Pub Date : 2003-07-07DOI: 10.1109/CCC.2003.1214427
D. Melkebeek, R. Santhanam
We derive a stronger consequence of EXP having polynomial-size circuits than was known previously, namely that there is a simulation of P in MAPOLYLOG that fools all deterministic polynomial-time adversaries. Using the connection between circuit lower bounds and derandomization, we obtain uniform assumptions for derandomizing BPP. Our results strengthen the space-randomness tradeoffs of Sipser, Nisan and Wigderson, and Lu. We show a partial converse: oracle circuit lower bounds for EXP imply that there are efficient simulations of P that fool deterministic polynomial-time adversaries. We also consider a more quantitative notion of simulation, where the measure of success of the simulation is the fraction of inputs of a given length on which the simulation works. Among other results, we show that if there is no polynomial time bound t such that P can be simulated well by MATIME(t), then for any /spl epsi/>0 there is a simulation of BPP in P that works for all but 2/sup n/spl epsi// inputs of length n. This is a uniform strengthening of a recent result of Goldreich and Wigderson. Finally, we give an unconditional simulation of multitape Turing machines operating in probabilistic time t by Turing machines operating in deterministic time O(2/sup t/). We show similar results for randomized NC/sup 1/ circuits. Our proofs are based on a combination of techniques in the theory of derandomization with results on holographic proofs.
{"title":"Holographic proofs and derandomization","authors":"D. Melkebeek, R. Santhanam","doi":"10.1109/CCC.2003.1214427","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214427","url":null,"abstract":"We derive a stronger consequence of EXP having polynomial-size circuits than was known previously, namely that there is a simulation of P in MAPOLYLOG that fools all deterministic polynomial-time adversaries. Using the connection between circuit lower bounds and derandomization, we obtain uniform assumptions for derandomizing BPP. Our results strengthen the space-randomness tradeoffs of Sipser, Nisan and Wigderson, and Lu. We show a partial converse: oracle circuit lower bounds for EXP imply that there are efficient simulations of P that fool deterministic polynomial-time adversaries. We also consider a more quantitative notion of simulation, where the measure of success of the simulation is the fraction of inputs of a given length on which the simulation works. Among other results, we show that if there is no polynomial time bound t such that P can be simulated well by MATIME(t), then for any /spl epsi/>0 there is a simulation of BPP in P that works for all but 2/sup n/spl epsi// inputs of length n. This is a uniform strengthening of a recent result of Goldreich and Wigderson. Finally, we give an unconditional simulation of multitape Turing machines operating in probabilistic time t by Turing machines operating in deterministic time O(2/sup t/). We show similar results for randomized NC/sup 1/ circuits. Our proofs are based on a combination of techniques in the theory of derandomization with results on holographic proofs.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"205 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":"122502736","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.2003.1214414
Amit Chakrabarti, Subhash Khot, Xiaodong Sun
We study the communication complexity of the set disjointness problem in the general multiparty model. For t players, each holding a subset of a universe of size n, we establish a near-optimal lower bound of /spl Omega/(n/(t log t)) on the communication complexity of the problem of determining whether their sets are disjoint. In the more restrictive one-way communication model, in which the players are required to speak in a predetermined order, we improve our bound to an optimal /spl Omega/(n/t). These results improve upon the earlier bounds of /spl Omega/(n/t/sup 2/) in the general model, and /spl Omega/((/spl epsiv//sup 2/n)/t/sup 1+/spl epsiv//) in the one-way model, due to Bar-Yossef, Jayram, Kumar, and Sivakumar (2002). As in the case of earlier results, our bounds apply to the unique intersection promise problem. This communication problem is known to have connections with the space complexity of approximating frequency moments in the data stream model. Our results lead to an improved space complexity lower bound of /spl Omega/(n/sup 1-2/k//log n) for approximating the k/sup th/ frequency moment with a constant number of passes over the input, and a technical improvement to /spl Omega/(n/sup 1-2/k/) if only one pass over the input is permitted. Our proofs rely on the information theoretic direct sum decomposition paradigm of Bar-Yossef et al. [2002]. Our improvements stem from novel analytical techniques, as opposed to earlier techniques based on Hellinger and related distances, for estimating the information cost of protocols for one-bit functions.
