Pub Date : 2003-07-07DOI: 10.1109/CCC.2003.1214424
Albert Atserias, V. Dalmau
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.
{"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}
Pub Date : 2003-07-07DOI: 10.1109/CCC.2003.1214430
Christian Glaßer, A. Selman, Samik Sengupta, Liyu Zhang
We study the question of whether the class DisNP of disjoint pairs (A, B) of NP-sets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NP-sets that is NP-hard. We show under reasonable hypotheses that nonsymmetric disjoint NP-pairs exist, which provide additional evidence for the existence of P-inseparable disjoint NP-pairs. We construct an oracle relative to which the class of disjoint NP-pairs does not have a complete pair, an oracle relative to which optimal proof systems exist, hence complete pairs exist, but no pair is NP-hard, and an oracle relative to which complete pairs exist, but optimal proof systems do not exist.
{"title":"Disjoint NP-pairs","authors":"Christian Glaßer, A. Selman, Samik Sengupta, Liyu Zhang","doi":"10.1109/CCC.2003.1214430","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214430","url":null,"abstract":"We study the question of whether the class DisNP of disjoint pairs (A, B) of NP-sets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NP-sets that is NP-hard. We show under reasonable hypotheses that nonsymmetric disjoint NP-pairs exist, which provide additional evidence for the existence of P-inseparable disjoint NP-pairs. We construct an oracle relative to which the class of disjoint NP-pairs does not have a complete pair, an oracle relative to which optimal proof systems exist, hence complete pairs exist, but no pair is NP-hard, and an oracle relative to which complete pairs exist, but optimal proof systems do not exist.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"40 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":"123180171","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.2003.1214426
J. Håstad
The purpose of this talk is to give an overview of the status of some problems in approximability.
这次演讲的目的是对近似性中一些问题的现状进行概述。
{"title":"Inapproximability - some history and some open problems","authors":"J. Håstad","doi":"10.1109/CCC.2003.1214426","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214426","url":null,"abstract":"The purpose of this talk is to give an overview of the status of some problems in approximability.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"75 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":"115738287","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.2003.1214432
Alan Nash, R. Impagliazzo, J. Remmel
We define and study strong diagonalization and compare it to weak diagonalization, implicit in the work of D. Kozen (1980). Kozen's result shows that virtually every separation can be recast as weak diagonalization. We show that there are classes of languages, which cannot be separated by strong diagonalization and provide evidence that strong diagonalization does not relativize. We also define two kinds of indirect diagonalization and study their power: Since we define strong diagonalization in terms of universal languages, we study their complexity. We distinguish and compare weak and strict universal languages. Finally we analyze some apparently weaker variants of universal languages, which we call pseudouniversal languages, and show that under weak closure conditions they easily yield universal languages.
{"title":"Universal languages and the power of diagonalization","authors":"Alan Nash, R. Impagliazzo, J. Remmel","doi":"10.1109/CCC.2003.1214432","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214432","url":null,"abstract":"We define and study strong diagonalization and compare it to weak diagonalization, implicit in the work of D. Kozen (1980). Kozen's result shows that virtually every separation can be recast as weak diagonalization. We show that there are classes of languages, which cannot be separated by strong diagonalization and provide evidence that strong diagonalization does not relativize. We also define two kinds of indirect diagonalization and study their power: Since we define strong diagonalization in terms of universal languages, we study their complexity. We distinguish and compare weak and strict universal languages. Finally we analyze some apparently weaker variants of universal languages, which we call pseudouniversal languages, and show that under weak closure conditions they easily yield universal languages.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"72 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":"132718098","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.2003.1214422
Andrei E. Romashchenko
We say that the mutual information of a triple of binary strings a, b, c can be extracted if there exists a string d such that a, b, and c are independent given d, and d is simple conditional to each of the strings a, b, c. This is an analog of the well-known Gacs-Korner (1973) definition of extrability of the mutual information for a pair of binary strings. We prove that (in contrast to the case of two strings) there exists a criterion of extrability of the mutual information for a triple a, b, c in terms of complexities involving a, b, c. Roughly speaking, the mutual information between a, b, c can be extracted if and only if the conditional mutual informations I(a:b|c), I(a:c|b), I(b:c|a) are negligible. Our proof of the main result is based on a nonShannon-type information inequality, which is a generalization of the recently discovered Zhang-Yeung inequality.
