Submodular maximization generalizes many important problems including Max Cut in directed/undirected graphs and hypergraphs, certain constraint satisfaction problems and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular functions is NP-hard.
{"title":"Maximizing Non-Monotone Submodular Functions","authors":"U. Feige, V. Mirrokni, J. Vondrák","doi":"10.1109/FOCS.2007.29","DOIUrl":"https://doi.org/10.1109/FOCS.2007.29","url":null,"abstract":"Submodular maximization generalizes many important problems including Max Cut in directed/undirected graphs and hypergraphs, certain constraint satisfaction problems and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular functions is NP-hard.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130313977","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 reexamine, what it means to compute Nash equilibria and, more, generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Gamma, with 3 or more players, and given epsiv > 0, compute a vector x' (a mixed strategy profile) that is within distance e (say in t^) of some (exact) Nash equilibrium. We show that approximation of an (actual) Nash equilibrium for games with 3 players, even to within any non-trivial constant additive factor epsiv < 1/2 in just one desired coordinate, is at least as hard as the long standing square-root sum problem, as well as more general arithmetic circuit decision problems, and thus that even placing the approximation problem in NP would-resolve a major open problem in the complexity of numerical computation. Furthermore, we show that the (exact or approximate) computation of Nash equilibria for 3 or more players is complete for the class of search problems, which we call FIXP, that can be cast as fixed point computation problems for functions represented by algebraic circuits (straight line programs) over basis {+, *, -, /, max, min}, with rational constants. We show that the linear fragment of FIXP equals PPAD. Many problems in game theory, economics, and probability theory, can be cast as fixed point problems for such algebraic functions. We discuss several important such problems: computing the value of Shapley's stochastic games, and the simpler games of Condon, extinction probabilities of branching processes, termination probabilities of stochastic context-free grammars, and of Recursive Markov Chains. We show that for some of them, the approximation, or even exact computation, problem can be placed-in PPAD, while for others, they are at least as hard as the square-root sum and arithmetic circuit decision problems.
{"title":"On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract)","authors":"K. Etessami, M. Yannakakis","doi":"10.1137/080720826","DOIUrl":"https://doi.org/10.1137/080720826","url":null,"abstract":"We reexamine, what it means to compute Nash equilibria and, more, generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Gamma, with 3 or more players, and given epsiv > 0, compute a vector x' (a mixed strategy profile) that is within distance e (say in t^) of some (exact) Nash equilibrium. We show that approximation of an (actual) Nash equilibrium for games with 3 players, even to within any non-trivial constant additive factor epsiv < 1/2 in just one desired coordinate, is at least as hard as the long standing square-root sum problem, as well as more general arithmetic circuit decision problems, and thus that even placing the approximation problem in NP would-resolve a major open problem in the complexity of numerical computation. Furthermore, we show that the (exact or approximate) computation of Nash equilibria for 3 or more players is complete for the class of search problems, which we call FIXP, that can be cast as fixed point computation problems for functions represented by algebraic circuits (straight line programs) over basis {+, *, -, /, max, min}, with rational constants. We show that the linear fragment of FIXP equals PPAD. Many problems in game theory, economics, and probability theory, can be cast as fixed point problems for such algebraic functions. We discuss several important such problems: computing the value of Shapley's stochastic games, and the simpler games of Condon, extinction probabilities of branching processes, termination probabilities of stochastic context-free grammars, and of Recursive Markov Chains. We show that for some of them, the approximation, or even exact computation, problem can be placed-in PPAD, while for others, they are at least as hard as the square-root sum and arithmetic circuit decision problems.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114858866","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}
Pub Date : 2008-07-01DOI: 10.1109/focs.2007.4389503
Subhash Khot, A. Naor
We design a randomized polynomial time algorithm which, given a 3-tensor of real numbers A={aijk}ij,k=1n such that for all i,j,kisin{1,...,n} we have aijk=aikj=akji=ajik=akij=akji and aiik=aijj=aiji=0, computes a number Alg(A) which satisfies with probability at least 1/2, Omega(radic(logn/n))ldrmaxxisin{-1,1}n Sigmai,j,k=1naijkxixjxklesAlg(A)lesmaxxisin{-1,1}n Sigmai,j,k=1naijkxixjxk. On the other hand, we show via a simple reduction from a result of Hastad and Venkatesh that under the assumption NPnsubeDTIME(n(logn)O(1)),for every epsiv>0 there is no algorithm that approximates maxxisin{-1,1}n Sigmai,j,k=1naijkxixjxk within a factor of 2(logn)t-epsiv in time 2(logn)O(1). Our algorithm is based on a reduction to the problem of computing the diameter of a convex body in Rn with respect to the L1 norm. We show that it is possible to do so up to a multiplicative error of O(radic(n/logn)), while no randomized polynomial time algorithm can achieve accuracy O(radic(n/logn)). This resolves a question posed by Brieden, Gritzmann, Kantian, Klee, Lovasz and Simonos. We apply our new algorithm improve the algorithm of Hastad and Venkatesh or the Max-E3-Lin-2 problem. Given an over-determined system epsiv of N linear equations modulo 2 in nlesN Boolean variables, such that in each equation appear only three distinct variables, the goal is to approximate in polynomial time the maximum number of satisfiable equations in epsiv minus N/2 (i.e. we subtract the expected number of satisfied equations in a random assignment). Hastad and Venkatesh obtained an algorithm which approximates this value up to a factor of O(radicN). We obtain a O(radic(n/logn)) approximation algorithm. By relating this problem to the refutation problem for random 3-CNF formulas we give evidence that obtaining a significant improvement over this approximation factor is likely to be difficult.
