I. Caragiannis, A. Fanelli, N. Gravin, Alexander Skopalik
Congestion games constitute an important class of games in which computing an exact or even approximate pure Nash equilibrium is in general {sf PLS}-complete. We present a surprisingly simple polynomial-time algorithm that computes $O(1)$-approximate Nash equilibria in these games. In particular, for congestion games with linear latency functions, our algorithm computes $(2+epsilon)$-approximate pure Nash equilibria in time polynomial in the number of players, the number of resources and $1/epsilon$. It also applies to games with polynomial latency functions with constant maximum degree $d$; there, the approximation guarantee is $d^{O(d)}$. The algorithm essentially identifies a polynomially long sequence of best-response moves that lead to an approximate equilibrium, the existence of such short sequences is interesting in itself. These are the first positive algorithmic results for approximate equilibria in non-symmetric congestion games. We strengthen them further by proving that, for congestion games that deviate from our mild assumptions, computing $rho$-approximate equilibria is {sf PLS}-complete for any polynomial-time computable $rho$.
{"title":"Efficient Computation of Approximate Pure Nash Equilibria in Congestion Games","authors":"I. Caragiannis, A. Fanelli, N. Gravin, Alexander Skopalik","doi":"10.1109/FOCS.2011.50","DOIUrl":"https://doi.org/10.1109/FOCS.2011.50","url":null,"abstract":"Congestion games constitute an important class of games in which computing an exact or even approximate pure Nash equilibrium is in general {sf PLS}-complete. We present a surprisingly simple polynomial-time algorithm that computes $O(1)$-approximate Nash equilibria in these games. In particular, for congestion games with linear latency functions, our algorithm computes $(2+epsilon)$-approximate pure Nash equilibria in time polynomial in the number of players, the number of resources and $1/epsilon$. It also applies to games with polynomial latency functions with constant maximum degree $d$; there, the approximation guarantee is $d^{O(d)}$. The algorithm essentially identifies a polynomially long sequence of best-response moves that lead to an approximate equilibrium, the existence of such short sequences is interesting in itself. These are the first positive algorithmic results for approximate equilibria in non-symmetric congestion games. We strengthen them further by proving that, for congestion games that deviate from our mild assumptions, computing $rho$-approximate equilibria is {sf PLS}-complete for any polynomial-time computable $rho$.","PeriodicalId":326048,"journal":{"name":"2011 IEEE 52nd Annual Symposium on Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114440505","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}
Positive semi definite programs are an important subclass of semi definite programs in which all matrices involved in the specification of the problem are positive semi definite and all scalars involved are non-negative. We present a parallel algorithm, which given an instance of a positive semi definite program of size N and an approximation factor e >, 0, runs in (parallel) time poly(1/e) polylog(N), using poly(N) processors, and outputs a value which is within multiplicative factor of (1+ e) to the optimal. Our result generalizes analogous result of Luby and Nisan (1993) for positive linear programs and our algorithm is inspired by their algorithm of [10].
{"title":"A Parallel Approximation Algorithm for Positive Semidefinite Programming","authors":"Rahul Jain, Penghui Yao","doi":"10.1109/FOCS.2011.25","DOIUrl":"https://doi.org/10.1109/FOCS.2011.25","url":null,"abstract":"Positive semi definite programs are an important subclass of semi definite programs in which all matrices involved in the specification of the problem are positive semi definite and all scalars involved are non-negative. We present a parallel algorithm, which given an instance of a positive semi definite program of size N and an approximation factor e >, 0, runs in (parallel) time poly(1/e) polylog(N), using poly(N) processors, and outputs a value which is within multiplicative factor of (1+ e) to the optimal. Our result generalizes analogous result of Luby and Nisan (1993) for positive linear programs and our algorithm is inspired by their algorithm of [10].","PeriodicalId":326048,"journal":{"name":"2011 IEEE 52nd Annual Symposium on Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122873592","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 discuss a psuedorandom generator to $epsilon$-fool degree-d polynomial threshold functions with respect to the Gaussian distribution with seed length $2^O_c(d)log(n) epsilon^{-4-c}$. Our analysis involves several new ideas including what we term the "noisy derivative" of a function and a stronger version of standard anticoncentration results.