{"title":"Near-optimal lower bounds on the multi-party communication complexity of set disjointness","authors":"Amit Chakrabarti, Subhash Khot, Xiaodong Sun","doi":"10.1109/CCC.2003.1214414","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214414","url":null,"abstract":"We study the communication complexity of the set disjointness problem in the general multiparty model. For t players, each holding a subset of a universe of size n, we establish a near-optimal lower bound of /spl Omega/(n/(t log t)) on the communication complexity of the problem of determining whether their sets are disjoint. In the more restrictive one-way communication model, in which the players are required to speak in a predetermined order, we improve our bound to an optimal /spl Omega/(n/t). These results improve upon the earlier bounds of /spl Omega/(n/t/sup 2/) in the general model, and /spl Omega/((/spl epsiv//sup 2/n)/t/sup 1+/spl epsiv//) in the one-way model, due to Bar-Yossef, Jayram, Kumar, and Sivakumar (2002). As in the case of earlier results, our bounds apply to the unique intersection promise problem. This communication problem is known to have connections with the space complexity of approximating frequency moments in the data stream model. Our results lead to an improved space complexity lower bound of /spl Omega/(n/sup 1-2/k//log n) for approximating the k/sup th/ frequency moment with a constant number of passes over the input, and a technical improvement to /spl Omega/(n/sup 1-2/k/) if only one pass over the input is permitted. Our proofs rely on the information theoretic direct sum decomposition paradigm of Bar-Yossef et al. [2002]. Our improvements stem from novel analytical techniques, as opposed to earlier techniques based on Hellinger and related distances, for estimating the information cost of protocols for one-bit functions.","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":"128738603","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.2003.1214421
E. Allender, M. Koucký, Detlef Ronneburger, Sambuddha Roy
We continue an investigation of resource-bounded Kolmogorov complexity and derandomization techniques begun in [E. Allender (2001), E. Allender et al., (2002)]. We introduce nondeterministic time-bounded Kolmogorov complexity measures (KNt and KNT) and examine the properties of these measures using constructions of hitting set generators for nondeterministic circuits [P. B. Miltersen et al., (1999), R. Shaltiel et al., (2001)]. We observe that KNt bears many similarities to the nondeterministic distinguishing complexity CND of [H. Buhrman et al., (2002)]. This motivates the definition of a new notion of time-bounded distinguishing complexity KDt, as an intermediate notion with connections to the class FewEXP. The set of KDt-random strings is complete for EXP under P/poly reductions. Most of the notions of resource-bounded Kolmogorov complexity discussed here and in [E. Allender (2001), E. Allender et al., (2002)] have close connections to circuit size (on different types of circuits). We extend this framework to define notions of Kolmogorov complexity KB and KF that are related to branching program size and formula size, respectively. The sets of KB- and KF-random strings lie in coNP; we show that oracle access to these sets enables one to factor Blum integers. We obtain related intractability results for approximating minimum formula size, branching program size, and circuit size. The NEXP/spl sube/NC and NEXP/spl sube/L/poly questions are shown to be equivalent to conditions about the KF and KB complexity of sets in P.
我们继续研究资源有限的Kolmogorov复杂性和非随机化技术，开始于[E]。Allender (2001)， E. Allender等人，(2002)]。我们引入了不确定的有界Kolmogorov复杂度测度(KNt和KNt)，并利用不确定电路的碰撞集生成器构造检验了这些测度的性质[P]。B. Miltersen et al.， (1999); R. Shaltiel et al.，(2001)。我们观察到KNt与[H]的不确定性区分复杂性CND有许多相似之处。Buhrman et al.，(2002)。这激发了对有时间限制的区分复杂度KDt的新概念的定义，作为与类FewEXP有连接的中间概念。对于P/poly约简下的EXP, kdt -随机字符串集是完备的。本文和文献[E]中讨论的资源界Kolmogorov复杂性的大多数概念。Allender (2001)， E. Allender et al.，(2002)]与电路尺寸有密切联系(在不同类型的电路上)。我们扩展这个框架来定义Kolmogorov复杂度KB和KF的概念，它们分别与分支程序大小和公式大小相关。KB-和kf -随机字符串的集合在coNP中;我们将展示oracle对这些集合的访问使我们能够对Blum整数进行因式分解。我们得到了近似最小公式大小、分支程序大小和电路大小的相关难解性结果。证明了NEXP/spl sub /NC和NEXP/spl sub /L/poly问题等价于P中集合的KF和KB复杂度的条件。
{"title":"Derandomization and distinguishing complexity","authors":"E. Allender, M. Koucký, Detlef Ronneburger, Sambuddha Roy","doi":"10.1109/CCC.2003.1214421","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214421","url":null,"abstract":"We continue an investigation of resource-bounded Kolmogorov complexity and derandomization techniques begun in [E. Allender (2001), E. Allender et al., (2002)]. We introduce nondeterministic time-bounded Kolmogorov complexity measures (KNt and KNT) and examine the properties of these measures using constructions of hitting set generators for nondeterministic circuits [P. B. Miltersen et al., (1999), R. Shaltiel et al., (2001)]. We observe that KNt bears many similarities to the nondeterministic distinguishing complexity CND of [H. Buhrman et al., (2002)]. This motivates the definition of a new notion of time-bounded distinguishing complexity KDt, as an intermediate notion with connections to the class FewEXP. The set of KDt-random strings is complete for EXP under P/poly reductions. Most of the notions of resource-bounded Kolmogorov complexity discussed here and in [E. Allender (2001), E. Allender et al., (2002)] have close connections to circuit size (on different types of circuits). We extend this framework to define notions of Kolmogorov complexity KB and KF that are related to branching program size and formula size, respectively. The sets of KB- and KF-random strings lie in coNP; we show that oracle access to these sets enables one to factor Blum integers. We obtain related intractability results for approximating minimum formula size, branching program size, and circuit size. The NEXP/spl sube/NC and NEXP/spl sube/L/poly questions are shown to be equivalent to conditions about the KF and KB complexity of sets in P.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"26 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":"125883618","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.2003.1214420
