We give a polynomial-time quantum reduction from worst-case (ideal) lattice problems directly to decision (Ring-)LWE. This extends to decision all the worst-case hardness results that were previously known for the search version, for the same or even better parameters and with no algebraic restrictions on the modulus or number field. Indeed, our reduction is the first that works for decision Ring-LWE with any number field and any modulus.
{"title":"Pseudorandomness of ring-LWE for any ring and modulus","authors":"Chris Peikert, O. Regev, Noah Stephens-Davidowitz","doi":"10.1145/3055399.3055489","DOIUrl":"https://doi.org/10.1145/3055399.3055489","url":null,"abstract":"We give a polynomial-time quantum reduction from worst-case (ideal) lattice problems directly to decision (Ring-)LWE. This extends to decision all the worst-case hardness results that were previously known for the search version, for the same or even better parameters and with no algebraic restrictions on the modulus or number field. Indeed, our reduction is the first that works for decision Ring-LWE with any number field and any modulus.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83719433","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 talk surveys a series of works that lift the rich semantics and structure of graphs, and the experience of the formal-verification community in reasoning about them, to classical graph-theoretical problems.
{"title":"Examining classical graph-theory problems from the viewpoint of formal-verification methods (invited talk)","authors":"O. Kupferman","doi":"10.1145/3055399.3079075","DOIUrl":"https://doi.org/10.1145/3055399.3079075","url":null,"abstract":"The talk surveys a series of works that lift the rich semantics and structure of graphs, and the experience of the formal-verification community in reasoning about them, to classical graph-theoretical problems.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"192 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78744892","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}
Non-malleable codes were introduced by Dziembowski, Pietrzak and Wichs as an elegant relaxation of error correcting codes, where the motivation is to handle more general forms of tampering while still providing meaningful guarantees. This has led to many elegant constructions and applications in cryptography. However, most works so far only studied tampering in the split-state model where different parts of the codeword are tampered independently, and thus do not apply to many other natural classes of tampering functions. The only exceptions are the work of Agrawal et al. which studied non-malleable codes against bit permutation composed with bit-wise tampering, and the works of Faust et al. and Ball et al., which studied non-malleable codes against local functions. However, in both cases each tampered bit only depends on a subset of input bits. In this work, we study the problem of constructing non-malleable codes against more general tampering functions that act on the entire codeword. We give the first efficient constructions of non-malleable codes against tampering functions and affine tampering functions. These are the first explicit non-malleable codes against tampering functions where each tampered bit can depend on all input bits. We also give efficient non-malleable codes against t-local functions for t=o(√n), where a t-local function has the property that any output bit depends on at most t input bits. In the case of deterministic decoders, this improves upon the results of Ball et al, which can handle t≤ n1/4. All our results on non-malleable codes are obtained by using the connection between non-malleable codes and seedless non-malleable extractors discovered by Cheraghchi and Guruswami. Therefore, we also give the first efficient constructions of seedless non-malleable extractors against tampering functions, t-local tampering functions for t=o(√n), and affine tampering functions. To derive our results on non-malleable codes, we design efficient algorithms to almost uniformly sample from the pre-image of any given output of our non-malleable extractor.
{"title":"Non-malleable codes and extractors for small-depth circuits, and affine functions","authors":"Eshan Chattopadhyay, Xin Li","doi":"10.1145/3055399.3055483","DOIUrl":"https://doi.org/10.1145/3055399.3055483","url":null,"abstract":"Non-malleable codes were introduced by Dziembowski, Pietrzak and Wichs as an elegant relaxation of error correcting codes, where the motivation is to handle more general forms of tampering while still providing meaningful guarantees. This has led to many elegant constructions and applications in cryptography. However, most works so far only studied tampering in the split-state model where different parts of the codeword are tampered independently, and thus do not apply to many other natural classes of tampering functions. The only exceptions are the work of Agrawal et al. which studied non-malleable codes against bit permutation composed with bit-wise tampering, and the works of Faust et al. and Ball et al., which studied non-malleable codes against local functions. However, in both cases each tampered bit only depends on a subset of input bits. In this work, we study the problem of constructing non-malleable codes against more general tampering functions that act on the entire codeword. We give the first efficient constructions of non-malleable codes against tampering functions and affine tampering functions. These are the first explicit non-malleable codes against tampering functions where each tampered bit can depend on all input bits. We also give efficient non-malleable codes against t-local functions for t=o(√n), where a t-local function has the property that any output bit depends on at most t input bits. In the case of deterministic decoders, this improves upon the results of Ball et al, which can handle t≤ n1/4. All our results on non-malleable codes are obtained by using the connection between non-malleable codes and seedless non-malleable extractors discovered by Cheraghchi and Guruswami. Therefore, we also give the first efficient constructions of seedless non-malleable extractors against tampering functions, t-local tampering functions for t=o(√n), and affine tampering functions. To derive our results on non-malleable codes, we design efficient algorithms to almost uniformly sample from the pre-image of any given output of our non-malleable extractor.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79918829","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 the strong-convexity assumption of regularization-based methods for solving bilinear saddle point problems may be relaxed to a weaker notion of area-convexity with respect to an alternating bilinear form. This allows bypassing the infamous '' barrier for strongly convex regularizers that has stalled progress on a number of algorithmic problems. Applying area-convex regularization, we present a nearly-linear time approximation algorithm for solving matrix inequality systems A X ≤ B over right-stochastic matrices X. By combining that algorithm with existing work on preconditioning maximum-flow, we obtain a nearly-linear time approximation algorithm for maximum concurrent flow in undirected graphs: given an undirected, capacitated graph with m edges and k demand vectors, the algorithm takes Õ(mkε'1) time and outputs k flows routing the specified demands with total congestion at most (1+ε) times optimal.