{"title":"A Small PRG for Polynomial Threshold Functions of Gaussians","authors":"D. Kane","doi":"10.1109/FOCS.2011.16","DOIUrl":"https://doi.org/10.1109/FOCS.2011.16","url":null,"abstract":"We discuss a psuedorandom generator to $epsilon$-fool degree-d polynomial threshold functions with respect to the Gaussian distribution with seed length $2^O_c(d)log(n) epsilon^{-4-c}$. Our analysis involves several new ideas including what we term the \"noisy derivative\" of a function and a stronger version of standard anticoncentration results.","PeriodicalId":326048,"journal":{"name":"2011 IEEE 52nd Annual Symposium on Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133052571","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}
M. Braverman, K. Makarychev, Yury Makarychev, A. Naor
The classical Grothendieck constant, denoted $K_G$, is equal to the integrality gap of the natural semi definite relaxation of the problem of computing$$max {sum_{i=1}^msum_{j=1}^n a_{ij} epsilon_idelta_j: {epsilon_i}_{i=1}^m,{delta_j}_{j=1}^nsubseteq {-1,1}},$$a generic and well-studied optimization problem with many applications. Krivine proved in 1977 that $K_Gleq pi / (2log(1+sqrt{2}))$ and conjectured that his estimate is sharp. We obtain a sharper Grothendieck inequality, showing that $K_G 0$. Our main contribution is conceptual: despite dealing with a binary rounding problem, random 2-dimensional projections combined with a careful partition of $R^2$ in order to round the projected vectors, beat the random hyper plane technique, contrary to Krivine's long-standing conjecture.
经典的格罗滕迪克常数，记为$K_G$，等于计算问题$$max {sum_{i=1}^msum_{j=1}^n a_{ij} epsilon_idelta_j: {epsilon_i}_{i=1}^m,{delta_j}_{j=1}^nsubseteq {-1,1}},$$的自然半确定松弛的完整性间隙，这是一个广泛研究的优化问题，有许多应用。Krivine在1977年证明了$K_Gleq pi / (2log(1+sqrt{2}))$，并推测他的估计是准确的。我们得到一个更清晰的格罗滕迪克不等式，表明$K_G 0$。我们的主要贡献是概念上的:尽管处理了二进制舍入问题，但随机的二维投影结合$R^2$的仔细划分，以舍入投影向量，击败了随机超平面技术，这与Krivine长期以来的猜想相反。
{"title":"The Grothendieck Constant is Strictly Smaller than Krivine's Bound","authors":"M. Braverman, K. Makarychev, Yury Makarychev, A. Naor","doi":"10.1017/fmp.2013.4","DOIUrl":"https://doi.org/10.1017/fmp.2013.4","url":null,"abstract":"The classical Grothendieck constant, denoted $K_G$, is equal to the integrality gap of the natural semi definite relaxation of the problem of computing$$max {sum_{i=1}^msum_{j=1}^n a_{ij} epsilon_idelta_j: {epsilon_i}_{i=1}^m,{delta_j}_{j=1}^nsubseteq {-1,1}},$$a generic and well-studied optimization problem with many applications. Krivine proved in 1977 that $K_Gleq pi / (2log(1+sqrt{2}))$ and conjectured that his estimate is sharp. We obtain a sharper Grothendieck inequality, showing that $K_G 0$. Our main contribution is conceptual: despite dealing with a binary rounding problem, random 2-dimensional projections combined with a careful partition of $R^2$ in order to round the projected vectors, beat the random hyper plane technique, contrary to Krivine's long-standing conjecture.","PeriodicalId":326048,"journal":{"name":"2011 IEEE 52nd Annual Symposium on Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121150390","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}
The PPSZ algorithm by Paturi, Pudl'ak, Saks, and Zane [1998] is the fastest known algorithm for Unique k-SAT, where the input formula does not have more than one satisfying assignment. For k>=5 the same bounds hold for general k-SAT. We show that this is also the case for k=3,4, using a slightly modified PPSZ algorithm. We do the analysis by defining a cost for satisfiable CNF formulas, which we prove to decrease in each PPSZ step by a certain amount. This improves our previous best bounds with Moser and Scheder [2011] for 3-SAT to O(1.308^n) and for 4-SAT to O(1.469^n).
