Pub Date : 2003-07-07DOI: 10.1109/CCC.2003.1214419
H. Barnum, M. Saks, M. Szegedy
We reformulate quantum query complexity in terms of inequalities and equations for a set of positive semidefinite matrices. Using the new formulation we: 1) show that the workspace of a quantum computer can be limited to at most n+k qubits (where n and k are the number of input and output bits respectively) without reducing the computational power of the model; 2) give an algorithm that on input the truth table of a partial Boolean function and an integer t runs in time polynomial in the size of the truth table and estimates, to any desired accuracy, the minimum probability of error that can be attained by a quantum query algorithm attempts to evaluate f in t queries; 3) use semidefinite programming duality to formulate a dual SDP P/spl circ/(f, t, /spl epsi/) that is feasible if and only if f cannot be evaluated within error /spl epsi/ by a t-step quantum query algorithm. Using this SDP, we derive a general lower bound for quantum query complexity that encompasses a lower bound method of Ambainis and its generalizations; 4) give an interpretation of a generalized form of branching in quantum computation.
{"title":"Quantum query complexity and semi-definite programming","authors":"H. Barnum, M. Saks, M. Szegedy","doi":"10.1109/CCC.2003.1214419","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214419","url":null,"abstract":"We reformulate quantum query complexity in terms of inequalities and equations for a set of positive semidefinite matrices. Using the new formulation we: 1) show that the workspace of a quantum computer can be limited to at most n+k qubits (where n and k are the number of input and output bits respectively) without reducing the computational power of the model; 2) give an algorithm that on input the truth table of a partial Boolean function and an integer t runs in time polynomial in the size of the truth table and estimates, to any desired accuracy, the minimum probability of error that can be attained by a quantum query algorithm attempts to evaluate f in t queries; 3) use semidefinite programming duality to formulate a dual SDP P/spl circ/(f, t, /spl epsi/) that is feasible if and only if f cannot be evaluated within error /spl epsi/ by a t-step quantum query algorithm. Using this SDP, we derive a general lower bound for quantum query complexity that encompasses a lower bound method of Ambainis and its generalizations; 4) give an interpretation of a generalized form of branching in quantum computation.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116078234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2003-07-07DOI: 10.1109/CCC.2003.1214436
Jonas Holmerin, Subhash Khot
As a problem with similar properties to minimum bisection, we consider the following: given a homogeneous system of linear equations over Z/sub 2/, with exactly k variables in each equation, find a balanced assignment that minimizes the number of satisfied equations. A balanced assignment is one which contains an equal number of 0s and 1s. When k=2, this is the minimum bisection problem. We consider the case k=3. In this case, it is NP-complete to determine whether the object function is zero [U. Feige, (2003)], so the problem is not approximable at all. However, we prove that it is NP-hard to determine distinguish between the cases that all but a fraction /spl epsi/ of the equations can be satisfied and that at least a fraction 1/4-/spl epsi/ of all equations cannot be satisfied. A similar result for minimum bisection would imply that the problem is hard to approximate within any constant. For the problem of approximating the maximum number of equations satisfied by a balanced assignment, this implies that the problem is NP-hard to approximate within 4/3-/spl epsi/, for any /spl epsi/>0.