{"title":"Area-convexity, l∞ regularization, and undirected multicommodity flow","authors":"Jonah Sherman","doi":"10.1145/3055399.3055501","DOIUrl":"https://doi.org/10.1145/3055399.3055501","url":null,"abstract":"We show the strong-convexity assumption of regularization-based methods for solving bilinear saddle point problems may be relaxed to a weaker notion of area-convexity with respect to an alternating bilinear form. This allows bypassing the infamous '' barrier for strongly convex regularizers that has stalled progress on a number of algorithmic problems. Applying area-convex regularization, we present a nearly-linear time approximation algorithm for solving matrix inequality systems A X ≤ B over right-stochastic matrices X. By combining that algorithm with existing work on preconditioning maximum-flow, we obtain a nearly-linear time approximation algorithm for maximum concurrent flow in undirected graphs: given an undirected, capacitated graph with m edges and k demand vectors, the algorithm takes Õ(mkε'1) time and outputs k flows routing the specified demands with total congestion at most (1+ε) times optimal.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"15 6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76450752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","authors":"Hamed Hatami, P. McKenzie, Valerie King","doi":"10.1145/3055399","DOIUrl":"https://doi.org/10.1145/3055399","url":null,"abstract":"","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77555691","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 de Morgan formula over Boolean variables x1,…,xn is a binary tree whose internal nodes are marked with AND or OR gates and whose leaves are marked with variables or their negation. We define the size of the formula as the number of leaves in it. Proving that some explicit function (in P or NP) requires a large formula is a central open question in computational complexity. While we believe that some explicit functions require exponential formula size, currently the best lower bound for an explicit function is the Ω(n3) lower bound for Andreev's function. A long line of work in quantum query complexity, culminating in the work of Reichardt [SODA, 2011], proved that for any formula of size s, there exists a polynomial of degree at most O(√s) that approximates the formula up to a small point-wise error. This is a classical theorem, arguing about polynomials and formulae, however the only known proof for it involves quantum algorithms. We apply Reichardt result to obtain the following: (1) We show how to trade average-case hardness in exchange for size. More precisely, we show that if a function f cannot be computed correctly on more than 1/2 + 2-k of the inputs by any formula of size at most s, then computing f exactly requires formula size at least Ω(k) · s. As an application, we improve the state of the art formula size lower bounds for explicit functions by a factor of Ω(logn). (2) We prove that the bipartite formula size of the Inner-Product function is Ω(n2). (A bipartite formula on Boolean variables x1,…,xn and y1, …, yn is a binary tree whose internal nodes are marked with AND or OR gates and whose leaves can compute any function of either the x or y variables.) We show that any bipartite formula for the Inner-Product modulo 2 function, namely IP(x,y) = Σi=1n xi yi (mod 2), must be of size Ω(n2), which is tight up to logarithmic factors. To the best of our knowledge, this is the first super-linear lower bound on the bipartite formula complexity of any explicit function.