Paturi, Pudl ak, Saks, and Zane[1998]提出的PPSZ算法是目前已知的最快的Unique k-SAT算法，该算法的输入公式不会有多于一个满意的赋值。对于k>=5，一般k- sat也有相同的界。我们使用稍微修改的PPSZ算法证明，对于k=3,4也是如此。我们通过定义一个可满足CNF公式的成本来进行分析，我们证明了每个PPSZ步骤都会减少一定数量的成本。这改进了我们之前使用Moser和Scheder[2011]对3-SAT到O(1.308^n)和4-SAT到O(1.469^n)的最佳界限。
{"title":"3-SAT Faster and Simpler - Unique-SAT Bounds for PPSZ Hold in General","authors":"Timon Hertli","doi":"10.1137/120868177","DOIUrl":"https://doi.org/10.1137/120868177","url":null,"abstract":"The PPSZ algorithm by Paturi, Pudl'ak, Saks, and Zane [1998] is the fastest known algorithm for Unique k-SAT, where the input formula does not have more than one satisfying assignment. For k>=5 the same bounds hold for general k-SAT. We show that this is also the case for k=3,4, using a slightly modified PPSZ algorithm. We do the analysis by defining a cost for satisfiable CNF formulas, which we prove to decrease in each PPSZ step by a certain amount. This improves our previous best bounds with Moser and Scheder [2011] for 3-SAT to O(1.308^n) and for 4-SAT to O(1.469^n).","PeriodicalId":326048,"journal":{"name":"2011 IEEE 52nd Annual Symposium on Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124762423","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 algorithm for maintaining maximal matching in a graph under addition and deletion of edges. Our data structure is randomized that takes $O( log n)$ expected amortized time for each edge update where $n$ is the number of vertices in the graph. While there is a trivial $O(n)$ algorithm for edge update, the previous best known result for this problem was due to Ivkovi'c and Llyodcite{llyod}. For a graph with $n$ vertices and $m$ edges, they give an $O( {(n+ m)}^{0.7072})$ update time algorithm which is sub linear only for a sparse graph. %To the best of our knowledge this %is the first polylog update time for maximal matching that implies an % exponential improvement from the previous results. For the related problem of maximum matching, Onak and Rubinfeld cite{onak} designed a randomized data structure that achieves $O(log^2 n)$ expected amortized time for each update for maintaining a $c$-approximate maximum matching for some large constant $c$. In contrast, we can maintain a factor two approximate maximum matching in $O(log n )$ expected amortized time per update as a direct corollary of the maximal matching scheme. This in turn also implies a two approximate vertex cover maintenance scheme that takes $O(log n )$expected amortized time per update.
提出了一种图在加边和删边情况下保持最大匹配的算法。我们的数据结构是随机化的，每次边缘更新需要$O( log n)$预期平摊时间，其中$n$是图中顶点的数量。虽然有一个微不足道的$O(n)$边缘更新算法，但这个问题之前最著名的结果是由于ivkovovic和lloyd cite{llyod}。对于具有$n$顶点和$m$边的图，他们给出了$O( {(n+ m)}^{0.7072})$更新时间算法，该算法仅对稀疏图是次线性的。 %To the best of our knowledge this %is the first polylog update time for maximal matching that implies an % exponential improvement from the previous results. For the related problem of maximum matching, Onak and Rubinfeld cite{onak} designed a randomized data structure that achieves $O(log^2 n)$ expected amortized time for each update for maintaining a $c$-approximate maximum matching for some large constant $c$. In contrast, we can maintain a factor two approximate maximum matching in $O(log n )$ expected amortized time per update as a direct corollary of the maximal matching scheme. This in turn also implies a two approximate vertex cover maintenance scheme that takes $O(log n )$expected amortized time per update.
