Let C be a class of distributions over {0, 1}n. A deterministic randomness extractor for C is a function E : {0, 1}n rarr {0, 1}m such that for any X in C the distribution E(X) is statistically close to the uniform distribution. A long line of research deals with explicit constructions of such extractors for various classes C while trying to maximize m. In this paper we give a general transformation that transforms a deterministic extractor E that extracts "few" bits into an extractor E' that extracts "almost all the bits present in the source distribution". More precisely, we prove a general theorem saying that if E and C satisfy certain properties, then we can transform E into an extractor E'. Our methods build on (and generalize) a technique of Gabizon, Raz and Shaltiel (FOCS 2004) that present such a transformation for the very restricted class C of "oblivious bit-fixing sources". Loosely speaking the high level idea is to find properties of E and C which allow "recycling" the output of E so that it can be "reused" to operate on the source distribution. An obvious obstacle is that the output of E is correlated with the source distribution. Using our transformation we give an explicit construction of a two-source extractor E : {0, 1}n times {0, 1}n rarr {0, 1}m such that for every two independent distributions X1 and X2 over {0, 1}n with min-entropy at least k = (1/2 + delta)n, E(X1, X2) is epsi-close to the uniform distribution on m = 2k - Cdeltalog(1/epsi) bits. This result is optimal except for the precise constant Cdelta and improves previous results by Chor and Goldreich (SICOMP 1988), Vazirani (Combinatorica 1987) and Dodis et al. (RANDOM 2004). We also give explicit constructions of extractors for samplable distributions that extract many bits even out of "low-entropy" samplable distributions. This improves some previous results by Trevisan and Vadhan (FOCS 2000)
{"title":"How to Get More Mileage from Randomness Extractors","authors":"Ronen Shaltiel","doi":"10.1002/rsa.20207","DOIUrl":"https://doi.org/10.1002/rsa.20207","url":null,"abstract":"Let C be a class of distributions over {0, 1}n. A deterministic randomness extractor for C is a function E : {0, 1}n rarr {0, 1}m such that for any X in C the distribution E(X) is statistically close to the uniform distribution. A long line of research deals with explicit constructions of such extractors for various classes C while trying to maximize m. In this paper we give a general transformation that transforms a deterministic extractor E that extracts \"few\" bits into an extractor E' that extracts \"almost all the bits present in the source distribution\". More precisely, we prove a general theorem saying that if E and C satisfy certain properties, then we can transform E into an extractor E'. Our methods build on (and generalize) a technique of Gabizon, Raz and Shaltiel (FOCS 2004) that present such a transformation for the very restricted class C of \"oblivious bit-fixing sources\". Loosely speaking the high level idea is to find properties of E and C which allow \"recycling\" the output of E so that it can be \"reused\" to operate on the source distribution. An obvious obstacle is that the output of E is correlated with the source distribution. Using our transformation we give an explicit construction of a two-source extractor E : {0, 1}n times {0, 1}n rarr {0, 1}m such that for every two independent distributions X1 and X2 over {0, 1}n with min-entropy at least k = (1/2 + delta)n, E(X1, X2) is epsi-close to the uniform distribution on m = 2k - Cdeltalog(1/epsi) bits. This result is optimal except for the precise constant Cdelta and improves previous results by Chor and Goldreich (SICOMP 1988), Vazirani (Combinatorica 1987) and Dodis et al. (RANDOM 2004). We also give explicit constructions of extractors for samplable distributions that extract many bits even out of \"low-entropy\" samplable distributions. This improves some previous results by Trevisan and Vadhan (FOCS 2000)","PeriodicalId":325664,"journal":{"name":"21st Annual IEEE Conference on Computational Complexity (CCC'06)","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131554702","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 establish a close connection between (sub)exponential time complexity and parameterized complexity by proving that the so-called miniaturization mapping is a reduction preserving isomorphism between the two theories