我们设计了一个随机多项式时间算法，给定实数的3张量a ={aijk}ij,k=1 n，使得对于所有i,j,kisin{1，…，n}我们有aijk=aikj=akji=ajik=akij=akji和aiik=aijj=aiji= akji，计算一个数Alg(a)它满足概率至少为1/2,Omega(radic(logn/n))ldrmaxxisin{-1,1} n Sigmai,j,k=1 naijkxixjkklesalg (a)lesmaxxisin{-1,1} n Sigmai,j,k=1 naijkxixjxxjxk。另一方面，我们通过对Hastad和Venkatesh的结果的简单简化表明，在假设NPnsubeDTIME(n(logn) O(1))下，对于每个epsiv>0，没有算法可以在2(logn) O(1)时间内在2(logn)t-epsiv的因子内逼近maxxisin{-1,1} n Sigmai,j,k=1 naijkxixjxk。我们的算法是基于一个简化的问题，计算一个凸体的直径在Rn相对于L1范数。我们证明，它可以达到O(radic(n/logn))的乘法误差，而没有随机多项式时间算法可以达到O(radic(n/logn))的精度。这就解决了布里登、格里茨曼、康德、克利、洛瓦兹和西蒙诺斯提出的一个问题。我们应用我们的新算法改进了haad和Venkatesh的算法或Max-E3-Lin-2问题。给定一个过度确定的系统epsiv (N个线性方程，在nlesN布尔变量中取模2)，使得在每个方程中只出现三个不同的变量，目标是在多项式时间内近似epsiv - N/2中可满足方程的最大数量(即我们减去随机分配中预期的满足方程的数量)。Hastad和Venkatesh得到了一种算法，该算法将该值逼近到0 (radicN)的因数。我们得到了一个O(radic(n/logn))近似算法。通过将这个问题与随机3-CNF公式的反驳问题联系起来，我们给出了证据，证明在这个近似因子上获得显着改进可能是困难的。
{"title":"Linear Equations Modulo 2 and the L1 Diameter of Convex Bodies","authors":"Subhash Khot, A. Naor","doi":"10.1109/focs.2007.4389503","DOIUrl":"https://doi.org/10.1109/focs.2007.4389503","url":null,"abstract":"We design a randomized polynomial time algorithm which, given a 3-tensor of real numbers A={a<sub>ijk</sub>}<sub>ij,k=1</sub> <sup>n</sup> such that for all i,j,kisin{1,...,n} we have a<sub>ijk</sub>=a<sub>ikj</sub>=a<sub>kji</sub>=a<sub>jik</sub>=a<sub>kij</sub>=a<sub>kji</sub> and a<sub>iik</sub>=a<sub>ijj</sub>=a<sub>iji</sub>=0, computes a number Alg(A) which satisfies with probability at least 1/2, Omega(radic(logn/n))ldrmax<sub>xisin{-1,1}</sub> <sup>n</sup> Sigma<sub>i,j,k=1</sub> <sup>n</sup>a<sub>ijk</sub>x<sub>i</sub>x<sub>j</sub>x<sub>k</sub>lesAlg(A)lesmax<sub>xisin{-1,1}</sub> <sup>n</sup> Sigma<sub>i,j,k=1</sub> <sup>n</sup>a<sub>ijk</sub>x<sub>i</sub>x<sub>j</sub>x<sub>k</sub>. On the other hand, we show via a simple reduction from a result of Hastad and Venkatesh that under the assumption NPnsubeDTIME(n<sup>(logn)</sup> <sup>O(1)</sup>),for every epsiv>0 there is no algorithm that approximates max<sub>xisin{-1,1}</sub> <sup>n</sup> Sigma<sub>i,j,k=1</sub> <sup>n</sup>a<sub>ijk</sub>x<sub>i</sub>x<sub>j</sub>x<sub>k</sub> within a factor of 2(logn)<sup>t-epsiv</sup> in time 2<sup>(logn)</sup> <sup>O(1)</sup>. Our algorithm is based on a reduction to the problem of computing the diameter of a convex body in R<sup>n</sup> with respect to the L<sub>1</sub> norm. We show that it is possible to do so up to a multiplicative error of O(radic(n/logn)), while no randomized polynomial time algorithm can achieve accuracy O(radic(n/logn)). This resolves a question posed by Brieden, Gritzmann, Kantian, Klee, Lovasz and Simonos. We apply our new algorithm improve the algorithm of Hastad and Venkatesh or the Max-E3-Lin-2 problem. Given an over-determined system epsiv of N linear equations modulo 2 in nlesN Boolean variables, such that in each equation appear only three distinct variables, the goal is to approximate in polynomial time the maximum number of satisfiable equations in epsiv minus N/2 (i.e. we subtract the expected number of satisfied equations in a random assignment). Hastad and Venkatesh obtained an algorithm which approximates this value up to a factor of O(radicN). We obtain a O(radic(n/logn)) approximation algorithm. By relating this problem to the refutation problem for random 3-CNF formulas we give evidence that obtaining a significant improvement over this approximation factor is likely to be difficult.