{"title":"A strong inapproximability gap for a generalization of minimum bisection","authors":"Jonas Holmerin, Subhash Khot","doi":"10.1109/CCC.2003.1214436","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214436","url":null,"abstract":"As a problem with similar properties to minimum bisection, we consider the following: given a homogeneous system of linear equations over Z/sub 2/, with exactly k variables in each equation, find a balanced assignment that minimizes the number of satisfied equations. A balanced assignment is one which contains an equal number of 0s and 1s. When k=2, this is the minimum bisection problem. We consider the case k=3. In this case, it is NP-complete to determine whether the object function is zero [U. Feige, (2003)], so the problem is not approximable at all. However, we prove that it is NP-hard to determine distinguish between the cases that all but a fraction /spl epsi/ of the equations can be satisfied and that at least a fraction 1/4-/spl epsi/ of all equations cannot be satisfied. A similar result for minimum bisection would imply that the problem is hard to approximate within any constant. For the problem of approximating the maximum number of equations satisfied by a balanced assignment, this implies that the problem is NP-hard to approximate within 4/3-/spl epsi/, for any /spl epsi/>0.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"2000 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116759204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2003-07-07DOI: 10.1109/CCC.2003.1214423
W. Merkle
We observe that known results on the Kolmogorov complexity of prefixes of effectively stochastic sequences extend to corresponding random sequences. First, there are recursively random sequences such that for any nondecreasing and unbounded computable function f and for almost all n, the uniform complexity of the length n prefix of the sequence is bounded by f(n). Second, a similar result with bounds of the form f(n) log n holds for partially-recursive random sequences. Furthermore, we show that there is no Mises-Wald-Church stochastic sequence such that the prefixes of the sequence have Kolmogorov complexity O(log n). This result implies a sharp bound for the complexity of the prefixes of Mises-Wald-Church stochastic and of partially-recursive random sequences. As an immediate corollary to our results, we obtain the known separation of the classes of recursively random and of Mises-Wald-Church stochastic sequences.
{"title":"The complexity of stochastic sequences","authors":"W. Merkle","doi":"10.1109/CCC.2003.1214423","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214423","url":null,"abstract":"We observe that known results on the Kolmogorov complexity of prefixes of effectively stochastic sequences extend to corresponding random sequences. First, there are recursively random sequences such that for any nondecreasing and unbounded computable function f and for almost all n, the uniform complexity of the length n prefix of the sequence is bounded by f(n). Second, a similar result with bounds of the form f(n) log n holds for partially-recursive random sequences. Furthermore, we show that there is no Mises-Wald-Church stochastic sequence such that the prefixes of the sequence have Kolmogorov complexity O(log n). This result implies a sharp bound for the complexity of the prefixes of Mises-Wald-Church stochastic and of partially-recursive random sequences. As an immediate corollary to our results, we obtain the known separation of the classes of recursively random and of Mises-Wald-Church stochastic sequences.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115980755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2003-07-07DOI: 10.1109/CCC.2003.1214428
Lars Engebretsen, Jonas Holmerin
We study nonBoolean PCPs that have perfect completeness and read three positions from the proof. For the case when the proof consists of values from a domain of size d for some integer constant d/spl ges/2, we construct a nonadaptive PCP with perfect completeness and soundness d/sup -1/+d/sup -2/+/spl epsiv/, for any constant /spl epsiv/>0, and an adaptive PCP with perfect completeness and soundness d/sup -1/+/spl epsiv/, for any constant /spl epsiv/>0. The latter PCP can be converted into a nonadaptive PCP with perfect completeness and soundness d/sup -1/+/spl epsiv/, for any constant /spl epsiv/>0, where four positions are read from the proof. These results match the best known constructions for the case d=2 and our proofs also show that the particular predicates we use in our PCPs are nonapproximable beyond the random assignment threshold.
{"title":"Three-query PCPs with perfect completeness over non-Boolean domains","authors":"Lars Engebretsen, Jonas Holmerin","doi":"10.1109/CCC.2003.1214428","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214428","url":null,"abstract":"We study nonBoolean PCPs that have perfect completeness and read three positions from the proof. For the case when the proof consists of values from a domain of size d for some integer constant d/spl ges/2, we construct a nonadaptive PCP with perfect completeness and soundness d/sup -1/+d/sup -2/+/spl epsiv/, for any constant /spl epsiv/>0, and an adaptive PCP with perfect completeness and soundness d/sup -1/+/spl epsiv/, for any constant /spl epsiv/>0. The latter PCP can be converted into a nonadaptive PCP with perfect completeness and soundness d/sup -1/+/spl epsiv/, for any constant /spl epsiv/>0, where four positions are read from the proof. These results match the best known constructions for the case d=2 and our proofs also show that the particular predicates we use in our PCPs are nonapproximable beyond the random assignment threshold.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"712 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116970598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2003-07-07DOI: 10.1109/CCC.2003.1214410
Emanuele Viola
We study the complexity of building pseudorandom generators (PRGs) with logarithmic seed length from hard functions. We show that, starting from a function f:{0,1}/sup l//spl rarr/{0,1} that is mildly hard on average, i.e. every circuit of size 2/sup /spl Omega/(l)/ fails to compute f on at least a 1/poly(l) fraction of inputs, we can build a PRG: {0,1}/sup O(logn)//spl rarr/{0,1}/sup n/ computable in ATIME(O(1), logn)=alternating time O(logn) with O(1) alternations. Such a PRG implies BP/spl middot/AC/sub 0/=AC/sub 0/ under DLOGTIME-uniformity. On the negative side, we prove a tight lower bound on black-box PRG constructions that are based on worst-case hard functions. We also prove a tight lower bound on black-box worst-case hardness amplification, which is the problem of producing an average-case hard function starting from a worst-case hard one. These lower bounds are obtained by showing that constant depth circuits cannot compute extractors and list-decodable codes.