{"title":"An isomorphism between subexponential and parameterized complexity theory","authors":"Yijia Chen, Martin Grohe","doi":"10.1137/070687153","DOIUrl":"https://doi.org/10.1137/070687153","url":null,"abstract":"We establish a close connection between (sub)exponential time complexity and parameterized complexity by proving that the so-called miniaturization mapping is a reduction preserving isomorphism between the two theories","PeriodicalId":325664,"journal":{"name":"21st Annual IEEE Conference on Computational Complexity (CCC'06)","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121155929","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 the relationship between the complexities of k-SAT and those of SAT restricted to formulas of constant density. Let sk be the infimum of those c ges 0 such that k-SAT on n variables can be decided in time O(2cn) and dDelta be the infimum of those c ges 0 such that SAT on n variables and les Deltan clauses can be decided in time O(2cn). We show that limkrarrinfin sk = limDeltararrinfindDelta. So, for any epsi > 0, k-SAT can be solved in 2(1-epsi)n time independent of k if and only if the same is true for SAT with any fixed density of clauses to variables. We derive some interesting consequences from this. For example, assuming that 3-SAT is exponentially hard (that is, s3 > 0), SAT of any fixed density can be solved in time whose exponent is strictly less than that for general SAT. We also give an improvement to the sparsification lemma of Impagliazzo et al. (1998) showing that instances of k-SAT of density slightly more than exponential in k are almost the hardest instances of k-SAT. The previous result showed this for densities doubly exponential in k
{"title":"A duality between clause width and clause density for SAT","authors":"Chris Calabro, R. Impagliazzo, R. Paturi","doi":"10.1109/CCC.2006.6","DOIUrl":"https://doi.org/10.1109/CCC.2006.6","url":null,"abstract":"We consider the relationship between the complexities of k-SAT and those of SAT restricted to formulas of constant density. Let sk be the infimum of those c ges 0 such that k-SAT on n variables can be decided in time O(2cn) and dDelta be the infimum of those c ges 0 such that SAT on n variables and les Deltan clauses can be decided in time O(2cn). We show that limkrarrinfin sk = limDeltararrinfindDelta. So, for any epsi > 0, k-SAT can be solved in 2(1-epsi)n time independent of k if and only if the same is true for SAT with any fixed density of clauses to variables. We derive some interesting consequences from this. For example, assuming that 3-SAT is exponentially hard (that is, s3 > 0), SAT of any fixed density can be solved in time whose exponent is strictly less than that for general SAT. We also give an improvement to the sparsification lemma of Impagliazzo et al. (1998) showing that instances of k-SAT of density slightly more than exponential in k are almost the hardest instances of k-SAT. The previous result showed this for densities doubly exponential in k","PeriodicalId":325664,"journal":{"name":"21st Annual IEEE Conference on Computational Complexity (CCC'06)","volume":"116 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124301376","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 study the power of quantum fingerprints in the simultaneous message passing (SMP) setting of communication complexity. Yao recently showed how to simulate, with exponential overhead, classical shared-randomness SMP protocols by means of quantum SMP protocols without shared randomness (Qpar-protocols). Our first result is to extend Yao's simulation to the strongest possible model: every many-round quantum protocol with unlimited shared entanglement can be simulated, with exponential overhead, by Qpar-protocols. We apply our technique to obtain an efficient Qpar-protocol for a function which cannot be efficiently solved through more restricted simulations. Second, we tightly characterize the power of the quantum fingerprinting technique by making a connection to arrangements of homogeneous halfspaces with maximal margin. These arrangements have been well studied in computational learning theory, and we use some strong results obtained in this area to exhibit weaknesses of quantum fingerprinting. In particular, this implies that for almost all functions, quantum fingerprinting protocols are exponentially worse than classical deterministic SMP protocols