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132082412","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 construct an explicit polynomial f(x1,..., xn), with coefficients in {0, 1}, such that the size of any syntactically multilinear arithmetic circuit computing f is at least Omega{n4/3 log2 n} The lower bound holds over any field.
{"title":"A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits","authors":"R. Raz, Amir Shpilka, A. Yehudayoff","doi":"10.1137/070707932","DOIUrl":"https://doi.org/10.1137/070707932","url":null,"abstract":"We construct an explicit polynomial f(x<sub>1</sub>,..., x<sub>n</sub>), with coefficients in {0, 1}, such that the size of any syntactically multilinear arithmetic circuit computing f is at least Omega{n<sup>4/3</sup> log<sup>2</sup> n} The lower bound holds over any field.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131059467","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 any deterministic data-stream algorithm that, makes a constant number of passes over the input and gives a constant, factor approximation of the length of the longest increasing subsequence in a sequence of length n must use space Omega(radicn). This proves a conjecture made by Gopalan, Jayram, Krauthgamer and Kumar |10| who proved a matching upper bound. Our results yield asymptotically tight tower bounds for all approximation factors, thus resolving the main open problem, from their paper. Our proof is based on analyzing a related communication problem and proving a direct sum type property for it.
{"title":"Lower Bounds on Streaming Algorithms for Approximating the Length of the Longest Increasing Subsequence","authors":"A. Gál, Parikshit Gopalan","doi":"10.1137/090770801","DOIUrl":"https://doi.org/10.1137/090770801","url":null,"abstract":"We show that any deterministic data-stream algorithm that, makes a constant number of passes over the input and gives a constant, factor approximation of the length of the longest increasing subsequence in a sequence of length n must use space Omega(radicn). This proves a conjecture made by Gopalan, Jayram, Krauthgamer and Kumar |10| who proved a matching upper bound. Our results yield asymptotically tight tower bounds for all approximation factors, thus resolving the main open problem, from their paper. Our proof is based on analyzing a related communication problem and proving a direct sum type property for it.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121975627","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 analyze-a fairly standard idealization of Pollard's rho algorithm for finding the discrete logarithm in acyclic group G. It is found that, with high probability, a collision occurs in O(radic( |G|log|G|log log|G|)) steps, not far from the widely conjectured value of Theta(radic|G|). Tins improves upon a recent result of Miller-Venkalesan which showed an upper bound of O(radic|G|log3|G|). Our proof is based on analyzing an appropriate nonreversible, non-lazy random walk on a discrete cycle of (odd) length |G|, and showing that the mixing time of the corresponding walk is O(log|G|log log|G|).
