Assume that for every derandomization result for logspace algorithms, there is a pseudorandom generator strong enough to nearly recover the derandomization by iterating over all seeds and taking a majority vote. We prove under a precise version of this assumption that BPL ⊆ ∩α > 0 DSPACE(log1 + α n). We strengthen the theorem to an equivalence by considering two generalizations of the concept of a pseudorandom generator against logspace. A targeted pseudorandom generator against logspace takes as input a short uniform random seed and a finite automaton; it outputs a long bitstring that looks random to that particular automaton. A simulation advice generator for logspace stretches a small uniform random seed into a long advice string; the requirement is that there is some logspace algorithm that, given a finite automaton and this advice string, simulates the automaton reading a long uniform random input. We prove that ∩α > 0 prBPSPACE(log1 + α n) = ∩α > 0 prDSPACE(log1 + α n) if and only if for every targeted pseudorandom generator against logspace, there is a simulation advice generator for logspace with similar parameters. Finally, we observe that in a certain uniform setting (namely, if we only worry about sequences of automata that can be generated in logspace), targeted pseudorandom generators against logspace can be transformed into simulation advice generators with similar parameters.
{"title":"Targeted pseudorandom generators, simulation advice generators, and derandomizing logspace","authors":"William M. Hoza, C. Umans","doi":"10.1145/3055399.3055414","DOIUrl":"https://doi.org/10.1145/3055399.3055414","url":null,"abstract":"Assume that for every derandomization result for logspace algorithms, there is a pseudorandom generator strong enough to nearly recover the derandomization by iterating over all seeds and taking a majority vote. We prove under a precise version of this assumption that BPL ⊆ ∩α > 0 DSPACE(log1 + α n). We strengthen the theorem to an equivalence by considering two generalizations of the concept of a pseudorandom generator against logspace. A targeted pseudorandom generator against logspace takes as input a short uniform random seed and a finite automaton; it outputs a long bitstring that looks random to that particular automaton. A simulation advice generator for logspace stretches a small uniform random seed into a long advice string; the requirement is that there is some logspace algorithm that, given a finite automaton and this advice string, simulates the automaton reading a long uniform random input. We prove that ∩α > 0 prBPSPACE(log1 + α n) = ∩α > 0 prDSPACE(log1 + α n) if and only if for every targeted pseudorandom generator against logspace, there is a simulation advice generator for logspace with similar parameters. Finally, we observe that in a certain uniform setting (namely, if we only worry about sequences of automata that can be generated in logspace), targeted pseudorandom generators against logspace can be transformed into simulation advice generators with similar parameters.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78596620","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}
Yi-Jun Chang, T. Kopelowitz, S. Pettie, Ruosong Wang, Wei Zhan
Energy is often the most constrained resource for battery-powered wireless devices and the lion's share of energy is often spent on transceiver usage (sending/receiving packets), not on computation. In this paper we study the energy complexity of Leader Election and Approximate Counting in several models of wireless radio networks. It turns out that energy complexity is very sensitive to whether the devices can generate random bits and their ability to detect collisions. We consider four collision-detection models: Strong-CD (in which transmitters and listeners detect collisions), Sender-CD and Receiver-CD (in which only transmitters or only listeners detect collisions), and No-CD (in which no one detects collisions.) The take-away message of our results is quite surprising. For randomized Leader Election algorithms, there is an exponential gap between the energy complexity of Sender-CD and Receiver-CD: No-CD = Sender-CD ⪢ Receiver-CD = Strong-CD and for deterministic Leader Election algorithms, there is another exponential gap in energy complexity, but in the reverse direction: No-CD = Receiver-CD ⪢ Sender-CD = Strong-CD In particular, the randomized energy complexity of Leader Election is Θ(log* n) in Sender-CD but Θ(log(log* n)) in Receiver-CD, where n is the (unknown) number of devices. Its deterministic complexity is ⏶(logN) in Receiver-CD but Θ(loglogN) in Sender-CD, where N is the (known) size of the devices' ID space. There is a tradeoff between time and energy. We give a new upper bound on the time-energy tradeoff curve for randomized Leader Election and Approximate Counting. A critical component of this algorithm is a new deterministic Leader Election algorithm for dense instances, when n=Θ(N), with inverse-Ackermann-type (O(α(N))) energy complexity.