布尔变量x1，…，xn上的de Morgan公式是一棵二叉树，其内部节点用与或门标记，其叶子用变量或它们的负值标记。我们将公式的大小定义为其中叶节点的数量。证明某些显式函数(在P或NP中)需要一个大的公式是计算复杂性中的一个中心开放问题。虽然我们认为一些显式函数需要指数公式大小，但目前显式函数的最佳下界是Andreev函数的Ω(n3)下界。在量子查询复杂性方面的大量工作，最终以Reichardt [SODA, 2011]的工作为高潮，证明了对于任何大小为s的公式，存在一个至多为O(√s)次的多项式，该多项式近似于该公式，直至小的逐点误差。这是一个经典定理，讨论多项式和公式，然而唯一已知的证明涉及量子算法。我们应用Reichardt结果得到以下结果:(1)我们展示了如何用平均硬度交换尺寸。更准确地说，我们表明，如果一个函数f不能通过任何大小最多为s的公式在超过1/2 + 2-k的输入上正确计算，那么精确计算f至少需要公式大小Ω(k)·s。作为一个应用，我们将显式函数的公式大小下界的最新状态提高了Ω(logn)。(2)证明了内积函数的二部公式大小为Ω(n2)。(布尔变量x1，…，xn和y1，…，yn的二部公式是一棵二叉树，其内部节点标记为与或门，其叶子可以计算x或y变量的任何函数。)我们证明了内积模2函数的任何二部公式，即IP(x,y) = Σi=1n xi yi (mod 2)，其大小必须是Ω(n2)，它紧于对数因子。据我们所知，这是任何显式函数的二部公式复杂度的第一个超线性下界。
{"title":"Formula lower bounds via the quantum method","authors":"Avishay Tal","doi":"10.1145/3055399.3055472","DOIUrl":"https://doi.org/10.1145/3055399.3055472","url":null,"abstract":"A de Morgan formula over Boolean variables x1,…,xn is a binary tree whose internal nodes are marked with AND or OR gates and whose leaves are marked with variables or their negation. We define the size of the formula as the number of leaves in it. Proving that some explicit function (in P or NP) requires a large formula is a central open question in computational complexity. While we believe that some explicit functions require exponential formula size, currently the best lower bound for an explicit function is the Ω(n3) lower bound for Andreev's function. A long line of work in quantum query complexity, culminating in the work of Reichardt [SODA, 2011], proved that for any formula of size s, there exists a polynomial of degree at most O(√s) that approximates the formula up to a small point-wise error. This is a classical theorem, arguing about polynomials and formulae, however the only known proof for it involves quantum algorithms. We apply Reichardt result to obtain the following: (1) We show how to trade average-case hardness in exchange for size. More precisely, we show that if a function f cannot be computed correctly on more than 1/2 + 2-k of the inputs by any formula of size at most s, then computing f exactly requires formula size at least Ω(k) · s. As an application, we improve the state of the art formula size lower bounds for explicit functions by a factor of Ω(logn). (2) We prove that the bipartite formula size of the Inner-Product function is Ω(n2). (A bipartite formula on Boolean variables x1,…,xn and y1, …, yn is a binary tree whose internal nodes are marked with AND or OR gates and whose leaves can compute any function of either the x or y variables.) We show that any bipartite formula for the Inner-Product modulo 2 function, namely IP(x,y) = Σi=1n xi yi (mod 2), must be of size Ω(n2), which is tight up to logarithmic factors. To the best of our knowledge, this is the first super-linear lower bound on the bipartite formula complexity of any explicit function.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88014712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a universal constant α > 0 we prove size lower bounds of 2α(n) for an explicit function in monotone NP in the following models of computation: monotone formulas, monotone switching networks, monotone span programs, and monotone comparator circuits, where n is the number of variables of the underlying function. Our lower bounds improve on the best previous bounds in each of these models, and are the best possible for any function up to constant factors in the exponent. Moreover, we give one unified proof that is short and fairly elementary.
{"title":"Strongly exponential lower bounds for monotone computation","authors":"T. Pitassi, Robert Robere","doi":"10.1145/3055399.3055478","DOIUrl":"https://doi.org/10.1145/3055399.3055478","url":null,"abstract":"For a universal constant α > 0 we prove size lower bounds of 2α(n) for an explicit function in monotone NP in the following models of computation: monotone formulas, monotone switching networks, monotone span programs, and monotone comparator circuits, where n is the number of variables of the underlying function. Our lower bounds improve on the best previous bounds in each of these models, and are the best possible for any function up to constant factors in the exponent. Moreover, we give one unified proof that is short and fairly elementary.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"71 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90490550","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}
Lattice-based cryptography offers some of the most attractive primitives believed to be resistant to quantum computers. This work introduces "Frodo" - a concrete instantiation of a key agreement mechanism based on hard problems in generic lattices.
{"title":"Practical post-quantum key agreement from generic lattices (invited talk)","authors":"V. Nikolaenko","doi":"10.1145/3055399.3079078","DOIUrl":"https://doi.org/10.1145/3055399.3079078","url":null,"abstract":"Lattice-based cryptography offers some of the most attractive primitives believed to be resistant to quantum computers. This work introduces \"Frodo\" - a concrete instantiation of a key agreement mechanism based on hard problems in generic lattices.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"146 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77725777","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 a new strongly polynomial algorithm for generalized flow maximization. The first strongly polynomial algorithm for this problem was given very recently by Végh; our new algorithm is much simpler, and much faster. The complexity bound O((m+nlogn)mnlog(n2/m)) improves on the previous estimate obtained by Végh by almost a factor O(n2). Even for small numerical parameter values, our algorithm is essentially as fast as the best weakly polynomial algorithms. The key new technical idea is relaxing primal feasibility conditions. This allows us to work almost exclusively with integral flows, in contrast to all previous algorithms.