{"title":"Strengths and weaknesses of quantum fingerprinting","authors":"Dmitry Gavinsky, J. Kempe, R. D. Wolf","doi":"10.1109/CCC.2006.39","DOIUrl":"https://doi.org/10.1109/CCC.2006.39","url":null,"abstract":"We study the power of quantum fingerprints in the simultaneous message passing (SMP) setting of communication complexity. Yao recently showed how to simulate, with exponential overhead, classical shared-randomness SMP protocols by means of quantum SMP protocols without shared randomness (Qpar-protocols). Our first result is to extend Yao's simulation to the strongest possible model: every many-round quantum protocol with unlimited shared entanglement can be simulated, with exponential overhead, by Qpar-protocols. We apply our technique to obtain an efficient Qpar-protocol for a function which cannot be efficiently solved through more restricted simulations. Second, we tightly characterize the power of the quantum fingerprinting technique by making a connection to arrangements of homogeneous halfspaces with maximal margin. These arrangements have been well studied in computational learning theory, and we use some strong results obtained in this area to exhibit weaknesses of quantum fingerprinting. In particular, this implies that for almost all functions, quantum fingerprinting protocols are exponentially worse than classical deterministic SMP protocols","PeriodicalId":325664,"journal":{"name":"21st Annual IEEE Conference on Computational Complexity (CCC'06)","volume":"23 16","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131805749","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}
Theoretical computer scientists have been debating the role of oracles since the 1970's. This paper illustrates both that oracles can give us nontrivial insights about the barrier problems in circuit complexity, and that they need not prevent us from trying to solve those problems. First, we give an oracle relative to which PP has linear-sized circuits, by proving a new lower bound for perceptrons and low-degree threshold polynomials. This oracle settles a longstanding open question, and generalizes earlier results due to Beigel and to Buhrman, Fortnow, and Thierauf. More importantly, it implies the first provably nonrelativizing separation of "traditional" complexity classes, as opposed to interactive proof classes such as MIP and MAEXP. For Vinodchandran showed, by a nonrelativizing argument, that PP does not have circuits of size nk for any fixed k. We present an alternative proof of this fact, which shows that PP does not even have quantum circuits of size nk with quantum advice. To our knowledge, this is the first nontrivial lower bound on quantum circuit size. Second, we study a beautiful algorithm of Bshouty et al. for learning Boolean circuits in ZPPNP. We show that the NP queries in this algorithm cannot be parallelized by any relativizing technique, by giving an oracle relative to which ZPPNP par and even BPPNP par have linear-size circuits. On the other hand, we also show that the NP queries could be parallelized if P = NP. Thus, classes such as ZPPNP par inhabit a "twilight zone", where we need to distinguish between relativizing and black-box techniques. Our results on this subject have implications for computational learning theory as well as for the circuit minimization problem
{"title":"Oracles are subtle but not malicious","authors":"S. Aaronson","doi":"10.1109/CCC.2006.32","DOIUrl":"https://doi.org/10.1109/CCC.2006.32","url":null,"abstract":"Theoretical computer scientists have been debating the role of oracles since the 1970's. This paper illustrates both that oracles can give us nontrivial insights about the barrier problems in circuit complexity, and that they need not prevent us from trying to solve those problems. First, we give an oracle relative to which PP has linear-sized circuits, by proving a new lower bound for perceptrons and low-degree threshold polynomials. This oracle settles a longstanding open question, and generalizes earlier results due to Beigel and to Buhrman, Fortnow, and Thierauf. More importantly, it implies the first provably nonrelativizing separation of \"traditional\" complexity classes, as opposed to interactive proof classes such as MIP and MAEXP. For Vinodchandran showed, by a nonrelativizing argument, that PP does not have circuits of size nk for any fixed k. We present an alternative proof of this fact, which shows that PP does not even have quantum circuits of size nk with quantum advice. To our knowledge, this is the first nontrivial lower bound on quantum circuit size. Second, we study a beautiful algorithm of Bshouty et al. for learning Boolean circuits in ZPPNP. We show that the NP queries in this algorithm cannot be parallelized by any relativizing technique, by giving an oracle relative to which ZPPNP par and even BPPNP par have linear-size circuits. On the other hand, we also show that the NP queries could be parallelized if P = NP. Thus, classes such as ZPPNP par inhabit a \"twilight zone\", where we need to distinguish between relativizing and black-box techniques. Our results on this subject have implications for computational learning theory as well as for the circuit minimization problem","PeriodicalId":325664,"journal":{"name":"21st Annual IEEE Conference on Computational Complexity (CCC'06)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126198545","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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