{"title":"Exponential separations in the energy complexity of leader election","authors":"Yi-Jun Chang, T. Kopelowitz, S. Pettie, Ruosong Wang, Wei Zhan","doi":"10.1145/3055399.3055481","DOIUrl":"https://doi.org/10.1145/3055399.3055481","url":null,"abstract":"Energy is often the most constrained resource for battery-powered wireless devices and the lion's share of energy is often spent on transceiver usage (sending/receiving packets), not on computation. In this paper we study the energy complexity of Leader Election and Approximate Counting in several models of wireless radio networks. It turns out that energy complexity is very sensitive to whether the devices can generate random bits and their ability to detect collisions. We consider four collision-detection models: Strong-CD (in which transmitters and listeners detect collisions), Sender-CD and Receiver-CD (in which only transmitters or only listeners detect collisions), and No-CD (in which no one detects collisions.) The take-away message of our results is quite surprising. For randomized Leader Election algorithms, there is an exponential gap between the energy complexity of Sender-CD and Receiver-CD: No-CD = Sender-CD ⪢ Receiver-CD = Strong-CD and for deterministic Leader Election algorithms, there is another exponential gap in energy complexity, but in the reverse direction: No-CD = Receiver-CD ⪢ Sender-CD = Strong-CD In particular, the randomized energy complexity of Leader Election is Θ(log* n) in Sender-CD but Θ(log(log* n)) in Receiver-CD, where n is the (unknown) number of devices. Its deterministic complexity is ⏶(logN) in Receiver-CD but Θ(loglogN) in Sender-CD, where N is the (known) size of the devices' ID space. There is a tradeoff between time and energy. We give a new upper bound on the time-energy tradeoff curve for randomized Leader Election and Approximate Counting. A critical component of this algorithm is a new deterministic Leader Election algorithm for dense instances, when n=Θ(N), with inverse-Ackermann-type (O(α(N))) energy complexity.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88729344","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 constant ϵ, we prove a (N) lower bound on the (randomized) communication complexity of ϵ-Nash equilibrium in two-player N x N games. For n-player binary-action games we prove an exp(n) lower bound for the (randomized) communication complexity of (ϵ,ϵ)-weak approximate Nash equilibrium, which is a profile of mixed actions such that at least (1-ϵ)-fraction of the players are ϵ-best replying.
对于一个常数ε，我们证明了在双玩家N x N博弈中ϵ-Nash均衡的(随机)通信复杂度的(N)下界。对于n人二元动作游戏，我们证明了(随机)通信复杂度(λ， λ)-弱近似纳什均衡的exp(n)下界，这是混合动作的一个特征，使得至少(1- λ)-部分玩家回复ϵ-best。
{"title":"Communication complexity of approximate Nash equilibria","authors":"Y. Babichenko, A. Rubinstein","doi":"10.1145/3055399.3055407","DOIUrl":"https://doi.org/10.1145/3055399.3055407","url":null,"abstract":"For a constant ϵ, we prove a (N) lower bound on the (randomized) communication complexity of ϵ-Nash equilibrium in two-player N x N games. For n-player binary-action games we prove an exp(n) lower bound for the (randomized) communication complexity of (ϵ,ϵ)-weak approximate Nash equilibrium, which is a profile of mixed actions such that at least (1-ϵ)-fraction of the players are ϵ-best replying.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"80 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87983548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we give improved constructions of several central objects in the literature of randomness extraction and tamper-resilient cryptography. Our main results are: (1) An explicit seeded non-malleable extractor with error ε and seed length d=O(logn)+O(log(1/ε)loglog(1/ε)), that supports min-entropy k=Ω(d) and outputs Ω(k) bits. Combined with the protocol by Dodis and Wichs, this gives a two round privacy amplification protocol with optimal entropy loss in the presence of an active adversary, for all security parameters up to Ω(k/logk), where k is the min-entropy of the shared weak random source. Previously, the best known seeded non-malleable extractors require seed length and min-entropy O(logn)+log(1/ε)2O√loglog(1/ε), and only give two round privacy amplification protocols with optimal entropy loss for security parameter up to k/2O(√logk). (2) An explicit non-malleable two-source extractor for min entropy k ≥ (1 - Υ)n, some constant Υ>0, that outputs Ω(k) bits with error 2-Ω(n/logn). We further show that we can efficiently uniformly sample from the pre-image of any output of the extractor. Combined with the connection found by Cheraghchi and Guruswami this gives a non-malleable code in the two-split-state model with relative rate Ω(1/logn). This exponentially improves previous constructions, all of which only achieve rate n-Ω(1). (3) Combined with the techniques by Ben-Aroya et. al, our non-malleable extractors give a two-source extractor for min-entropy O(logn loglogn), which also implies a K-Ramsey graph on N vertices with K=(logN)O(logloglogN). Previously the best known two-source extractor by Ben-Aroya et. al requires min-entropy logn 2O(√logn), which gives a Ramsey graph with K=(logN)2O(√logloglogN). We further show a way to reduce the problem of constructing seeded non-malleable extractors to the problem of constructing non-malleable independent source extractors. Using the non-malleable 10-source extractor with optimal error by Chattopadhyay and Zuckerman, we give a 10-source extractor for min-entropy O(logn). Previously the best known extractor for such min-entropy by Cohen and Schulman requires O(loglogn) sources. Independent of our work, Cohen obtained similar results to (1) and the two-source extractor, except the dependence on ε is log(1/ε)poly loglog(1/ε) and the two-source extractor requires min-entropy logn poly loglogn.