{"title":"Fully Dynamic Maximal Matching in O (log n) Update Time","authors":"Surender Baswana, Manoj Gupta, Sandeep Sen","doi":"10.1137/130914140","DOIUrl":"https://doi.org/10.1137/130914140","url":null,"abstract":"We present an algorithm for maintaining maximal matching in a graph under addition and deletion of edges. Our data structure is randomized that takes $O( log n)$ expected amortized time for each edge update where $n$ is the number of vertices in the graph. While there is a trivial $O(n)$ algorithm for edge update, the previous best known result for this problem was due to Ivkovi'c and Llyodcite{llyod}. For a graph with $n$ vertices and $m$ edges, they give an $O( {(n+ m)}^{0.7072})$ update time algorithm which is sub linear only for a sparse graph. %To the best of our knowledge this %is the first polylog update time for maximal matching that implies an % exponential improvement from the previous results. For the related problem of maximum matching, Onak and Rubinfeld cite{onak} designed a randomized data structure that achieves $O(log^2 n)$ expected amortized time for each update for maintaining a $c$-approximate maximum matching for some large constant $c$. In contrast, we can maintain a factor two approximate maximum matching in $O(log n )$ expected amortized time per update as a direct corollary of the maximal matching scheme. This in turn also implies a two approximate vertex cover maintenance scheme that takes $O(log n )$expected amortized time per update.","PeriodicalId":326048,"journal":{"name":"2011 IEEE 52nd Annual Symposium on Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122125964","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 consider low-rank reconstruction of a matrix using a subset of its columns and we present asymptotically optimal algorithms for both spectral norm and Frobenius norm reconstruction. The main tools we introduce to obtain our results are: (i) the use of fast approximate SVD-like decompositions for column-based matrix reconstruction, and (ii) two deterministic algorithms for selecting rows from matrices with orthonormal columns, building upon the sparse representation theorem for decompositions of the identity that appeared in [1].
{"title":"Near Optimal Column-Based Matrix Reconstruction","authors":"Christos Boutsidis, P. Drineas, M. Magdon-Ismail","doi":"10.1137/12086755X","DOIUrl":"https://doi.org/10.1137/12086755X","url":null,"abstract":"We consider low-rank reconstruction of a matrix using a subset of its columns and we present asymptotically optimal algorithms for both spectral norm and Frobenius norm reconstruction. The main tools we introduce to obtain our results are: (i) the use of fast approximate SVD-like decompositions for column-based matrix reconstruction, and (ii) two deterministic algorithms for selecting rows from matrices with orthonormal columns, building upon the sparse representation theorem for decompositions of the identity that appeared in [1].","PeriodicalId":326048,"journal":{"name":"2011 IEEE 52nd Annual Symposium on Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126287708","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}
Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, J. M. M. Van Rooij, J. O. Wojtaszczyk
For the vast majority of local problems on graphs of small tree width (where by local we mean that a solution can be verified by checking separately the neighbourhood of each vertex), standard dynamic programming techniques give c^tw |V|^O(1) time algorithms, where tw is the tree width of the input graph G = (V, E) and c is a constant. On the other hand, for problems with a global requirement (usually connectivity) the best -- known algorithms were naive dynamic programming schemes running in at least tw^tw time. We breach this gap by introducing a technique we named Cut&Count that allows to produce c^tw |V|^O(1) time Monte Carlo algorithms for most connectivity-type problems, including Hamiltonian Path, Steiner Tree, Feedback Vertex Set and Connected Dominating Set. These results have numerous consequences in various fields, like parameterized complexity, exact and approximate algorithms on planar and H-minor-free graphs and exact algorithms on graphs of bounded degree. The constant c in our algorithms is in all cases small, and in several cases we are able to show that improving those constants would cause the Strong Exponential Time Hypothesis to fail. In contrast to the problems aiming to minimize the number of connected components that we solve using Cut&Count as mentioned above, we show that, assuming the Exponential Time Hypothesis, the aforementioned gap cannot be breached for some problems that aim to maximize the number of connected components like Cycle Packing.
