We study the problem of computing an ε-Nash equilibrium in repeated games. Earlier work by Borgs et al. [2010] suggests that this problem is intractable. We show that if we make a slight change to their model---modeling the players as polynomial-time Turing machines that maintain state (rather than stateless polynomial-time Turing machines)---and make some standard cryptographic hardness assumptions (the existence of public key encryption), the problem can actually be solved in polynomial time.
{"title":"The truth behind the myth of the folk theorem","authors":"Joseph Y. Halpern, R. Pass, Lior Seeman","doi":"10.1145/2554797.2554847","DOIUrl":"https://doi.org/10.1145/2554797.2554847","url":null,"abstract":"We study the problem of computing an ε-Nash equilibrium in repeated games. Earlier work by Borgs et al. [2010] suggests that this problem is intractable. We show that if we make a slight change to their model---modeling the players as polynomial-time Turing machines that maintain state (rather than stateless polynomial-time Turing machines)---and make some standard cryptographic hardness assumptions (the existence of public key encryption), the problem can actually be solved in polynomial time.","PeriodicalId":382856,"journal":{"name":"Proceedings of the 5th conference on Innovations in theoretical computer science","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130063191","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}
High triangle density -- the graph property stating that a constant fraction of two-hop paths belong to a triangle -- is a common signature of social networks. This paper studies triangle-dense graphs from a structural perspective. We prove constructively that significant portions of a triangle-dense graph are contained in a disjoint union of dense, radius 2 subgraphs. This result quantifies the extent to which triangle-dense graphs resemble unions of cliques. We also show that our algorithm recovers planted clusterings in approximation-stable k-median instances.
{"title":"Decompositions of triangle-dense graphs","authors":"Rishi Gupta, T. Roughgarden, C. Seshadhri","doi":"10.1145/2554797.2554840","DOIUrl":"https://doi.org/10.1145/2554797.2554840","url":null,"abstract":"High triangle density -- the graph property stating that a constant fraction of two-hop paths belong to a triangle -- is a common signature of social networks. This paper studies triangle-dense graphs from a structural perspective. We prove constructively that significant portions of a triangle-dense graph are contained in a disjoint union of dense, radius 2 subgraphs. This result quantifies the extent to which triangle-dense graphs resemble unions of cliques. We also show that our algorithm recovers planted clusterings in approximation-stable k-median instances.","PeriodicalId":382856,"journal":{"name":"Proceedings of the 5th conference on Innovations in theoretical computer science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134430553","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 a seminal paper, Valiant (2006) introduced a computational model for evolution to address the question of complexity that can arise through Darwinian mechanisms. Valiant views evolution as a restricted form of computational learning, where the goal is to evolve a hypothesis that is close to the ideal function. Feldman (2008) showed that (correlational) statistical query learning algorithms could be framed as evolutionary mechanisms in Valiant's model. P. Valiant (2012) considered evolvability of real-valued functions and also showed that weak-optimization algorithms that use weak-evaluation oracles could be converted to evolutionary mechanisms. In this work, we focus on the complexity of representations of evolutionary mechanisms. In general, the reductions of Feldman and P. Valiant may result in intermediate representations that are arbitrarily complex polynomial-sized circuits). We argue that biological constraints often dictate that the representations have low complexity, such as constant depth and fan-in circuits. We give mechanisms for evolving sparse linear functions under a large class of smooth distributions. These evolutionary algorithms are attribute-efficient in the sense that the size of the representations and the number of generations required depend only on the sparsity of the target function and the accuracy parameter, but have no dependence on the total number of attributes.