{"title":"Improved non-malleable extractors, non-malleable codes and independent source extractors","authors":"Xin Li","doi":"10.1145/3055399.3055486","DOIUrl":"https://doi.org/10.1145/3055399.3055486","url":null,"abstract":"In this paper we give improved constructions of several central objects in the literature of randomness extraction and tamper-resilient cryptography. Our main results are: (1) An explicit seeded non-malleable extractor with error ε and seed length d=O(logn)+O(log(1/ε)loglog(1/ε)), that supports min-entropy k=Ω(d) and outputs Ω(k) bits. Combined with the protocol by Dodis and Wichs, this gives a two round privacy amplification protocol with optimal entropy loss in the presence of an active adversary, for all security parameters up to Ω(k/logk), where k is the min-entropy of the shared weak random source. Previously, the best known seeded non-malleable extractors require seed length and min-entropy O(logn)+log(1/ε)2O√loglog(1/ε), and only give two round privacy amplification protocols with optimal entropy loss for security parameter up to k/2O(√logk). (2) An explicit non-malleable two-source extractor for min entropy k ≥ (1 - Υ)n, some constant Υ>0, that outputs Ω(k) bits with error 2-Ω(n/logn). We further show that we can efficiently uniformly sample from the pre-image of any output of the extractor. Combined with the connection found by Cheraghchi and Guruswami this gives a non-malleable code in the two-split-state model with relative rate Ω(1/logn). This exponentially improves previous constructions, all of which only achieve rate n-Ω(1). (3) Combined with the techniques by Ben-Aroya et. al, our non-malleable extractors give a two-source extractor for min-entropy O(logn loglogn), which also implies a K-Ramsey graph on N vertices with K=(logN)O(logloglogN). Previously the best known two-source extractor by Ben-Aroya et. al requires min-entropy logn 2O(√logn), which gives a Ramsey graph with K=(logN)2O(√logloglogN). We further show a way to reduce the problem of constructing seeded non-malleable extractors to the problem of constructing non-malleable independent source extractors. Using the non-malleable 10-source extractor with optimal error by Chattopadhyay and Zuckerman, we give a 10-source extractor for min-entropy O(logn). Previously the best known extractor for such min-entropy by Cohen and Schulman requires O(loglogn) sources. Independent of our work, Cohen obtained similar results to (1) and the two-source extractor, except the dependence on ε is log(1/ε)poly loglog(1/ε) and the two-source extractor requires min-entropy logn poly loglogn.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"90 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79738448","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}
Gopal Pandurangan, Peter Robinson, Michele Scquizzato
This paper presents a randomized (Las Vegas) distributed algorithm that constructs a minimum spanning tree (MST) in weighted networks with optimal (up to polylogarithmic factors) time and message complexity. This algorithm runs in Õ(D + √n) time and exchanges Õ(m) messages (both with high probability), where n is the number of nodes of the network, D is the diameter, and m is the number of edges. This is the first distributed MST algorithm that matches simultaneously the time lower bound of Ω(D + √n) [Elkin, SIAM J. Comput. 2006] and the message lower bound of Ω(m) [Kutten et al., J. ACM 2015], which both apply to randomized Monte Carlo algorithms. The prior time and message lower bounds are derived using two completely different graph constructions; the existing lower bound construction that shows one lower bound does not work for the other. To complement our algorithm, we present a new lower bound graph construction for which any distributed MST algorithm requires both Ω(D + √n) rounds and Ω(m) messages.
