削减吗?我们通过提出一种算法来回答这个问题，该算法将 real ^3中任意n个不相交三角形的集合切割成O(n^{7/4}polylog n)个允许深度阶的三角形碎片。算法的运行时间为O(n^{3.69})。我们还证明了一个精细化的界，它依赖于三角形边在xy平面上的投影之间的交点的个数K:我们证明了O(n^{1+≥} + n^{1/4} K^{3/4} polylogn)片段足以获得一个深度阶。这一结果推广到一般位置上以常数有界次代数弧为界的任意单调曲面斑块，构成了曲面斑块的第一次二次界。最后，作为我们方法的副产品，我们得到了一个更快的算法，将一组线切割成O(n^{3/2}polylog n)个允许深度顺序的片段。我们的行算法运行时间为O(n^{5.38})，而之前的算法运行时间为O(n^{8.77})。
{"title":"Removing Depth-Order Cycles among Triangles: An Efficient Algorithm Generating Triangular Fragments","authors":"M. D. Berg","doi":"10.1109/FOCS.2017.33","DOIUrl":"https://doi.org/10.1109/FOCS.2017.33","url":null,"abstract":"More than 25 years ago, inspired by applications in computer graphics, Chazelle etal (FOCS 1991) studied the following question: Is it possible to cut any set of n lines or other objects in Reals^3 into a subquadratic number of fragments such that the resulting fragments admit a depth order? They managed to prove an O(n^{9/4}) bound on the number of fragments, but only for the very special case of bipartite weavings of lines. Since then only little progress was made, until a recent breakthrough by Aronov and Sharir (STOC 2016) who showed that O(n^{3/2}polylog n) fragments suffice for any set of lines. In a follow-up paper Aronov, Miller and Sharir (SODA 2017) proved an O(n^{3/2+≥}) bound for triangles, but their method uses high-degree algebraic arcs to perform the cuts. Hence, the resulting pieces have curved boundaries. Moreover, their method uses polynomial partitions, for which currently no algorithm is known. Thus the most natural version of the problem is still wide open: Is it possible to cut any collection of n disjoint triangles in Reals^3 into a subquadratic number of triangular fragments that admit a depth order? And if so, can we compute the cuts efficiently?We answer this question by presenting an algorithm that cuts any set of n disjoint triangles in Reals^3 into O(n^{7/4}polylog n) triangular fragments that admit a depth order. The running time of our algorithm is O(n^{3.69}). We also prove a refined bound that depends on the number, K, of intersections between the projections of the triangle edges onto the xy-plane: we show that O(n^{1+≥} + n^{1/4} K^{3/4}polylog n) fragments suffice to obtain a depth order. This result extends to xy-monotone surface patches bounded by a constant number of bounded-degree algebraic arcs in general position, constituting the first subquadratic bound for surface patches. Finally, as a byproduct of our approach we obtain a faster algorithm to cut a set of lines into O(n^{3/2}polylog n) fragments that admit a depth order. Our algorithm for lines runs in O(n^{5.38}) time, while the previous algorithm uses O(n^{8.77}) time.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114719429","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}
Sara Ahmadian, A. Norouzi-Fard, O. Svensson, Justin Ward
Clustering is a classic topic in optimization with k-means being one of the most fundamental such problems. In the absence of any restrictions on the input, the best known algorithm for k-means with a provable guarantee is a simple local search heuristic yielding an approximation guarantee of 9+≥ilon, a ratio that is known to be tight with respect to such methods.We overcome this barrier by presenting a new primal-dual approach that allows us to (1) exploit the geometric structure of k-means and (2) to satisfy the hard constraint that at most k clusters are selected without deteriorating the approximation guarantee. Our main result is a 6.357-approximation algorithm with respect to the standard LP relaxation. Our techniques are quite general and we also show improved guarantees for the general version of k-means where the underlying metric is not required to be Euclidean and for k-median in Euclidean metrics.
{"title":"Better Guarantees for k-Means and Euclidean k-Median by Primal-Dual Algorithms","authors":"Sara Ahmadian, A. Norouzi-Fard, O. Svensson, Justin Ward","doi":"10.1109/FOCS.2017.15","DOIUrl":"https://doi.org/10.1109/FOCS.2017.15","url":null,"abstract":"Clustering is a classic topic in optimization with k-means being one of the most fundamental such problems. In the absence of any restrictions on the input, the best known algorithm for k-means with a provable guarantee is a simple local search heuristic yielding an approximation guarantee of 9+≥ilon, a ratio that is known to be tight with respect to such methods.We overcome this barrier by presenting a new primal-dual approach that allows us to (1) exploit the geometric structure of k-means and (2) to satisfy the hard constraint that at most k clusters are selected without deteriorating the approximation guarantee. Our main result is a 6.357-approximation algorithm with respect to the standard LP relaxation. Our techniques are quite general and we also show improved guarantees for the general version of k-means where the underlying metric is not required to be Euclidean and for k-median in Euclidean metrics.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"130 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115365082","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}
Paul Dütting, M. Feldman, Thomas Kesselheim, Brendan Lucier
We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms, and is used to derive new and improved results for combinatorial markets (with and without complements), multi-dimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees.