{"title":"Attribute-efficient evolvability of linear functions","authors":"E. Angelino, Varun Kanade","doi":"10.1145/2554797.2554824","DOIUrl":"https://doi.org/10.1145/2554797.2554824","url":null,"abstract":"In a seminal paper, Valiant (2006) introduced a computational model for evolution to address the question of complexity that can arise through Darwinian mechanisms. Valiant views evolution as a restricted form of computational learning, where the goal is to evolve a hypothesis that is close to the ideal function. Feldman (2008) showed that (correlational) statistical query learning algorithms could be framed as evolutionary mechanisms in Valiant's model. P. Valiant (2012) considered evolvability of real-valued functions and also showed that weak-optimization algorithms that use weak-evaluation oracles could be converted to evolutionary mechanisms. In this work, we focus on the complexity of representations of evolutionary mechanisms. In general, the reductions of Feldman and P. Valiant may result in intermediate representations that are arbitrarily complex polynomial-sized circuits). We argue that biological constraints often dictate that the representations have low complexity, such as constant depth and fan-in circuits. We give mechanisms for evolving sparse linear functions under a large class of smooth distributions. These evolutionary algorithms are attribute-efficient in the sense that the size of the representations and the number of generations required depend only on the sparsity of the target function and the accuracy parameter, but have no dependence on the total number of attributes.","PeriodicalId":382856,"journal":{"name":"Proceedings of the 5th conference on Innovations in theoretical computer science","volume":"117 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116225474","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 propose a new conjecture on some exponential sums. These particular sums have not apparently been considered in the literature. Subject to the conjecture we obtain the first effective construction of asymptotically good tree codes. The available numerical evidence is consistent with the conjecture and is sufficient to certify codes for significant-length communications.
{"title":"Tree codes and a conjecture on exponential sums","authors":"Cristopher Moore, L. Schulman","doi":"10.1145/2554797.2554813","DOIUrl":"https://doi.org/10.1145/2554797.2554813","url":null,"abstract":"We propose a new conjecture on some exponential sums. These particular sums have not apparently been considered in the literature. Subject to the conjecture we obtain the first effective construction of asymptotically good tree codes. The available numerical evidence is consistent with the conjecture and is sufficient to certify codes for significant-length communications.","PeriodicalId":382856,"journal":{"name":"Proceedings of the 5th conference on Innovations in theoretical computer science","volume":"91 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126045564","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give two new characterizations of ( 2-linear, smooth) locally testable error-correcting codes in terms of Cayley graphs over Fh2: A locally testable code is equivalent to a Cayley graph over h2 whose set of generators is significantly larger than h and has no short linear dependencies, bbut yields a shortest-path metric that embeds into l with constant distortion. This extends and gives a converse to a result of Khot and Naor (2006), which showed that codes with large dual distance imply Cayley graphs that have no low-distortion embeddings into l . A locally testable code is equivalent to a Cayley graph over Fh2 that has significantly more than h eigenvalues near 1, which have no short linear dependencies among them and which "explain" all of the large eigenvalues. This extends and gives a converse to a recent construction of Barak et al. (2012), which showed that locally testable codes imply Cayley graphs that are small-set expanders but have many large eigenvalues.
{"title":"Locally testable codes and cayley graphs","authors":"Parikshit Gopalan, S. Vadhan, Yuan Zhou","doi":"10.1145/2554797.2554807","DOIUrl":"https://doi.org/10.1145/2554797.2554807","url":null,"abstract":"We give two new characterizations of ( 2-linear, smooth) locally testable error-correcting codes in terms of Cayley graphs over Fh2: A locally testable code is equivalent to a Cayley graph over h2 whose set of generators is significantly larger than h and has no short linear dependencies, bbut yields a shortest-path metric that embeds into l with constant distortion. This extends and gives a converse to a result of Khot and Naor (2006), which showed that codes with large dual distance imply Cayley graphs that have no low-distortion embeddings into l . A locally testable code is equivalent to a Cayley graph over Fh2 that has significantly more than h eigenvalues near 1, which have no short linear dependencies among them and which \"explain\" all of the large eigenvalues. This extends and gives a converse to a recent construction of Barak et al. (2012), which showed that locally testable codes imply Cayley graphs that are small-set expanders but have many large eigenvalues.","PeriodicalId":382856,"journal":{"name":"Proceedings of the 5th conference on Innovations in theoretical computer science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125796359","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 multireference alignment problem consists of estimating a signal from multiple noisy shifted observations. Inspired by existing Unique-Games approximation algorithms, we provide a semidefinite program (SDP) based relaxation which approximates the maximum likelihood estimator (MLE) for the multireference alignment problem. Although we show this MLE problem is Unique-Games hard to approximate within any constant, we observe that our poly-time approximation algorithm for this problem appears to perform quite well in typical instances, outperforming existing methods. In an attempt to explain this behavior we provide stability guarantees for our SDP under a random noise model on the observations. This case is more challenging to analyze than traditional semi-random instances of Unique-Games: the noise model is on vertices of a graph and translates into dependent noise on the edges. Interestingly, we show that if certain positivity constraints in the relaxation are dropped, its solution becomes equivalent to performing phase correlation, a popular method used for pairwise alignment in imaging applications. Finally, we describe how symmetry reduction techniques from matrix representation theory can greatly decrease the computational cost of the SDP considered.