本文提出了一种随机(Las Vegas)分布式算法，该算法在时间和消息复杂度最优的加权网络中构造最小生成树(MST)。该算法运行时间为Õ(D +√n)，交换消息为Õ(m)条(均为大概率)，其中n为网络节点数，D为直径，m为边数。这是第一个同时匹配Ω(D +√n)的时间下界[Elkin, SIAM J. Comput. 2006]和Ω(m)的消息下界[Kutten et al.， J. ACM 2015]的分布式MST算法，两者都适用于随机蒙特卡罗算法。使用两种完全不同的图结构推导了先验时间和消息下界;显示一个下界的现有下界构造不适用于另一个下界。为了补充我们的算法，我们提出了一个新的下界图构造，其中任何分布式MST算法都需要Ω(D +√n)轮和Ω(m)消息。
{"title":"A time- and message-optimal distributed algorithm for minimum spanning trees","authors":"Gopal Pandurangan, Peter Robinson, Michele Scquizzato","doi":"10.1145/3055399.3055449","DOIUrl":"https://doi.org/10.1145/3055399.3055449","url":null,"abstract":"This paper presents a randomized (Las Vegas) distributed algorithm that constructs a minimum spanning tree (MST) in weighted networks with optimal (up to polylogarithmic factors) time and message complexity. This algorithm runs in Õ(D + √n) time and exchanges Õ(m) messages (both with high probability), where n is the number of nodes of the network, D is the diameter, and m is the number of edges. This is the first distributed MST algorithm that matches simultaneously the time lower bound of Ω(D + √n) [Elkin, SIAM J. Comput. 2006] and the message lower bound of Ω(m) [Kutten et al., J. ACM 2015], which both apply to randomized Monte Carlo algorithms. The prior time and message lower bounds are derived using two completely different graph constructions; the existing lower bound construction that shows one lower bound does not work for the other. To complement our algorithm, we present a new lower bound graph construction for which any distributed MST algorithm requires both Ω(D + √n) rounds and Ω(m) messages.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76243952","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 the geodesic walk for sampling Riemannian manifolds and apply it to the problem of generating uniform random points from the interior of polytopes in ℝn specified by m inequalities. The walk is a discrete-time simulation of a stochastic differential equation (SDE) on the Riemannian manifold equipped with the metric induced by the Hessian of a convex function; each step is the solution of an ordinary differential equation (ODE). The resulting sampling algorithm for polytopes mixes in O*(mn3/4) steps. This is the first walk that breaks the quadratic barrier for mixing in high dimension, improving on the previous best bound of O*(mn) by Kannan and Narayanan for the Dikin walk. We also show that each step of the geodesic walk (solving an ODE) can be implemented efficiently, thus improving the time complexity for sampling polytopes. Our analysis of the geodesic walk for general Hessian manifolds does not assume positive curvature and might be of independent interest.