{"title":"Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Non-Stochastic Inputs","authors":"Paul Dütting, M. Feldman, Thomas Kesselheim, Brendan Lucier","doi":"10.1109/FOCS.2017.56","DOIUrl":"https://doi.org/10.1109/FOCS.2017.56","url":null,"abstract":"We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms, and is used to derive new and improved results for combinatorial markets (with and without complements), multi-dimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"111 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116233366","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 the KLS constant for n-dimensional isotropic logconcavemeasures is O(n^{1/4}), improving on the current best bound ofO(n^{1/3}√{log n}). As corollaries we obtain the same improvedbound on the thin-shell estimate, Poincare constant and Lipschitzconcentration constant and an alternative proof of this bound forthe isotropic constant; it also follows that the ball walk for samplingfrom an isotropic logconcave density in R^{n} converges in O^{*}(n^{2.5})steps from a warm start.
{"title":"Eldan's Stochastic Localization and the KLS Hyperplane Conjecture: An Improved Lower Bound for Expansion","authors":"Y. Lee, S. Vempala","doi":"10.1109/FOCS.2017.96","DOIUrl":"https://doi.org/10.1109/FOCS.2017.96","url":null,"abstract":"We show that the KLS constant for n-dimensional isotropic logconcavemeasures is O(n^{1/4}), improving on the current best bound ofO(n^{1/3}√{log n}). As corollaries we obtain the same improvedbound on the thin-shell estimate, Poincare constant and Lipschitzconcentration constant and an alternative proof of this bound forthe isotropic constant; it also follows that the ball walk for samplingfrom an isotropic logconcave density in R^{n} converges in O^{*}(n^{2.5})steps from a warm start.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114489664","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 frustration-free local Hamiltonian has the property that its ground state minimises the energy of all local terms simultaneously. In general, even deciding whether a Hamiltonian is frustration-free is a hard task, as it is closely related to the QMA1-complete quantum satisfiability problem (QSAT) – the quantum analogue of SAT, which is the archetypal NP-complete problem in classical computer science. This connection shows that the frustration-free property is not only relevant to physics but also to computer science.The Quantum Lovász Local Lemma (QLLL) provides a sufficient condition for frustration-freeness. Is there an efficient way to prepare a frustration-free state under the conditions of the QLLL? Previous results showed that the answer is positive if all local terms commute. These works were based on Mosers compression argument which was the original analysis technique of the celebrated resampling algorithm. We generalise and simplify the compression argument, so that it provides a simplified version of the previous quantum results, and improves on some classical results as well.More importantly, we improve on the previous constructive results by designing an algorithm that works efficiently for non-commuting terms as well, assuming that the system is uniformly gapped, by which we mean that the system and all its subsystems have an inverse polynomial energy gap. Similarly to the previous results, our algorithm has the charming feature that it uses only local measurement operations corresponding to the local Hamiltonian terms.
{"title":"On Preparing Ground States of Gapped Hamiltonians: An Efficient Quantum Lovász Local Lemma","authors":"A. Gilyén, Or Sattath","doi":"10.1109/FOCS.2017.47","DOIUrl":"https://doi.org/10.1109/FOCS.2017.47","url":null,"abstract":"A frustration-free local Hamiltonian has the property that its ground state minimises the energy of all local terms simultaneously. In general, even deciding whether a Hamiltonian is frustration-free is a hard task, as it is closely related to the QMA1-complete quantum satisfiability problem (QSAT) – the quantum analogue of SAT, which is the archetypal NP-complete problem in classical computer science. This connection shows that the frustration-free property is not only relevant to physics but also to computer science.The Quantum Lovász Local Lemma (QLLL) provides a sufficient condition for frustration-freeness. Is there an efficient way to prepare a frustration-free state under the conditions of the QLLL? Previous results showed that the answer is positive if all local terms commute. These works were based on Mosers compression argument which was the original analysis technique of the celebrated resampling algorithm. We generalise and simplify the compression argument, so that it provides a simplified version of the previous quantum results, and improves on some classical results as well.More importantly, we improve on the previous constructive results by designing an algorithm that works efficiently for non-commuting terms as well, assuming that the system is uniformly gapped, by which we mean that the system and all its subsystems have an inverse polynomial energy gap. Similarly to the previous results, our algorithm has the charming feature that it uses only local measurement operations corresponding to the local Hamiltonian terms.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130316898","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}
V. Bhattiprolu, Mrinalkanti Ghosh, V. Guruswami, Euiwoong Lee, Madhur Tulsiani