{"title":"Multireference alignment using semidefinite programming","authors":"A. Bandeira, M. Charikar, A. Singer, Andy Zhu","doi":"10.1145/2554797.2554839","DOIUrl":"https://doi.org/10.1145/2554797.2554839","url":null,"abstract":"The multireference alignment problem consists of estimating a signal from multiple noisy shifted observations. Inspired by existing Unique-Games approximation algorithms, we provide a semidefinite program (SDP) based relaxation which approximates the maximum likelihood estimator (MLE) for the multireference alignment problem. Although we show this MLE problem is Unique-Games hard to approximate within any constant, we observe that our poly-time approximation algorithm for this problem appears to perform quite well in typical instances, outperforming existing methods. In an attempt to explain this behavior we provide stability guarantees for our SDP under a random noise model on the observations. This case is more challenging to analyze than traditional semi-random instances of Unique-Games: the noise model is on vertices of a graph and translates into dependent noise on the edges. Interestingly, we show that if certain positivity constraints in the relaxation are dropped, its solution becomes equivalent to performing phase correlation, a popular method used for pairwise alignment in imaging applications. Finally, we describe how symmetry reduction techniques from matrix representation theory can greatly decrease the computational cost of the SDP considered.","PeriodicalId":382856,"journal":{"name":"Proceedings of the 5th conference on Innovations in theoretical computer science","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130488135","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}
It is shown that there exists a sequence of 3-regular graphs {Gn}∞n=1 and a Hadamard space X such that {Gn}∞n=1 forms an expander sequence with respect to {X{, yet random regular graphs are not expanders with respect to {X{. This answers a question of [31]. {Gn}∞n=1 are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear time constant factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an auxiliary continuous geometric object that allows for the implementation of martingale methods. This extended abstract does not contain proofs. The full version of this paper can be found at arXiv:1306.5434.
{"title":"Expanders with respect to Hadamard spaces and random graphs: extended abstract","authors":"M. Mendel, A. Naor","doi":"10.1145/2554797.2554829","DOIUrl":"https://doi.org/10.1145/2554797.2554829","url":null,"abstract":"It is shown that there exists a sequence of 3-regular graphs {Gn}∞n=1 and a Hadamard space X such that {Gn}∞n=1 forms an expander sequence with respect to {X{, yet random regular graphs are not expanders with respect to {X{. This answers a question of [31]. {Gn}∞n=1 are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear time constant factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an auxiliary continuous geometric object that allows for the implementation of martingale methods. This extended abstract does not contain proofs. The full version of this paper can be found at arXiv:1306.5434.","PeriodicalId":382856,"journal":{"name":"Proceedings of the 5th conference on Innovations in theoretical computer science","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132454627","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}
One-time memories (OTM's) are simple tamper-resistant cryptographic devices, which can be used to implement one-time programs, a very general form of software protection and program obfuscation. Here we investigate the possibility of building OTM's using quantum mechanical devices. It is known that OTM's cannot exist in a fully-quantum world or in a fully-classical world. Instead, we propose a new model based on isolated qubits - qubits that can only be accessed using local operations and classical communication (LOCC). This model combines a quantum resource (single-qubit measurements) with a classical restriction (on communication between qubits), and can be implemented using current technologies, such as nitrogen vacancy centers in diamond. In this model, we construct OTM's that are information-theoretically secure against one-pass LOCC adversaries that use 2-outcome measurements. Our construction resembles Wiesner's old idea of quantum conjugate coding, implemented using random error-correcting codes; our proof of security uses entropy chaining to bound the supremum of a suitable empirical process. In addition, we conjecture that our random codes can be replaced by some class of efficiently-decodable codes, to get computationally-efficient OTM's that are secure against computationally-bounded LOCC adversaries. In addition, we construct data-hiding states, which allow an LOCC sender to encode an (n-O(1))-bit messsage into n qubits, such that at most half of the message can be extracted by a one-pass LOCC receiver, but the whole message can be extracted by a general quantum receiver.