{"title":"Hardness vs. randomness within alternating time","authors":"Emanuele Viola","doi":"10.1109/CCC.2003.1214410","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214410","url":null,"abstract":"We study the complexity of building pseudorandom generators (PRGs) with logarithmic seed length from hard functions. We show that, starting from a function f:{0,1}/sup l//spl rarr/{0,1} that is mildly hard on average, i.e. every circuit of size 2/sup /spl Omega/(l)/ fails to compute f on at least a 1/poly(l) fraction of inputs, we can build a PRG: {0,1}/sup O(logn)//spl rarr/{0,1}/sup n/ computable in ATIME(O(1), logn)=alternating time O(logn) with O(1) alternations. Such a PRG implies BP/spl middot/AC/sub 0/=AC/sub 0/ under DLOGTIME-uniformity. On the negative side, we prove a tight lower bound on black-box PRG constructions that are based on worst-case hard functions. We also prove a tight lower bound on black-box worst-case hardness amplification, which is the problem of producing an average-case hard function starting from a worst-case hard one. These lower bounds are obtained by showing that constant depth circuits cannot compute extractors and list-decodable codes.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132058778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2003-07-07DOI: 10.1109/CCC.2003.1214413
Navin Goyal, M. Saks, Venkatesh Srinivasan
We consider the problem of evaluating a Boolean function on PRAMs. We exhibit a Boolean function f:{0,1}/sup n//spl rarr/{0,1} that can be evaluated in time O(log log n) in a deterministic CROW (concurrent read owner write) PRAM model, but requires time /spl Omega/(log n) in EROW (exclusive read owner write) PRAM. Our lower bound also holds in the randomized Monte Carlo EROW model. This Boolean function is derived from the well-known pointer chasing problem, and was first considered by Nisan and Bar-Yossef (1997). Our lower bound improves a special case of the previous result of Nisan and Bar-Yossef, who proved a lower bound of /spl Omega/(/spl radic/(log n)) for this function in the deterministic EREW model (and hence in the EROW model). Our result is the first to achieve the best possible separation between the CROW and EROW PRAM models for functions on complete domains (Boolean or nonBoolean), improving the previous results (E. Gafni et al., 1989; F. Fich et al., 1990; N. Nisan et al., 1997).