{"title":"Near Optimal Bounds for Collision in Pollard Rho for Discrete Log","authors":"J. Kim, R. Montenegro, P. Tetali","doi":"10.1109/FOCS.2007.44","DOIUrl":"https://doi.org/10.1109/FOCS.2007.44","url":null,"abstract":"We analyze-a fairly standard idealization of Pollard's rho algorithm for finding the discrete logarithm in acyclic group G. It is found that, with high probability, a collision occurs in O(radic( |G|log|G|log log|G|)) steps, not far from the widely conjectured value of Theta(radic|G|). Tins improves upon a recent result of Miller-Venkalesan which showed an upper bound of O(radic|G|log3|G|). Our proof is based on analyzing an appropriate nonreversible, non-lazy random walk on a discrete cycle of (odd) length |G|, and showing that the mixing time of the corresponding walk is O(log|G|log log|G|).","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":"77 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127558781","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}
This note provides a brief summary of how a new algebraic tool, bilinear groups, is transforming public-key cryptography. For the examples mentioned, the best solutions without bilinear groups either do not exist or are far less efficient. Many of the systems discussed in this note were implemented by Lynn [45] in a software library freely available under the GPL.
{"title":"A Brief Look at Pairings Based Cryptography","authors":"D. Boneh","doi":"10.1109/FOCS.2007.5","DOIUrl":"https://doi.org/10.1109/FOCS.2007.5","url":null,"abstract":"This note provides a brief summary of how a new algebraic tool, bilinear groups, is transforming public-key cryptography. For the examples mentioned, the best solutions without bilinear groups either do not exist or are far less efficient. Many of the systems discussed in this note were implemented by Lynn [45] in a software library freely available under the GPL.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":"82 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122476595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We. introduce, a framework for studying semidefiniie programming (SOP) relaxations based on the Lasserre hierarchy in the context of approximation algorithms for combinatorial problems. As an application of our approach, we give, improved approximation algorithms for two problems. We show that for some fixed constant epsiv > 0, given a 3-uniform hypergraph containing an independent set of size (1/2 - epsiv)v, we can find an independent set of size Omega(nepsiv). This improves upon the result of Krivelevich, Nathaniel and Sitdakov, who gave an algorithm finding an independent set of size Omega(n6gamma-3) for hypergraphs with an independent set of size gamman (but no guarantee for gamma les 1/2). We also give an algorithm which finds an O(n0.2072)-coloring given a 3-colorable graph, improving upon the work of Aurora, Clamtac and Charikar. Our approach stands in contrast to a long series of inapproximability results in the Lovasz Schrijver linear programming (LP) and SDP hierarchies for other problems.
{"title":"Approximation Algorithms Using Hierarchies of Semidefinite Programming Relaxations","authors":"E. Chlamtáč","doi":"10.1109/FOCS.2007.13","DOIUrl":"https://doi.org/10.1109/FOCS.2007.13","url":null,"abstract":"We. introduce, a framework for studying semidefiniie programming (SOP) relaxations based on the Lasserre hierarchy in the context of approximation algorithms for combinatorial problems. As an application of our approach, we give, improved approximation algorithms for two problems. We show that for some fixed constant epsiv > 0, given a 3-uniform hypergraph containing an independent set of size (1/2 - epsiv)v, we can find an independent set of size Omega(nepsiv). This improves upon the result of Krivelevich, Nathaniel and Sitdakov, who gave an algorithm finding an independent set of size Omega(n6gamma-3) for hypergraphs with an independent set of size gamman (but no guarantee for gamma les 1/2). We also give an algorithm which finds an O(n0.2072)-coloring given a 3-colorable graph, improving upon the work of Aurora, Clamtac and Charikar. Our approach stands in contrast to a long series of inapproximability results in the Lovasz Schrijver linear programming (LP) and SDP hierarchies for other problems.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":"297 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132584878","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}
Iftach Haitner, Jonathan J. Hoch, Omer Reingold, G. Segev
We study the round complexity of various cryptographic protocols. Our main result is a tight lower bound on the round complexity of any fully-black-box construction of a statistically-hiding commitment scheme from oneway permutations, and even front trapdoor permutations. This lower bound matches the round complexity of the statistically-hiding commitment scheme due to Naor, Ostrovsky, Venkatesan and Yung (CRYPTO '92). As a corollary, we derive similar tight lower bounds for several other ctyptographicprotocols, such as single-server private information retrieval, interactive hashing, and oblivious transfer that guarantees statistical security for one of the parties. Our techniques extend the collision-finding oracle due to Simon (EUROCRYPT '98) to the setting of interactive protocols (our extension also implies an alternative proof for the main property of the original oracle). In addition, we substantially extend the reconstruction paradigm of Gennaro and Trevisan (FOCS '00). In both cases, our extensions are quite delicate and may be found useful in proving additional black-box separation results.