{"title":"Geodesic walks in polytopes","authors":"Y. Lee, S. Vempala","doi":"10.1145/3055399.3055416","DOIUrl":"https://doi.org/10.1145/3055399.3055416","url":null,"abstract":"We introduce the geodesic walk for sampling Riemannian manifolds and apply it to the problem of generating uniform random points from the interior of polytopes in ℝn specified by m inequalities. The walk is a discrete-time simulation of a stochastic differential equation (SDE) on the Riemannian manifold equipped with the metric induced by the Hessian of a convex function; each step is the solution of an ordinary differential equation (ODE). The resulting sampling algorithm for polytopes mixes in O*(mn3/4) steps. This is the first walk that breaks the quadratic barrier for mixing in high dimension, improving on the previous best bound of O*(mn) by Kannan and Narayanan for the Dikin walk. We also show that each step of the geodesic walk (solving an ODE) can be implemented efficiently, thus improving the time complexity for sampling polytopes. Our analysis of the geodesic walk for general Hessian manifolds does not assume positive curvature and might be of independent interest.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81426255","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. Arvind, Pushkar S. Joglekar, P. Mukhopadhyay, S. Raja
In this paper we show that black-box polynomial identity testing for noncommutative polynomials f∈𝔽⟨z1,z2,…,zn⟩ of degree D and sparsity t, can be done in randomized (n,logt,logD) time. As a consequence, given a circuit C of size s computing a polynomial f∈𝔽⟨ z1,z2,…,zn⟩ with at most t non-zero monomials, then testing if f is identically zero can be done by a randomized algorithm with running time polynomial in s and n and logt. This makes significant progress on a question that has been open for over ten years. Our algorithm is based on automata-theoretic ideas that can efficiently isolate a monomial in the given polynomial. In particular, we carry out the monomial isolation using nondeterministic automata. In general, noncommutative circuits of size s can compute polynomials of degree exponential in s and number of monomials double-exponential in s. In this paper, we consider a natural class of homogeneous noncommutative circuits, that we call +-regular circuits, and give a white-box polynomial time deterministic polynomial identity test. These circuits can compute noncommutative polynomials with number of monomials double-exponential in the circuit size. Our algorithm combines some new structural results for +-regular circuits with known results for noncommutative ABP identity testing, rank bound of commutative depth three identities, and equivalence testing problem for words. Finally, we consider the black-box identity testing problem for depth three +-regular circuits and give a randomized polynomial time identity test. In particular, we show if f∈𝔽Z⟩ is a nonzero noncommutative polynomial computed by a depth three +-regular circuit of size s, then f cannot be a polynomial identity for the matrix algebra 𝕄s(𝔽) when 𝔽 is sufficiently large depending on the degree of f.
{"title":"Randomized polynomial time identity testing for noncommutative circuits","authors":"V. Arvind, Pushkar S. Joglekar, P. Mukhopadhyay, S. Raja","doi":"10.1145/3055399.3055442","DOIUrl":"https://doi.org/10.1145/3055399.3055442","url":null,"abstract":"In this paper we show that black-box polynomial identity testing for noncommutative polynomials f∈𝔽⟨z1,z2,…,zn⟩ of degree D and sparsity t, can be done in randomized (n,logt,logD) time. As a consequence, given a circuit C of size s computing a polynomial f∈𝔽⟨ z1,z2,…,zn⟩ with at most t non-zero monomials, then testing if f is identically zero can be done by a randomized algorithm with running time polynomial in s and n and logt. This makes significant progress on a question that has been open for over ten years. Our algorithm is based on automata-theoretic ideas that can efficiently isolate a monomial in the given polynomial. In particular, we carry out the monomial isolation using nondeterministic automata. In general, noncommutative circuits of size s can compute polynomials of degree exponential in s and number of monomials double-exponential in s. In this paper, we consider a natural class of homogeneous noncommutative circuits, that we call +-regular circuits, and give a white-box polynomial time deterministic polynomial identity test. These circuits can compute noncommutative polynomials with number of monomials double-exponential in the circuit size. Our algorithm combines some new structural results for +-regular circuits with known results for noncommutative ABP identity testing, rank bound of commutative depth three identities, and equivalence testing problem for words. Finally, we consider the black-box identity testing problem for depth three +-regular circuits and give a randomized polynomial time identity test. In particular, we show if f∈𝔽Z⟩ is a nonzero noncommutative polynomial computed by a depth three +-regular circuit of size s, then f cannot be a polynomial identity for the matrix algebra 𝕄s(𝔽) when 𝔽 is sufficiently large depending on the degree of f.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"92 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80929706","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}
Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with n variables and m clauses, there is a value of m = Ω(n) beyond which the CSP will be unsatisfiable with high probability. Strong refutation is the problem of certifying that no variable assignment satisfies more than a constant fraction of clauses; this is the natural algorithmic problem in the unsatisfiable regime (when m/n = ω(1)). Intuitively, strong refutation should become easier as the clause density m/n grows, because the contradictions introduced by the random clauses become more locally apparent. For CSPs such as k-SAT and k-XOR, there is a long-standing gap between the clause density at which efficient strong refutation algorithms are known, m/n ≥ Ο(nk/2-1), and the clause density at which instances become unsatisfiable with high probability, m/n = ω (1). In this paper, we give spectral and sum-of-squares algorithms for strongly refuting random k-XOR instances with clause density m/n ≥ Ο(n(k/2-1)(1-δ)) in time exp(Ο(nδ)) or in Ο(nδ) rounds of the sum-of-squares hierarchy, for any δ ∈ [0,1) and any integer k ≥ 3. Our algorithms provide a smooth transition between the clause density at which polynomial-time algorithms are known at δ = 0, and brute-force refutation at the satisfiability threshold when δ = 1. We also leverage our k-XOR results to obtain strong refutation algorithms for SAT (or any other Boolean CSP) at similar clause densities. Our algorithms match the known sum-of-squares lower bounds due to Grigoriev and Schonebeck, up to logarithmic factors.