Ke Yang, Avrim Blum
We introduce a "statistical query sampling" model, in which the goal of an algorithm is to produce an element in a hidden set S/spl sube/{0,1}/sup n/ with reasonable probability. The algorithm gains information about S through oracle calls (statistical queries), where the algorithm submits a query function g(/spl middot/) and receives an approximation to Pr/sub x/spl isin/S/[g(x)=1]. We show how this model is related to NMR quantum computing, in which only statistical properties of an ensemble of quantum systems can be measured, and in particular to the question of whether one can translate standard quantum algorithms to the NMR setting without putting all of their classical postprocessing into the quantum system. Using Fourier analysis techniques developed in the related context of statistical query learning, we prove a number of lower bounds (both information-theoretic and cryptographic) on the ability of algorithms to produce an x/spl isin/S, even when the set S is fairly simple. These lower bounds point out a difficulty in efficiently applying NMR quantum computing to algorithms such as Shor's and Simon's algorithm that involve significant classical postprocessing. We also explicitly relate the notion of statistical query sampling to that of statistical query learning.
我们引入了一个“统计查询抽样”模型，其中算法的目标是以合理的概率在隐藏集S/spl sub /{0,1}/sup n/中产生一个元素。算法通过oracle调用(统计查询)获得关于S的信息，其中算法提交查询函数g(/spl middot/)并接收到Pr/sub x/spl isin/S/[g(x)=1]的近似值。我们展示了这个模型是如何与核磁共振量子计算相关的，在核磁共振量子计算中，只有量子系统集合的统计特性可以被测量，特别是是否可以将标准量子算法转换为核磁共振设置而不将所有经典后处理放入量子系统的问题。使用在统计查询学习相关背景下开发的傅里叶分析技术，我们证明了算法产生x/spl isin/S的能力的一些下界(包括信息论和密码学)，即使集合S相当简单。这些下界指出了将核磁共振量子计算有效地应用于诸如肖尔算法和西蒙算法等涉及大量经典后处理的算法的困难。我们还明确地将统计查询抽样的概念与统计查询学习的概念联系起来。
{"title":"On statistical query sampling and NMR quantum computing","authors":"Ke Yang, Avrim Blum","doi":"10.1109/CCC.2003.1214420","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214420","url":null,"abstract":"We introduce a \"statistical query sampling\" model, in which the goal of an algorithm is to produce an element in a hidden set S/spl sube/{0,1}/sup n/ with reasonable probability. The algorithm gains information about S through oracle calls (statistical queries), where the algorithm submits a query function g(/spl middot/) and receives an approximation to Pr/sub x/spl isin/S/[g(x)=1]. We show how this model is related to NMR quantum computing, in which only statistical properties of an ensemble of quantum systems can be measured, and in particular to the question of whether one can translate standard quantum algorithms to the NMR setting without putting all of their classical postprocessing into the quantum system. Using Fourier analysis techniques developed in the related context of statistical query learning, we prove a number of lower bounds (both information-theoretic and cryptographic) on the ability of algorithms to produce an x/spl isin/S, even when the set S is fairly simple. These lower bounds point out a difficulty in efficiently applying NMR quantum computing to algorithms such as Shor's and Simon's algorithm that involve significant classical postprocessing. We also explicitly relate the notion of statistical query sampling to that of statistical query learning.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"2009 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":"125650383","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.2003.1214429
V. Guruswami
Under list decoding of error-correcting codes, the decoding algorithm is allowed to output a small list of codewords that are close to the noisy received word. This relaxation permits recovery even under very high noise thresholds. We consider one possible scenario that would permit disambiguating between the elements of the list, namely where the sender of the message provides some hopefully small amount of side information about the transmitted message on a separate auxiliary channel that is noise-free. This setting becomes meaningful and useful when the amount of side information that needs to be communicated is much smaller than the length of the message. We study what kind of side information is necessary and sufficient in the above context. The short, conceptual answer is that the side information must be randomized and the message recovery is with a small failure probability. Specifically, we prove that deterministic schemes, which guarantee correct recovery of the message, provide no savings and essentially the entire message has to be sent as side information. However there exist randomized schemes, which only need side information of length logarithmic in the message length. In fact, in the limit of repeated communication of several messages, amortized amount of side information needed per message can be a constant independent of the message length or the failure probability. Concretely, we can correct up to a fraction (1/2-/spl gamma/) of errors for binary codes using only 2log(1//spl gamma/)+O(1) amortized bits of side information per message, and this is in fact the best possible (up to additive constant terms).