研究了一个布尔函数在pram上的求值问题。我们展示了一个布尔函数f:{0,1}/sup n//spl rarr/{0,1}，在确定性CROW(并发读所有者写)PRAM模型中可以在时间O(log log n)内求值，但在EROW(独占读所有者写)PRAM中需要时间/spl Omega/(log n)。我们的下界也适用于随机蒙特卡罗EROW模型。这个布尔函数来源于著名的指针追踪问题，最早是由Nisan和Bar-Yossef(1997)提出的。我们的下界改进了Nisan和Bar-Yossef先前结果的一个特例，他们在确定性的EREW模型中(因此在EROW模型中)证明了该函数的/spl ω /(/spl径向/(log n))的下界。我们的结果是第一个实现了完整域(布尔或非布尔)上函数的CROW和EROW PRAM模型之间的最佳分离，改进了以前的结果(E. Gafni等人，1989;F. Fich et al.， 1990;N. Nisan et al.， 1997)。
{"title":"Optimal separation of EROW and CROW PRAMs","authors":"Navin Goyal, M. Saks, Venkatesh Srinivasan","doi":"10.1109/CCC.2003.1214413","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214413","url":null,"abstract":"We consider the problem of evaluating a Boolean function on PRAMs. We exhibit a Boolean function f:{0,1}/sup n//spl rarr/{0,1} that can be evaluated in time O(log log n) in a deterministic CROW (concurrent read owner write) PRAM model, but requires time /spl Omega/(log n) in EROW (exclusive read owner write) PRAM. Our lower bound also holds in the randomized Monte Carlo EROW model. This Boolean function is derived from the well-known pointer chasing problem, and was first considered by Nisan and Bar-Yossef (1997). Our lower bound improves a special case of the previous result of Nisan and Bar-Yossef, who proved a lower bound of /spl Omega/(/spl radic/(log n)) for this function in the deterministic EREW model (and hence in the EROW model). Our result is the first to achieve the best possible separation between the CROW and EROW PRAM models for functions on complete domains (Boolean or nonBoolean), improving the previous results (E. Gafni et al., 1989; F. Fich et al., 1990; N. Nisan et al., 1997).","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123819027","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2003-07-07DOI: 10.1109/CCC.2003.1214425
P. Beame, R. Impagliazzo, T. Pitassi, Nathan Segerlind
A fruitful connection between algorithm design and proof complexity is the formalization of the DPLL approach to satisfiability testing in terms of tree-like resolution proofs. We consider extensions of the DPLL approach that add some version of memoization, remembering formulas the algorithm has previously shown unsatisfiable. Various versions of such formula caching algorithms have been suggested for satisfiability and stochastic satisfiability (S. M. Majercik et al., 1998; F. Bacchus et al., 2003). We formalize this method, and characterize the strength of various versions in terms of proof systems. These proof systems seem to be both new and simple, and have a rich structure. We compare their strength to several studied proof systems: tree-like resolution, regular resolution, general resolution, and Res(k). We give both simulations and separations.
算法设计和证明复杂性之间的一个富有成效的联系是将DPLL方法形式化，以树状分辨率证明的形式进行可满足性测试。我们考虑了DPLL方法的扩展，增加了一些版本的记忆，记住了算法以前显示的不满意的公式。对于可满足性和随机可满足性，已经提出了这种公式缓存算法的各种版本(S. M. Majercik等人，1998;F. Bacchus et al.， 2003)。我们将这种方法形式化，并根据证明系统描述了各种版本的强度。这些证明系统似乎既新颖又简单，结构丰富。我们将它们的强度与几个研究过的证明系统进行了比较:树状分辨率、常规分辨率、一般分辨率和Res(k)。我们给出了模拟和分离。
{"title":"Memoization and DPLL: formula caching proof systems","authors":"P. Beame, R. Impagliazzo, T. Pitassi, Nathan Segerlind","doi":"10.1109/CCC.2003.1214425","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214425","url":null,"abstract":"A fruitful connection between algorithm design and proof complexity is the formalization of the DPLL approach to satisfiability testing in terms of tree-like resolution proofs. We consider extensions of the DPLL approach that add some version of memoization, remembering formulas the algorithm has previously shown unsatisfiable. Various versions of such formula caching algorithms have been suggested for satisfiability and stochastic satisfiability (S. M. Majercik et al., 1998; F. Bacchus et al., 2003). We formalize this method, and characterize the strength of various versions in terms of proof systems. These proof systems seem to be both new and simple, and have a rich structure. We compare their strength to several studied proof systems: tree-like resolution, regular resolution, general resolution, and Res(k). We give both simulations and separations.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117174973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2003-07-07DOI: 10.1109/CCC.2003.1214437
Subhash Khot, O. Regev
Based on a conjecture regarding the power of unique 2-prover-1-round games presented in [S. Khot, (2002)], we show that vertex cover is hard to approximate within any constant factor better than 2. We actually show a stronger result, namely, based on the same conjecture, vertex cover on k-uniform hypergraphs is hard to approximate within any constant factor better than k.
