We present results in structural complexity theory concerned with the following interrelated topics: computation with postselection/restarting, closed timelike curves (CTCs), and approximate counting. The first result is a new characterization of the lesser known complexity class BPPpath in terms of more familiar concepts. Precisely, BPPpath is the class of problems that can be efficiently solved with a nonadaptive oracle for the approximate counting problem. Similarly, PP equals the class of problems that can be solved efficiently with nonadaptive queries for the related approximate difference problem. Another result is concerned with the computational power conferred by CTCs, or equivalently, the computational complexity of finding stationary distributions for quantum channels. Using the preceding characterization of PP, we show that any poly(n)-time quantum computation using a CTC of O(log n) qubits may as well just use a CTC of 1 classical bit. This result essentially amounts to showing that one can find a stationary distribution for a poly(n)-dimensional quantum channel in PP.
{"title":"The Weakness of CTC Qubits and the Power of Approximate Counting","authors":"R. O'Donnell, A. Say","doi":"10.1145/3196832","DOIUrl":"https://doi.org/10.1145/3196832","url":null,"abstract":"We present results in structural complexity theory concerned with the following interrelated topics: computation with postselection/restarting, closed timelike curves (CTCs), and approximate counting. The first result is a new characterization of the lesser known complexity class BPPpath in terms of more familiar concepts. Precisely, BPPpath is the class of problems that can be efficiently solved with a nonadaptive oracle for the approximate counting problem. Similarly, PP equals the class of problems that can be solved efficiently with nonadaptive queries for the related approximate difference problem. Another result is concerned with the computational power conferred by CTCs, or equivalently, the computational complexity of finding stationary distributions for quantum channels. Using the preceding characterization of PP, we show that any poly(n)-time quantum computation using a CTC of O(log n) qubits may as well just use a CTC of 1 classical bit. This result essentially amounts to showing that one can find a stationary distribution for a poly(n)-dimensional quantum channel in PP.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123883305","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}
The known proofs of MA ⊆ PP incur a quadratic overhead in the running time. We prove that this quadratic overhead is necessary for black-box simulations; in particular, we obtain an oracle relative to which MA-TIME (t) ⊈ P-TIME (o(t2)). We also show that 2-sided-error Merlin–Arthur games can be simulated by 1-sided-error Arthur–Merlin games with quadratic overhead. We also present a simple, query complexity based proof (provided by Mika Göös) that there is an oracle relative to which MA ⊈ NPBPP (which was previously known to hold by a proof using generics).
{"title":"Quadratic Simulations of Merlin–Arthur Games","authors":"Thomas Watson","doi":"10.1145/3389399","DOIUrl":"https://doi.org/10.1145/3389399","url":null,"abstract":"The known proofs of MA ⊆ PP incur a quadratic overhead in the running time. We prove that this quadratic overhead is necessary for black-box simulations; in particular, we obtain an oracle relative to which MA-TIME (t) ⊈ P-TIME (o(t2)). We also show that 2-sided-error Merlin–Arthur games can be simulated by 1-sided-error Arthur–Merlin games with quadratic overhead. We also present a simple, query complexity based proof (provided by Mika Göös) that there is an oracle relative to which MA ⊈ NPBPP (which was previously known to hold by a proof using generics).","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128390507","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}
Sushmita Gupta, P. Misra, Saket Saurabh, M. Zehavi
An input to the POPULAR MATCHING problem, in the roommates setting (as opposed to the marriage setting), consists of a graph G (not necessarily bipartite) where each vertex ranks its neighbors in strict order, known as its preference. In the POPULAR MATCHING problem the objective is to test whether there exists a matching M* such that there is no matching M where more vertices prefer their matched status in M (in terms of their preferences) over their matched status in M*. In this article, we settle the computational complexity of the POPULAR MATCHING problem in the roommates setting by showing that the problem is NP-complete. Thus, we resolve an open question that has been repeatedly and explicitly asked over the last decade.