{"title":"A simpler and faster strongly polynomial algorithm for generalized flow maximization","authors":"Neil Olver, László A. Végh","doi":"10.1145/3055399.3055439","DOIUrl":"https://doi.org/10.1145/3055399.3055439","url":null,"abstract":"We present a new strongly polynomial algorithm for generalized flow maximization. The first strongly polynomial algorithm for this problem was given very recently by Végh; our new algorithm is much simpler, and much faster. The complexity bound O((m+nlogn)mnlog(n2/m)) improves on the previous estimate obtained by Végh by almost a factor O(n2). Even for small numerical parameter values, our algorithm is essentially as fast as the best weakly polynomial algorithms. The key new technical idea is relaxing primal feasibility conditions. This allows us to work almost exclusively with integral flows, in contrast to all previous algorithms.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73887984","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}
Our first result shows membership in PPAD for the problem of computing approximate equilibria for an Arrow-Debreu exchange market for piecewise-linear concave (PLC) utility functions. As a corollary we also obtain membership in PPAD for Leontief utility functions. This settles an open question of Vazirani and Yannakakis (2011). Next we show FIXP-hardness of computing equilibria in Arrow-Debreu exchange markets under Leontief utility functions, and Arrow-Debreu markets under linear utility functions and Leontief production sets, thereby settling these open questions of Vazirani and Yannakakis (2011). As corollaries, we obtain FIXP-hardness for PLC utilities and for Arrow-Debreu markets under linear utility functions and polyhedral production sets. In all cases, as required under FIXP, the set of instances mapped onto will admit equilibria, i.e., will be "yes" instances. If all instances are under consideration, then in all cases we prove that the problem of deciding if a given instance admits an equilibrium is ETR-complete, where ETR is the class Existential Theory of Reals. As a consequence of the results stated above, and the fact that membership in FIXP has been established for PLC utilities, the entire computational difficulty of Arrow-Debreu markets under PLC utility functions lies in the Leontief utility subcase. This is perhaps the most unexpected aspect of our result, since Leontief utilities are meant for the case that goods are perfect complements, whereas PLC utilities are very general, capturing not only the cases when goods are complements and substitutes, but also arbitrary combinations of these and much more. Finally, we give a polynomial time algorithm for finding an equilibrium in Arrow-Debreu exchange markets under Leontief utility functions provided the number of agents is a constant. This settles part of an open problem of Devanur and Kannan (2008).
{"title":"Settling the complexity of Leontief and PLC exchange markets under exact and approximate equilibria","authors":"J. Garg, R. Mehta, V. Vazirani, Sadra Yazdanbod","doi":"10.1145/3055399.3055474","DOIUrl":"https://doi.org/10.1145/3055399.3055474","url":null,"abstract":"Our first result shows membership in PPAD for the problem of computing approximate equilibria for an Arrow-Debreu exchange market for piecewise-linear concave (PLC) utility functions. As a corollary we also obtain membership in PPAD for Leontief utility functions. This settles an open question of Vazirani and Yannakakis (2011). Next we show FIXP-hardness of computing equilibria in Arrow-Debreu exchange markets under Leontief utility functions, and Arrow-Debreu markets under linear utility functions and Leontief production sets, thereby settling these open questions of Vazirani and Yannakakis (2011). As corollaries, we obtain FIXP-hardness for PLC utilities and for Arrow-Debreu markets under linear utility functions and polyhedral production sets. In all cases, as required under FIXP, the set of instances mapped onto will admit equilibria, i.e., will be \"yes\" instances. If all instances are under consideration, then in all cases we prove that the problem of deciding if a given instance admits an equilibrium is ETR-complete, where ETR is the class Existential Theory of Reals. As a consequence of the results stated above, and the fact that membership in FIXP has been established for PLC utilities, the entire computational difficulty of Arrow-Debreu markets under PLC utility functions lies in the Leontief utility subcase. This is perhaps the most unexpected aspect of our result, since Leontief utilities are meant for the case that goods are perfect complements, whereas PLC utilities are very general, capturing not only the cases when goods are complements and substitutes, but also arbitrary combinations of these and much more. Finally, we give a polynomial time algorithm for finding an equilibrium in Arrow-Debreu exchange markets under Leontief utility functions provided the number of agents is a constant. This settles part of an open problem of Devanur and Kannan (2008).","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82147126","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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