{"title":"Extracting the mutual information for a triple of binary strings","authors":"Andrei E. Romashchenko","doi":"10.1109/CCC.2003.1214422","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214422","url":null,"abstract":"We say that the mutual information of a triple of binary strings a, b, c can be extracted if there exists a string d such that a, b, and c are independent given d, and d is simple conditional to each of the strings a, b, c. This is an analog of the well-known Gacs-Korner (1973) definition of extrability of the mutual information for a pair of binary strings. We prove that (in contrast to the case of two strings) there exists a criterion of extrability of the mutual information for a triple a, b, c in terms of complexities involving a, b, c. Roughly speaking, the mutual information between a, b, c can be extracted if and only if the conditional mutual informations I(a:b|c), I(a:c|b), I(b:c|a) are negligible. Our proof of the main result is based on a nonShannon-type information inequality, which is a generalization of the recently discovered Zhang-Yeung inequality.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"6 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":"130706254","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.2003.1214406
R. O'Donnell, R. Servedio
We give new extremal bounds on polynomial threshold function (PTF) representations of Boolean functions. Our results include the following: 1) Almost every Boolean function has PTF degree at most n/2+O(/spl radic/(n log n)). Together with results of Anthony and Alon, we establish a conjecture of Wang and Williams [1991] and Aspnes, Beigel, Furst, and Rudich [1994] up to lower order terms. 2) Every Boolean function has PTF density at most (1-1/O(n))2/sup n/. This improves a result of Gotsman [1989]. 3) Every Boolean function has weak PTF density at most O(1)2/sup n/. This gives a negative answer to a question posed by Saks [1993]. 4) PTF degree /spl lfloor/log/sub 2/m/spl rfloor/+1 is necessary and sufficient for Boolean functions with sparsity m. This answers a question of Beigel [2000].
{"title":"Extremal properties of polynomial threshold functions","authors":"R. O'Donnell, R. Servedio","doi":"10.1109/CCC.2003.1214406","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214406","url":null,"abstract":"We give new extremal bounds on polynomial threshold function (PTF) representations of Boolean functions. Our results include the following: 1) Almost every Boolean function has PTF degree at most n/2+O(/spl radic/(n log n)). Together with results of Anthony and Alon, we establish a conjecture of Wang and Williams [1991] and Aspnes, Beigel, Furst, and Rudich [1994] up to lower order terms. 2) Every Boolean function has PTF density at most (1-1/O(n))2/sup n/. This improves a result of Gotsman [1989]. 3) Every Boolean function has weak PTF density at most O(1)2/sup n/. This gives a negative answer to a question posed by Saks [1993]. 4) PTF degree /spl lfloor/log/sub 2/m/spl rfloor/+1 is necessary and sufficient for Boolean functions with sparsity m. This answers a question of Beigel [2000].","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"21 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":"121060767","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.2003.1214417
Rodney G. Downey
The goal is to provide a tourist guide, with an eye towards structural issues, to what I consider some of the major highlights of parameterized complexity.
我们的目标是提供一个导游，着眼于结构问题，我认为参数化复杂性的一些主要亮点。
{"title":"Parameterized complexity for the skeptic","authors":"Rodney G. Downey","doi":"10.1109/CCC.2003.1214417","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214417","url":null,"abstract":"The goal is to provide a tourist guide, with an eye towards structural issues, to what I consider some of the major highlights of parameterized complexity.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"164 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":"122020698","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 : 2002-10-02DOI: 10.1109/CCC.2003.1214418
S. Aaronson
Given a Boolean function f, we study two natural generalizations of the certificate complexity C(f): the randomized certificate complexity RC(f) and the quantum certificate complexity QC(f). Using Ambainis' adversary method, we exactly characterize QC(f) as the square root of RC(f). We then use this result to prove the new relation R/sub 0/(f)=O(Q/sub 2/(f)/sup 2/Q/sub 0/(f)log n) for total f, where R/sub 0/, Q/sub 2/, and Q/sub 0/ are zero-error randomized, bounded-error quantum, and zero-error quantum query complexities respectively. Finally we give asymptotic gaps between the measures, including a total f for which C(f) is superquadratic in QC(f), and a symmetric partial f for which QC(f)=O(1) yet Q/sub 2/(f)=/spl Omega/(n/log n).