{"title":"Finding Collisions in Interactive Protocols - A Tight Lower Bound on the Round Complexity of Statistically-Hiding Commitments","authors":"Iftach Haitner, Jonathan J. Hoch, Omer Reingold, G. Segev","doi":"10.1109/FOCS.2007.27","DOIUrl":"https://doi.org/10.1109/FOCS.2007.27","url":null,"abstract":"We study the round complexity of various cryptographic protocols. Our main result is a tight lower bound on the round complexity of any fully-black-box construction of a statistically-hiding commitment scheme from oneway permutations, and even front trapdoor permutations. This lower bound matches the round complexity of the statistically-hiding commitment scheme due to Naor, Ostrovsky, Venkatesan and Yung (CRYPTO '92). As a corollary, we derive similar tight lower bounds for several other ctyptographicprotocols, such as single-server private information retrieval, interactive hashing, and oblivious transfer that guarantees statistical security for one of the parties. Our techniques extend the collision-finding oracle due to Simon (EUROCRYPT '98) to the setting of interactive protocols (our extension also implies an alternative proof for the main property of the original oracle). In addition, we substantially extend the reconstruction paradigm of Gennaro and Trevisan (FOCS '00). In both cases, our extensions are quite delicate and may be found useful in proving additional black-box separation results.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":"227 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133608037","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 combinatorial complexity of the. union of n "fat" tetrahedra in 3-space (i.e., tetrahedra all of whose solid angles are at least .some fixed constant) of arbitrary sizes, is O(n2+epsiv),for any epsiv > 0: the bound is almost tight in the worst case, thus almost settling a conjecture of Pach el al. [24]. Our result extends, in a significant way, the result of Pach et al. [24] for the restricted case of nearly congruent cubes. The analysis uses cuttings, combined with the Dobkin-K'irkpatrick hierarchical decomposition of convex polytopes, in order to partition space into subcells, so that, on average, the overwhelming majority of the tetrahedra intersecting a subcell Delta behave as fat dihedral wedges in Delta. As an immediate corollary, we obtain that the combinatorial complexity of the union of n cubes in R3 having arbitrary side lengths, is O(n2+epsiv), for any epsiv > 0 again, significantly extending the result of [24]. Our analysis can easily he extended to yield a nearly-quadratic bound on the complexity of the union of arbitrarily oriented fat triangular prisms (whose cross-sections have, arbitrary sizes) in R3. Finally, we show that a simple variant of our analysis implies a nearly-linear bound on the complexity of the union of fat triangles in the plane.
{"title":"Almost Tight Bound for the Union of Fat Tetrahedra in Three Dimensions","authors":"Esther Ezra, M. Sharir","doi":"10.1109/FOCS.2007.9","DOIUrl":"https://doi.org/10.1109/FOCS.2007.9","url":null,"abstract":"We show that the combinatorial complexity of the. union of n \"fat\" tetrahedra in 3-space (i.e., tetrahedra all of whose solid angles are at least .some fixed constant) of arbitrary sizes, is O(n2+epsiv),for any epsiv > 0: the bound is almost tight in the worst case, thus almost settling a conjecture of Pach el al. [24]. Our result extends, in a significant way, the result of Pach et al. [24] for the restricted case of nearly congruent cubes. The analysis uses cuttings, combined with the Dobkin-K'irkpatrick hierarchical decomposition of convex polytopes, in order to partition space into subcells, so that, on average, the overwhelming majority of the tetrahedra intersecting a subcell Delta behave as fat dihedral wedges in Delta. As an immediate corollary, we obtain that the combinatorial complexity of the union of n cubes in R3 having arbitrary side lengths, is O(n2+epsiv), for any epsiv > 0 again, significantly extending the result of [24]. Our analysis can easily he extended to yield a nearly-quadratic bound on the complexity of the union of arbitrarily oriented fat triangular prisms (whose cross-sections have, arbitrary sizes) in R3. Finally, we show that a simple variant of our analysis implies a nearly-linear bound on the complexity of the union of fat triangles in the plane.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126742119","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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