{"title":"Strongly refuting random CSPs below the spectral threshold","authors":"P. Raghavendra, Satish Rao, T. Schramm","doi":"10.1145/3055399.3055417","DOIUrl":"https://doi.org/10.1145/3055399.3055417","url":null,"abstract":"Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with n variables and m clauses, there is a value of m = Ω(n) beyond which the CSP will be unsatisfiable with high probability. Strong refutation is the problem of certifying that no variable assignment satisfies more than a constant fraction of clauses; this is the natural algorithmic problem in the unsatisfiable regime (when m/n = ω(1)). Intuitively, strong refutation should become easier as the clause density m/n grows, because the contradictions introduced by the random clauses become more locally apparent. For CSPs such as k-SAT and k-XOR, there is a long-standing gap between the clause density at which efficient strong refutation algorithms are known, m/n ≥ Ο(nk/2-1), and the clause density at which instances become unsatisfiable with high probability, m/n = ω (1). In this paper, we give spectral and sum-of-squares algorithms for strongly refuting random k-XOR instances with clause density m/n ≥ Ο(n(k/2-1)(1-δ)) in time exp(Ο(nδ)) or in Ο(nδ) rounds of the sum-of-squares hierarchy, for any δ ∈ [0,1) and any integer k ≥ 3. Our algorithms provide a smooth transition between the clause density at which polynomial-time algorithms are known at δ = 0, and brute-force refutation at the satisfiability threshold when δ = 1. We also leverage our k-XOR results to obtain strong refutation algorithms for SAT (or any other Boolean CSP) at similar clause densities. Our algorithms match the known sum-of-squares lower bounds due to Grigoriev and Schonebeck, up to logarithmic factors.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"290 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77504118","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 a monopolist that is selling n items to a single additive buyer, where the buyer's values for the items are drawn according to independent distributions F1,F2,…,Fn that possibly have unbounded support. It is well known that - unlike in the single item case - the revenue-optimal auction (a pricing scheme) may be complex, sometimes requiring a continuum of menu entries. It is also known that simple auctions with a finite bounded number of menu entries can extract a constant fraction of the optimal revenue. Nonetheless, the question of the possibility of extracting an arbitrarily high fraction of the optimal revenue via a finite menu size remained open. In this paper, we give an affirmative answer to this open question, showing that for every n and for every ε>0, there exists a complexity bound C=C(n,ε) such that auctions of menu size at most C suffice for obtaining a (1-ε) fraction of the optimal revenue from any F1,…,Fn. We prove upper and lower bounds on the revenue approximation complexity C(n,ε), as well as on the deterministic communication complexity required to run an auction that achieves such an approximation.
{"title":"The menu-size complexity of revenue approximation","authors":"Moshe Babaioff, Yannai A. Gonczarowski, N. Nisan","doi":"10.1145/3055399.3055426","DOIUrl":"https://doi.org/10.1145/3055399.3055426","url":null,"abstract":"We consider a monopolist that is selling n items to a single additive buyer, where the buyer's values for the items are drawn according to independent distributions F1,F2,…,Fn that possibly have unbounded support. It is well known that - unlike in the single item case - the revenue-optimal auction (a pricing scheme) may be complex, sometimes requiring a continuum of menu entries. It is also known that simple auctions with a finite bounded number of menu entries can extract a constant fraction of the optimal revenue. Nonetheless, the question of the possibility of extracting an arbitrarily high fraction of the optimal revenue via a finite menu size remained open. In this paper, we give an affirmative answer to this open question, showing that for every n and for every ε>0, there exists a complexity bound C=C(n,ε) such that auctions of menu size at most C suffice for obtaining a (1-ε) fraction of the optimal revenue from any F1,…,Fn. We prove upper and lower bounds on the revenue approximation complexity C(n,ε), as well as on the deterministic communication complexity required to run an auction that achieves such an approximation.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83111493","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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