We consider the following basic problem: given an n-variate degree-d homogeneous polynomial f with real coefficients, compute a unit vector x in R{string^}n that maximizes abs(f(x)). Besides its fundamental nature, this problem arises in diverse contexts ranging from tensor and operator norms to graph expansion to quantum information theory. The homogeneous degree-2 case is efficiently solvable as it corresponds to computing the spectral norm of an associated matrix, but the higher degree case is NP-hard.We give approximation algorithms for this problem that offer a trade-off between the approximation ratio and running time: in n{string^}O(q) time, we get an approximation within factor (O(n),/,q){string^}(d/2-1) for arbitrary polynomials, (O(n),/,q){string^}(d/4-1/2) for polynomials with non-negative coefficients, and (m,/,q){string^}(1/2) for sparse polynomials with m monomials. The approximation guarantees are with respect to the optimum of the level-q sum-of-squares (SoS) SDP relaxation of the problem (though our algorithms do not rely on actually solving the SDP). Known polynomial time algorithms for this problem rely on decoupling lemmas. Such tools are not capable of offering a trade-off like our results as they blow up the number of variables by a factor equal to the degree. We develop new decoupling tools that are more efficient in the number of variables at the expense of less structure in the output polynomials. This enables us to harness the benefits of higher level SoS relaxations. Our decoupling methods also work with folded polynomials, which are polynomials with polynomials as coefficients. This allows us to exploit easy substructures (such as quadratics) by considering them as coefficients in our algorithms. %We complement our algorithmic results with some polynomially large integrality gaps for d-levels of the SoS relaxation. For general polynomials this follows from known results for random polynomials, which yield a gap of Ω(n){string^}(d/4-1/2). For polynomials with non-negative coefficients, we prove an Ω(n{string^}(1/6),/, polylogs) gap for the degree-4 case, based on a novel distribution of 4-uniform hypergraphs. We establish an n{string^}Ω(d) gap for general degree-d, albeit for a slightly weaker (but still very natural) relaxation. Toward this, we give a method to lift a level-4 solution matrix M to a higher level solution, under a mild technical condition on M.From a structural perspective, our work yields worst-case convergence results on the performance of the sum-of-squares hierarchy for polynomial optimization. Despite the popularity of SoS in this context, such results were previously only known for the case of q = Omega(n).
{"title":"Weak Decoupling, Polynomial Folds and Approximate Optimization over the Sphere","authors":"V. Bhattiprolu, Mrinalkanti Ghosh, V. Guruswami, Euiwoong Lee, Madhur Tulsiani","doi":"10.1109/FOCS.2017.97","DOIUrl":"https://doi.org/10.1109/FOCS.2017.97","url":null,"abstract":"We consider the following basic problem: given an n-variate degree-d homogeneous polynomial f with real coefficients, compute a unit vector x in R{string^}n that maximizes abs(f(x)). Besides its fundamental nature, this problem arises in diverse contexts ranging from tensor and operator norms to graph expansion to quantum information theory. The homogeneous degree-2 case is efficiently solvable as it corresponds to computing the spectral norm of an associated matrix, but the higher degree case is NP-hard.We give approximation algorithms for this problem that offer a trade-off between the approximation ratio and running time: in n{string^}O(q) time, we get an approximation within factor (O(n),/,q){string^}(d/2-1) for arbitrary polynomials, (O(n),/,q){string^}(d/4-1/2) for polynomials with non-negative coefficients, and (m,/,q){string^}(1/2) for sparse polynomials with m monomials. The approximation guarantees are with respect to the optimum of the level-q sum-of-squares (SoS) SDP relaxation of the problem (though our algorithms do not rely on actually solving the SDP). Known polynomial time algorithms for this problem rely on decoupling lemmas. Such tools are not capable of offering a trade-off like our results as they blow up the number of variables by a factor equal to the degree. We develop new decoupling tools that are more efficient in the number of variables at the expense of less structure in the output polynomials. This enables us to harness the benefits of higher level SoS relaxations. Our decoupling methods also work with folded polynomials, which are polynomials with polynomials as coefficients. This allows us to exploit easy substructures (such as quadratics) by considering them as coefficients in our algorithms. %We complement our algorithmic results with some polynomially large integrality gaps for d-levels of the SoS relaxation. For general polynomials this follows from known results for random polynomials, which yield a gap of Ω(n){string^}(d/4-1/2). For polynomials with non-negative coefficients, we prove an Ω(n{string^}(1/6),/, polylogs) gap for the degree-4 case, based on a novel distribution of 4-uniform hypergraphs. We establish an n{string^}Ω(d) gap for general degree-d, albeit for a slightly weaker (but still very natural) relaxation. Toward this, we give a method to lift a level-4 solution matrix M to a higher level solution, under a mild technical condition on M.From a structural perspective, our work yields worst-case convergence results on the performance of the sum-of-squares hierarchy for polynomial optimization. Despite the popularity of SoS in this context, such results were previously only known for the case of q = Omega(n).","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132372726","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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