{"title":"Quantum certificate complexity","authors":"S. Aaronson","doi":"10.1109/CCC.2003.1214418","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214418","url":null,"abstract":"Given a Boolean function f, we study two natural generalizations of the certificate complexity C(f): the randomized certificate complexity RC(f) and the quantum certificate complexity QC(f). Using Ambainis' adversary method, we exactly characterize QC(f) as the square root of RC(f). We then use this result to prove the new relation R/sub 0/(f)=O(Q/sub 2/(f)/sup 2/Q/sub 0/(f)log n) for total f, where R/sub 0/, Q/sub 2/, and Q/sub 0/ are zero-error randomized, bounded-error quantum, and zero-error quantum query complexities respectively. Finally we give asymptotic gaps between the measures, including a total f for which C(f) is superquadratic in QC(f), and a symmetric partial f for which QC(f)=O(1) yet Q/sub 2/(f)=/spl Omega/(n/log n).","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124049737","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 : 2002-08-05DOI: 10.1109/CCC.2003.1214415
H. Klauck
We investigate the power of the most important lower bound technique in randomized communication complexity, which is based on an evaluation of the maximal size of approximately monochromatic rectangles, with respect to arbitrary distributions on the inputs. While it is known that the 0-error version of this bound is polynomially tight for deterministic communication, nothing in this direction is known for constant error and randomized communication complexity. We first study a one-sided version of this bound and obtain that its value lies between the MA- and AM- complexities of the considered function. Hence the lower bound actually works for a (communication) complexity class between MA/spl cap/co - MA and AM/spl cap/co - AM, and allows to show that the MA-complexity of the disjointness problem is /spl Omega/(/spl radic/n). Following this we consider the conjecture that the lower bound method is polynomially tight for randomized communication complexity. First we disprove a distributional version of this conjecture. Then we give a combinatorial characterization of the value of the lower bound method, in which the optimization over all distributions is absent. This characterization is done by what we call a bounded error uniform threshold cover, and reduces showing tightness of the bound to the construction of an efficient protocol for a specific communication problem. We then study relaxations of bounded error uniform threshold covers, namely approximate majority covers and majority covers, and exhibit exponential separations between them. Each of these covers captures a lower bound method previously used for randomized communication complexity.
{"title":"Rectangle size bounds and threshold covers in communication complexity","authors":"H. Klauck","doi":"10.1109/CCC.2003.1214415","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214415","url":null,"abstract":"We investigate the power of the most important lower bound technique in randomized communication complexity, which is based on an evaluation of the maximal size of approximately monochromatic rectangles, with respect to arbitrary distributions on the inputs. While it is known that the 0-error version of this bound is polynomially tight for deterministic communication, nothing in this direction is known for constant error and randomized communication complexity. We first study a one-sided version of this bound and obtain that its value lies between the MA- and AM- complexities of the considered function. Hence the lower bound actually works for a (communication) complexity class between MA/spl cap/co - MA and AM/spl cap/co - AM, and allows to show that the MA-complexity of the disjointness problem is /spl Omega/(/spl radic/n). Following this we consider the conjecture that the lower bound method is polynomially tight for randomized communication complexity. First we disprove a distributional version of this conjecture. Then we give a combinatorial characterization of the value of the lower bound method, in which the optimization over all distributions is absent. This characterization is done by what we call a bounded error uniform threshold cover, and reduces showing tightness of the bound to the construction of an efficient protocol for a specific communication problem. We then study relaxations of bounded error uniform threshold covers, namely approximate majority covers and majority covers, and exhibit exponential separations between them. Each of these covers captures a lower bound method previously used for randomized communication complexity.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"215 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133097106","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：