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Quantum query complexity and semi-definite programming 量子查询复杂度与半确定规划
18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings. Pub Date : 2003-07-07 DOI: 10.1109/CCC.2003.1214419
H. Barnum, M. Saks, M. Szegedy
{"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}
引用次数: 94
A strong inapproximability gap for a generalization of minimum bisection 最小二分法推广的一个强不可逼近性间隙
18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings. Pub Date : 2003-07-07 DOI: 10.1109/CCC.2003.1214436
Jonas Holmerin, Subhash Khot
{"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}
引用次数: 5
The complexity of stochastic sequences 随机序列的复杂性
18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings. Pub Date : 2003-07-07 DOI: 10.1109/CCC.2003.1214423
W. Merkle
{"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}
引用次数: 26
Three-query PCPs with perfect completeness over non-Boolean domains 非布尔域上完全完备的三查询pcp
18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings. Pub Date : 2003-07-07 DOI: 10.1109/CCC.2003.1214428
Lars Engebretsen, Jonas Holmerin
{"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}
引用次数: 6
Hardness vs. randomness within alternating time 在交替的时间内，硬度vs随机性
18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings. Pub Date : 2003-07-07 DOI: 10.1109/CCC.2003.1214410
Emanuele Viola
{"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}
引用次数: 15
Optimal separation of EROW and CROW PRAMs EROW与CROW婴儿车的最佳分离
18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings. Pub Date : 2003-07-07 DOI: 10.1109/CCC.2003.1214413
Navin Goyal, M. Saks, Venkatesh Srinivasan
{"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}
引用次数: 1
Memoization and DPLL: formula caching proof systems 记忆和DPLL:公式缓存证明系统
18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings. Pub Date : 2003-07-07 DOI: 10.1109/CCC.2003.1214425
P. Beame, R. Impagliazzo, T. Pitassi, Nathan Segerlind
{"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}
引用次数: 37
Vertex cover might be hard to approximate to within 2-/spl epsiv/ 顶点覆盖可能很难接近于2-/spl / epsiv/
18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings. Pub Date : 2003-07-07 DOI: 10.1109/CCC.2003.1214437
Subhash Khot, O. Regev
{"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}
引用次数: 356
The complexity of unique k-SAT: an isolation lemma for k-CNFs 唯一k-SAT的复杂性:k-CNFs的一个隔离引理
18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings. Pub Date : 2003-07-07 DOI: 10.1109/CCC.2003.1214416
Chris Calabro, R. Impagliazzo, Valentine Kabanets, R. Paturi
{"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}
引用次数: 89
A combinatorial characterization of resolution width 分辨率宽度的组合表征
18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings. Pub Date : 2003-07-07 DOI: 10.1109/CCC.2003.1214424
Albert Atserias, V. Dalmau
{"title":"A combinatorial characterization of resolution width","authors":"Albert Atserias, V. Dalmau","doi":"10.1109/CCC.2003.1214424","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214424","url":null,"abstract":"We provide a characterization of the resolution width introduced in the context of propositional proof complexity in terms of the existential pebble game introduced in the context of finite model theory. The characterization is tight and purely combinatorial. Our first application of this result is a surprising proof that the minimum space of refuting a 3-CNF formula is always bounded from below by the minimum width of refuting it (minus 3). This solves a well-known open problem. The second application is the unification of several width lower bound arguments, and a new width lower bound for the dense linear order principle. Since we also show that this principle has resolution refutations of polynomial size, this provides yet another example showing that the size-width relationship is tight.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"16 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":"122167611","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}
引用次数: 169
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