{"title":"List decoding with side information","authors":"V. Guruswami","doi":"10.1109/CCC.2003.1214429","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214429","url":null,"abstract":"Under list decoding of error-correcting codes, the decoding algorithm is allowed to output a small list of codewords that are close to the noisy received word. This relaxation permits recovery even under very high noise thresholds. We consider one possible scenario that would permit disambiguating between the elements of the list, namely where the sender of the message provides some hopefully small amount of side information about the transmitted message on a separate auxiliary channel that is noise-free. This setting becomes meaningful and useful when the amount of side information that needs to be communicated is much smaller than the length of the message. We study what kind of side information is necessary and sufficient in the above context. The short, conceptual answer is that the side information must be randomized and the message recovery is with a small failure probability. Specifically, we prove that deterministic schemes, which guarantee correct recovery of the message, provide no savings and essentially the entire message has to be sent as side information. However there exist randomized schemes, which only need side information of length logarithmic in the message length. In fact, in the limit of repeated communication of several messages, amortized amount of side information needed per message can be a constant independent of the message length or the failure probability. Concretely, we can correct up to a fraction (1/2-/spl gamma/) of errors for binary codes using only 2log(1//spl gamma/)+O(1) amortized bits of side information per message, and this is in fact the best possible (up to additive constant terms).","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"28 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":"121472920","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.2003.1214434
Manindra Agrawal
We extract a paradigm for derandomizing tests for polynomial identities from the recent AKS primality testing algorithm. We then discuss its possible application to other tests.
{"title":"On derandomizing tests for certain polynomial identities","authors":"Manindra Agrawal","doi":"10.1109/CCC.2003.1214434","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214434","url":null,"abstract":"We extract a paradigm for derandomizing tests for polynomial identities from the recent AKS primality testing algorithm. We then discuss its possible application to other tests.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"2 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":"122045729","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.2003.1214411
P. Sen, Venkatesh Srinivasan
We consider a fundamental problem in data structures, static predecessor searching: Given a subset S of size n from the universe [m], store S so that queries of the form "What is the predecessor of x in S?" can be answered efficiently. We study this problem in the cell probe model introduced by Yao [1981]. Recently, Beame and Fich [2002] obtained optimal bounds on the number of probes needed by any deterministic query scheme if the associated storage scheme uses only n/sup O(1)/ cells of word size (log m)/sup O(1)/ bits. We give a new lower bound proof for this problem that matches the bounds of Beame and Fich. Our lower bound proof has the following advantages: it works for randomised query schemes too, while Beame and Fich's proof works for deterministic query schemes only. In addition, it is simpler than Beame and Fich's proof. We prove our lower bound using the round elimination approach of Miltersen, Nisan, Safra and Wigderson [1998]. Using tools from information theory, we prove a strong round elimination lemma for communication complexity that enables us to obtain a tight lower bound for the predecessor problem. We also use our round elimination lemma to obtain a rounds versus communication tradeoff for the 'greater-than' problem, improving on the tradeoff in [1998]. We believe that our round elimination lemma is of independent interest and should have other applications.