{"title":"Building one-time memories from isolated qubits: (extended abstract)","authors":"Yi-Kai Liu","doi":"10.1145/2554797.2554823","DOIUrl":"https://doi.org/10.1145/2554797.2554823","url":null,"abstract":"One-time memories (OTM's) are simple tamper-resistant cryptographic devices, which can be used to implement one-time programs, a very general form of software protection and program obfuscation. Here we investigate the possibility of building OTM's using quantum mechanical devices. It is known that OTM's cannot exist in a fully-quantum world or in a fully-classical world. Instead, we propose a new model based on isolated qubits - qubits that can only be accessed using local operations and classical communication (LOCC). This model combines a quantum resource (single-qubit measurements) with a classical restriction (on communication between qubits), and can be implemented using current technologies, such as nitrogen vacancy centers in diamond. In this model, we construct OTM's that are information-theoretically secure against one-pass LOCC adversaries that use 2-outcome measurements. Our construction resembles Wiesner's old idea of quantum conjugate coding, implemented using random error-correcting codes; our proof of security uses entropy chaining to bound the supremum of a suitable empirical process. In addition, we conjecture that our random codes can be replaced by some class of efficiently-decodable codes, to get computationally-efficient OTM's that are secure against computationally-bounded LOCC adversaries. In addition, we construct data-hiding states, which allow an LOCC sender to encode an (n-O(1))-bit messsage into n qubits, such that at most half of the message can be extracted by a one-pass LOCC receiver, but the whole message can be extracted by a general quantum receiver.","PeriodicalId":382856,"journal":{"name":"Proceedings of the 5th conference on Innovations in theoretical computer science","volume":"239 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134314208","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}
Karthekeyan Chandrasekaran, J. Thaler, Jonathan Ullman, A. Wan
We study the problem of answering k-way marginal queries on a database D ϵ ({0,1}d)n, while preserving differential privacy. The answer to a k-way marginal query is the fraction of the database's records x in {0,1}d with a given value in each of a given set of up to k columns. Marginal queries enable a rich class of statistical analyses on a dataset, and designing efficient algorithms for privately answering marginal queries has been identified as an important open problem in private data analysis. For any k, we give a differentially private online algorithm that runs in time poly (n, 2o(d)) per query and answers any sequence of poly(n) many k-way marginal queries with error at most ±0.01 on every query, provided n ≥ d0.51. To the best of our knowledge, this is the first algorithm capable of privately answering marginal queries with a non-trivial worst-case accuracy guarantee for databases containing poly(d, k) records in time exp(o(d)). Our algorithm runs the private multiplicative weights algorithm (Hardt and Rothblum, FOCS '10) on a new approximate polynomial representation of the database. We derive our representation for the database by approximating the OR function restricted to low Hamming weight inputs using low-degree polynomials with coefficients of bounded L1-norm. In doing so, we show new upper and lower bounds on the degree of such polynomials, which may be of independent approximation-theoretic interest.