{"title":"Solving Connectivity Problems Parameterized by Treewidth in Single Exponential Time","authors":"Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, J. M. M. Van Rooij, J. O. Wojtaszczyk","doi":"10.1145/3506707","DOIUrl":"https://doi.org/10.1145/3506707","url":null,"abstract":"For the vast majority of local problems on graphs of small tree width (where by local we mean that a solution can be verified by checking separately the neighbourhood of each vertex), standard dynamic programming techniques give c^tw |V|^O(1) time algorithms, where tw is the tree width of the input graph G = (V, E) and c is a constant. On the other hand, for problems with a global requirement (usually connectivity) the best -- known algorithms were naive dynamic programming schemes running in at least tw^tw time. We breach this gap by introducing a technique we named Cut&Count that allows to produce c^tw |V|^O(1) time Monte Carlo algorithms for most connectivity-type problems, including Hamiltonian Path, Steiner Tree, Feedback Vertex Set and Connected Dominating Set. These results have numerous consequences in various fields, like parameterized complexity, exact and approximate algorithms on planar and H-minor-free graphs and exact algorithms on graphs of bounded degree. The constant c in our algorithms is in all cases small, and in several cases we are able to show that improving those constants would cause the Strong Exponential Time Hypothesis to fail. In contrast to the problems aiming to minimize the number of connected components that we solve using Cut&Count as mentioned above, we show that, assuming the Exponential Time Hypothesis, the aforementioned gap cannot be breached for some problems that aim to maximize the number of connected components like Cycle Packing.","PeriodicalId":326048,"journal":{"name":"2011 IEEE 52nd Annual Symposium on Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128753266","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 studying how to communicate over a public channel with an active adversary, Dodis and Wichs introduced the notion of a non-malleable extractor. A non-malleable extractor dramatically strengthens the notion of a strong extractor. A strong extractor takes two inputs, a weakly-random $x$ and a uniformly random seed $y$, and outputs a string which appears uniform, even given $y$. For a non-malleable extractor $nm$, the output $nm(x,y)$ should appear uniform given $y$ as well as $nm(x,adv(y))$, where $adv$ is an arbitrary function with $adv(y) neq y$. We show that an extractor introduced by Chor and Gold reich is non-malleable when the entropy rate is above half. It outputs a linear number of bits when the entropy rate is $1/2 + alpha$, for any $alpha>0$. Previously, no nontrivial parameters were known for any non-malleable extractor. To achieve a polynomial running time when outputting many bits, we rely on a widely-believed conjecture about the distribution of prime numbers in arithmetic progressions. Our analysis involves a character sum estimate, which may be of independent interest. Using our non-malleable extractor, we obtain protocols for ``privacy amplification & quot;: key agreement between two parties who share a weakly-random secret. Our protocols work in the presence of an active adversary with unlimited computational power, and have asymptotically optimal entropy loss. When the secret has entropy rate greater than $1/2$, the protocol follows from a result of Dodis and Wichs, and takes two rounds. When the secret has entropy rate $delta$ for any constant~$delta>0$, our new protocol takes a constant (polynomial in $1/delta$) number of rounds. Our protocols run in polynomial time under the above well-known conjecture about primes.