{"title":"Lower bounds for predecessor searching in the cell probe model","authors":"P. Sen, Venkatesh Srinivasan","doi":"10.1109/CCC.2003.1214411","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214411","url":null,"abstract":"We consider a fundamental problem in data structures, static predecessor searching: Given a subset S of size n from the universe [m], store S so that queries of the form \"What is the predecessor of x in S?\" can be answered efficiently. We study this problem in the cell probe model introduced by Yao [1981]. Recently, Beame and Fich [2002] obtained optimal bounds on the number of probes needed by any deterministic query scheme if the associated storage scheme uses only n/sup O(1)/ cells of word size (log m)/sup O(1)/ bits. We give a new lower bound proof for this problem that matches the bounds of Beame and Fich. Our lower bound proof has the following advantages: it works for randomised query schemes too, while Beame and Fich's proof works for deterministic query schemes only. In addition, it is simpler than Beame and Fich's proof. We prove our lower bound using the round elimination approach of Miltersen, Nisan, Safra and Wigderson [1998]. Using tools from information theory, we prove a strong round elimination lemma for communication complexity that enables us to obtain a tight lower bound for the predecessor problem. We also use our round elimination lemma to obtain a rounds versus communication tradeoff for the 'greater-than' problem, improving on the tradeoff in [1998]. We believe that our round elimination lemma is of independent interest and should have other applications.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"246 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":"116442647","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.2003.1214412
Detlef Sieling
Decision trees are representations of discrete functions with widespread applications in, e.g., complexity theory and data mining and exploration. In these areas it is important to obtain decision trees of small size. The minimization problem for decision trees is known to be NP-hard. The problem is even hard to approximate up to any constant factor.
{"title":"Minimization of decision trees is hard to approximate","authors":"Detlef Sieling","doi":"10.1109/CCC.2003.1214412","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214412","url":null,"abstract":"Decision trees are representations of discrete functions with widespread applications in, e.g., complexity theory and data mining and exploration. In these areas it is important to obtain decision trees of small size. The minimization problem for decision trees is known to be NP-hard. The problem is even hard to approximate up to any constant factor.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"33 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":"133839434","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.2003.1214433
L. Fortnow, A. Pavan, Samik Sengupta
We show that if SAT does not have small circuits, then there must exist a small number of formulas such that every small circuit fails to compute satisfiability correctly on at least one of these formulas. We use this result to show that if P/sup NP[1]/=P/sup NP[2]/, then the polynomial-time hierarchy collapses to S/sub 2//sup P//spl sube//spl Sigma//sub 2//sup p//spl cap//spl Pi//sub 2//sup p/. Even showing that the hierarchy collapsed to /spl Sigma//sub 2//sup p/ remained open.
{"title":"Proving SAT does not have small circuits with an application to the two queries problem","authors":"L. Fortnow, A. Pavan, Samik Sengupta","doi":"10.1109/CCC.2003.1214433","DOIUrl":"https://doi.org/10.1109/CCC.2003.1214433","url":null,"abstract":"We show that if SAT does not have small circuits, then there must exist a small number of formulas such that every small circuit fails to compute satisfiability correctly on at least one of these formulas. We use this result to show that if P/sup NP[1]/=P/sup NP[2]/, then the polynomial-time hierarchy collapses to S/sub 2//sup P//spl sube//spl Sigma//sub 2//sup p//spl cap//spl Pi//sub 2//sup p/. Even showing that the hierarchy collapsed to /spl Sigma//sub 2//sup p/ remained open.","PeriodicalId":286846,"journal":{"name":"18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.","volume":"49 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":"129394361","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.2003.1214407
Yijia Chen, J. Flum, Martin Grohe
We give machine characterisations and logical descriptions of a number of parameterized complexity classes. The focus of our attention is the class W[P], which we characterise as the class of all parameterized problems decidable by a nondeterministic fixed-parameter tractable algorithm, whose use of nondeterminism is bounded in terms of the parameter. We give similar characterisations for AW[P], the "alternating version of W[P]", and various other parameterized complexity classes. We also give logical characterisations of the classes W[P] and AW[P] in terms of fragments of least fixed-point logic, thereby putting these two classes into a uniform framework that we have developed in earlier work. Furthermore, we investigate the relation between alternation and space in parameterized complexity theory. We prove that the compact Turing machine computation problem, shown to be hard for the class AW[SAT] in (K. A. Abrahamson et al., 1995) is complete for the class uniform-XNL.
我们给出了一些参数化复杂性类的机器特征和逻辑描述。我们关注的焦点是类W[P]，我们将其描述为可由非确定性固定参数可处理算法确定的所有参数化问题的类，其不确定性的使用在参数方面是有限的。我们对AW[P]、“W[P]的交替版本”和其他各种参数化复杂性类给出了类似的特征。我们还根据最小不动点逻辑的片段给出类W[P]和AW[P]的逻辑特征，从而将这两个类放入我们在早期工作中开发的统一框架中。进一步研究了参数化复杂性理论中交替与空间的关系。我们证明了在(K. A. Abrahamson et al.， 1995)中对于类AW[SAT]来说比较困难的紧凑型图灵机计算问题对于类uniform- xl来说是完全的。
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