{"title":"Faster private release of marginals on small databases","authors":"Karthekeyan Chandrasekaran, J. Thaler, Jonathan Ullman, A. Wan","doi":"10.1145/2554797.2554833","DOIUrl":"https://doi.org/10.1145/2554797.2554833","url":null,"abstract":"We study the problem of answering k-way marginal queries on a database D ϵ ({0,1}d)n, while preserving differential privacy. The answer to a k-way marginal query is the fraction of the database's records x in {0,1}d with a given value in each of a given set of up to k columns. Marginal queries enable a rich class of statistical analyses on a dataset, and designing efficient algorithms for privately answering marginal queries has been identified as an important open problem in private data analysis. For any k, we give a differentially private online algorithm that runs in time poly (n, 2o(d)) per query and answers any sequence of poly(n) many k-way marginal queries with error at most ±0.01 on every query, provided n ≥ d0.51. To the best of our knowledge, this is the first algorithm capable of privately answering marginal queries with a non-trivial worst-case accuracy guarantee for databases containing poly(d, k) records in time exp(o(d)). Our algorithm runs the private multiplicative weights algorithm (Hardt and Rothblum, FOCS '10) on a new approximate polynomial representation of the database. We derive our representation for the database by approximating the OR function restricted to low Hamming weight inputs using low-degree polynomials with coefficients of bounded L1-norm. In doing so, we show new upper and lower bounds on the degree of such polynomials, which may be of independent approximation-theoretic interest.","PeriodicalId":382856,"journal":{"name":"Proceedings of the 5th conference on Innovations in theoretical computer science","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123793764","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}
Local algorithms on graphs are algorithms that run in parallel on the nodes of a graph to compute some global structural feature of the graph. Such algorithms use only local information available at nodes to determine local aspects of the global structure, while also potentially using some randomness. Research over the years has shown that such algorithms can be surprisingly powerful in terms of computing structures like large independent sets in graphs locally. These algorithms have also been implicitly considered in the work on graph limits, where a conjecture due to Hatami, Lovász and Szegedy [17] implied that local algorithms may be able to compute near-maximum independent sets in (sparse) random d-regular graphs. In this paper we refute this conjecture and show that every independent set produced by local algorithms is smaller that the largest one by a multiplicative factor of at least 1/2+1/(2√2) ≈ .853, asymptotically as d → ∞. Our result is based on an important clustering phenomena predicted first in the literature on spin glasses, and recently proved rigorously for a variety of constraint satisfaction problems on random graphs. Such properties suggest that the geometry of the solution space can be quite intricate. The specific clustering property, that we prove and apply in this paper shows that typically every two large independent sets in a random graph either have a significant intersection, or have a nearly empty intersection. As a result, large independent sets are clustered according to the proximity to each other. While the clustering property was postulated earlier as an obstruction for the success of local algorithms, such as for example, the Belief Propagation algorithm, our result is the first one where the clustering property is used to formally prove limits on local algorithms.
{"title":"Limits of local algorithms over sparse random graphs","authors":"D. Gamarnik, M. Sudan","doi":"10.1145/2554797.2554831","DOIUrl":"https://doi.org/10.1145/2554797.2554831","url":null,"abstract":"Local algorithms on graphs are algorithms that run in parallel on the nodes of a graph to compute some global structural feature of the graph. Such algorithms use only local information available at nodes to determine local aspects of the global structure, while also potentially using some randomness. Research over the years has shown that such algorithms can be surprisingly powerful in terms of computing structures like large independent sets in graphs locally. These algorithms have also been implicitly considered in the work on graph limits, where a conjecture due to Hatami, Lovász and Szegedy [17] implied that local algorithms may be able to compute near-maximum independent sets in (sparse) random d-regular graphs. In this paper we refute this conjecture and show that every independent set produced by local algorithms is smaller that the largest one by a multiplicative factor of at least 1/2+1/(2√2) ≈ .853, asymptotically as d → ∞. Our result is based on an important clustering phenomena predicted first in the literature on spin glasses, and recently proved rigorously for a variety of constraint satisfaction problems on random graphs. Such properties suggest that the geometry of the solution space can be quite intricate. The specific clustering property, that we prove and apply in this paper shows that typically every two large independent sets in a random graph either have a significant intersection, or have a nearly empty intersection. As a result, large independent sets are clustered according to the proximity to each other. While the clustering property was postulated earlier as an obstruction for the success of local algorithms, such as for example, the Belief Propagation algorithm, our result is the first one where the clustering property is used to formally prove limits on local algorithms.","PeriodicalId":382856,"journal":{"name":"Proceedings of the 5th conference on Innovations in theoretical computer science","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121366893","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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