{"title":"Privacy Amplification and Non-malleable Extractors via Character Sums","authors":"Y. Dodis, Xin Li, T. Wooley, David Zuckerman","doi":"10.1137/120868414","DOIUrl":"https://doi.org/10.1137/120868414","url":null,"abstract":"In studying how to communicate over a public channel with an active adversary, Dodis and Wichs introduced the notion of a non-malleable extractor. A non-malleable extractor dramatically strengthens the notion of a strong extractor. A strong extractor takes two inputs, a weakly-random $x$ and a uniformly random seed $y$, and outputs a string which appears uniform, even given $y$. For a non-malleable extractor $nm$, the output $nm(x,y)$ should appear uniform given $y$ as well as $nm(x,adv(y))$, where $adv$ is an arbitrary function with $adv(y) neq y$. We show that an extractor introduced by Chor and Gold reich is non-malleable when the entropy rate is above half. It outputs a linear number of bits when the entropy rate is $1/2 + alpha$, for any $alpha>0$. Previously, no nontrivial parameters were known for any non-malleable extractor. To achieve a polynomial running time when outputting many bits, we rely on a widely-believed conjecture about the distribution of prime numbers in arithmetic progressions. Our analysis involves a character sum estimate, which may be of independent interest. Using our non-malleable extractor, we obtain protocols for ``privacy amplification & quot;: key agreement between two parties who share a weakly-random secret. Our protocols work in the presence of an active adversary with unlimited computational power, and have asymptotically optimal entropy loss. When the secret has entropy rate greater than $1/2$, the protocol follows from a result of Dodis and Wichs, and takes two rounds. When the secret has entropy rate $delta$ for any constant~$delta>0$, our new protocol takes a constant (polynomial in $1/delta$) number of rounds. Our protocols run in polynomial time under the above well-known conjecture about primes.","PeriodicalId":326048,"journal":{"name":"2011 IEEE 52nd Annual Symposium on Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114489962","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 improved algorithm for solving symmetrically diagonally dominant linear systems. On input of an $ntimes n$ symmetric diagonally dominant matrix $A$ with $m$ non-zero entries and a vector $b$ such that $Abar{x} = b$ for some (unknown) vector $bar{x}$, our algorithm computes a vector $x$ such that $| |{x}-bar{x}| |_A1 in time. O tiled (m log n log (1/epsilon))^2. The solver utilizes in a standard way a 'preconditioning' chain of progressively sparser graphs. To claim the faster running time we make a two-fold improvement in the algorithm for constructing the chain. The new chain exploits previously unknown properties of the graph sparsification algorithm given in [Koutis,Miller,Peng, FOCS 2010], allowing for stronger preconditioning properties.We also present an algorithm of independent interest that constructs nearly-tight low-stretch spanning trees in time Otiled (mlog n), a factor of O (log n) faster than the algorithm in [Abraham,Bartal,Neiman, FOCS 2008]. This speedup directly reflects on the construction time of the preconditioning chain.
{"title":"A Nearly-m log n Time Solver for SDD Linear Systems","authors":"I. Koutis, G. Miller, Richard Peng","doi":"10.1109/FOCS.2011.85","DOIUrl":"https://doi.org/10.1109/FOCS.2011.85","url":null,"abstract":"We present an improved algorithm for solving symmetrically diagonally dominant linear systems. On input of an $ntimes n$ symmetric diagonally dominant matrix $A$ with $m$ non-zero entries and a vector $b$ such that $Abar{x} = b$ for some (unknown) vector $bar{x}$, our algorithm computes a vector $x$ such that $| |{x}-bar{x}| |_A1 in time. O tiled (m log n log (1/epsilon))^2. The solver utilizes in a standard way a 'preconditioning' chain of progressively sparser graphs. To claim the faster running time we make a two-fold improvement in the algorithm for constructing the chain. The new chain exploits previously unknown properties of the graph sparsification algorithm given in [Koutis,Miller,Peng, FOCS 2010], allowing for stronger preconditioning properties.We also present an algorithm of independent interest that constructs nearly-tight low-stretch spanning trees in time Otiled (mlog n), a factor of O (log n) faster than the algorithm in [Abraham,Bartal,Neiman, FOCS 2008]. This speedup directly reflects on the construction time of the preconditioning chain.","PeriodicalId":326048,"journal":{"name":"2011 IEEE 52nd Annual Symposium on Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125140763","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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