{"title":"Vertex cover might be hard to approximate to within 2-/spl epsiv/","authors":"Subhash Khot, O. Regev","doi":"10.1109/CCC.2003.1214437","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214437","url":null,"abstract":"Based on a conjecture regarding the power of unique 2-prover-1-round games presented in [S. Khot, (2002)], we show that vertex cover is hard to approximate within any constant factor better than 2. We actually show a stronger result, namely, based on the same conjecture, vertex cover on k-uniform hypergraphs is hard to approximate within any constant factor better than k.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131217834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2003-07-07DOI: 10.1109/CCC.2003.1214416
Chris Calabro, R. Impagliazzo, Valentine Kabanets, R. Paturi
We provide some evidence that unique k-SAT is as hard to solve as general k-SAT, where k-SAT denotes the satisfiability problem for k-CNFs and unique k-SAT is the promise version where the given formula has 0 or 1 solutions. Namely, defining for each k/spl ges/1, s/sub k/=inf{/spl delta//spl ges/0|/spl exist/aO(2/sup /spl delta/n/)-time randomized algorithm for k-SAT} and, similarly, /spl sigma//sub k/=inf{/spl delta//spl ges/0|/spl exist/aO(2/sup /spl delta/n/)-time randomized algorithm for Unique k-SAT}, we show that lim/sub k/spl rarr//spl infin//s/sub k/=lim/sub k/spl rarr//spl infin///spl sigma//sub k/. As a corollary, we prove that, if Unique 3-SAT can be solved in time 2/sup /spl epsi/n/ for every /spl epsi/>0, then so can k-SAT for k/spl ges/3. Our main technical result is an isolation lemma for k-CNFs, which shows that a given satisfiable k-CNF can be efficiently probabilistically reduced to a uniquely satisfiable k-CNF, with nontrivial, albeit exponentially small, success probability.
{"title":"The complexity of unique k-SAT: an isolation lemma for k-CNFs","authors":"Chris Calabro, R. Impagliazzo, Valentine Kabanets, R. Paturi","doi":"10.1109/CCC.2003.1214416","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214416","url":null,"abstract":"We provide some evidence that unique k-SAT is as hard to solve as general k-SAT, where k-SAT denotes the satisfiability problem for k-CNFs and unique k-SAT is the promise version where the given formula has 0 or 1 solutions. Namely, defining for each k/spl ges/1, s/sub k/=inf{/spl delta//spl ges/0|/spl exist/aO(2/sup /spl delta/n/)-time randomized algorithm for k-SAT} and, similarly, /spl sigma//sub k/=inf{/spl delta//spl ges/0|/spl exist/aO(2/sup /spl delta/n/)-time randomized algorithm for Unique k-SAT}, we show that lim/sub k/spl rarr//spl infin//s/sub k/=lim/sub k/spl rarr//spl infin///spl sigma//sub k/. As a corollary, we prove that, if Unique 3-SAT can be solved in time 2/sup /spl epsi/n/ for every /spl epsi/>0, then so can k-SAT for k/spl ges/3. Our main technical result is an isolation lemma for k-CNFs, which shows that a given satisfiable k-CNF can be efficiently probabilistically reduced to a uniquely satisfiable k-CNF, with nontrivial, albeit exponentially small, success probability.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133767117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2003-07-07DOI: 10.1109/CCC.2003.1214409
R. Impagliazzo, Philippe Moser
We show that if RP has p-measure nonzero then ZPP=EXP. As corollaries, we obtain a zero-one law for RP, and that both probabilistic classes ZPP and RP have the same p-measure. Finally we prove that if NP has p-measure nonzero then NP=AM.
{"title":"A zero one law for RP","authors":"R. Impagliazzo, Philippe Moser","doi":"10.1109/CCC.2003.1214409","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214409","url":null,"abstract":"We show that if RP has p-measure nonzero then ZPP=EXP. As corollaries, we obtain a zero-one law for RP, and that both probabilistic classes ZPP and RP have the same p-measure. Finally we prove that if NP has p-measure nonzero then NP=AM.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130127926","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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