{"title":"Popular Matching in Roommates Setting Is NP-hard","authors":"Sushmita Gupta, P. Misra, Saket Saurabh, M. Zehavi","doi":"10.1145/3442354","DOIUrl":"https://doi.org/10.1145/3442354","url":null,"abstract":"An input to the POPULAR MATCHING problem, in the roommates setting (as opposed to the marriage setting), consists of a graph G (not necessarily bipartite) where each vertex ranks its neighbors in strict order, known as its preference. In the POPULAR MATCHING problem the objective is to test whether there exists a matching M* such that there is no matching M where more vertices prefer their matched status in M (in terms of their preferences) over their matched status in M*. In this article, we settle the computational complexity of the POPULAR MATCHING problem in the roommates setting by showing that the problem is NP-complete. Thus, we resolve an open question that has been repeatedly and explicitly asked over the last decade.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123461975","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}
Many important graph-theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article, we study the complexity of #pHOMSTOH, the problem of counting graph homomorphisms from an input graph to a graph H modulo a prime number p. Dyer and Greenhill proved a dichotomy stating that the tractability of non-modular counting graph homomorphisms depends on the structure of the target graph. Many intractable cases in non-modular counting become tractable in modular counting due to the common phenomenon of cancellation. In subsequent studies on counting modulo 2, however, the influence of the structure of H on the tractability was shown to persist, which yields similar dichotomies. Our main result states that for every tree H and every prime p the problem #pHOMSTOH is either polynomial time computable or #pP-complete. This relates to the conjecture of Faben and Jerrum stating that this dichotomy holds for every graph H when counting modulo 2. In contrast to previous results on modular counting, the tractable cases of #pHOMSTOH are essentially the same for all values of the modulo when H is a tree. To prove this result, we study the structural properties of a homomorphism. As an important interim result, our study yields a dichotomy for the problem of counting weighted independent sets in a bipartite graph modulo some prime p. These results are the first suggesting that such dichotomies hold not only for the modulo 2 case but also for the modular counting functions of all primes p.
{"title":"Counting Homomorphisms to Trees Modulo a Prime","authors":"Andreas Göbel, J. Lagodzinski, Karen Seidel","doi":"10.1145/3460958","DOIUrl":"https://doi.org/10.1145/3460958","url":null,"abstract":"Many important graph-theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article, we study the complexity of #pHOMSTOH, the problem of counting graph homomorphisms from an input graph to a graph H modulo a prime number p. Dyer and Greenhill proved a dichotomy stating that the tractability of non-modular counting graph homomorphisms depends on the structure of the target graph. Many intractable cases in non-modular counting become tractable in modular counting due to the common phenomenon of cancellation. In subsequent studies on counting modulo 2, however, the influence of the structure of H on the tractability was shown to persist, which yields similar dichotomies. Our main result states that for every tree H and every prime p the problem #pHOMSTOH is either polynomial time computable or #pP-complete. This relates to the conjecture of Faben and Jerrum stating that this dichotomy holds for every graph H when counting modulo 2. In contrast to previous results on modular counting, the tractable cases of #pHOMSTOH are essentially the same for all values of the modulo when H is a tree. To prove this result, we study the structural properties of a homomorphism. As an important interim result, our study yields a dichotomy for the problem of counting weighted independent sets in a bipartite graph modulo some prime p. These results are the first suggesting that such dichotomies hold not only for the modulo 2 case but also for the modular counting functions of all primes p.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133071297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We aim to understand inherent reasons for lower bounds for QBF proof systems and revisit and compare two previous approaches in this direction. The first of these relates size lower bounds for strong QBF Frege systems to circuit lower bounds via strategy extraction (Beyersdorff and Pich, LICS’16). Here, we show a refined version of strategy extraction and thereby for any QBF proof system obtain a trichotomy for hardness: (1) via circuit lower bounds, (2) via propositional Resolution lower bounds, or (3) “genuine” QBF lower bounds. The second approach tries to explain QBF lower bounds through quantifier alternations in a system called relaxing QU-Res (Chen, ACM TOCT 2017). We prove a strong lower bound for relaxing QU-Res, which at the same time exhibits significant shortcomings of that model. Prompted by this, we introduce a hierarchy of new systems that improve Chen’s model and prove a strict separation for the complexity of proofs in this hierarchy. We show that lower bounds in our new model correspond to the trichotomy obtained via strategy extraction.
{"title":"Reasons for Hardness in QBF Proof Systems","authors":"Olaf Beyersdorff, Luke Hinde, J. Pich","doi":"10.1145/3378665","DOIUrl":"https://doi.org/10.1145/3378665","url":null,"abstract":"We aim to understand inherent reasons for lower bounds for QBF proof systems and revisit and compare two previous approaches in this direction. The first of these relates size lower bounds for strong QBF Frege systems to circuit lower bounds via strategy extraction (Beyersdorff and Pich, LICS’16). Here, we show a refined version of strategy extraction and thereby for any QBF proof system obtain a trichotomy for hardness: (1) via circuit lower bounds, (2) via propositional Resolution lower bounds, or (3) “genuine” QBF lower bounds. The second approach tries to explain QBF lower bounds through quantifier alternations in a system called relaxing QU-Res (Chen, ACM TOCT 2017). We prove a strong lower bound for relaxing QU-Res, which at the same time exhibits significant shortcomings of that model. Prompted by this, we introduce a hierarchy of new systems that improve Chen’s model and prove a strict separation for the complexity of proofs in this hierarchy. We show that lower bounds in our new model correspond to the trichotomy obtained via strategy extraction.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"440 2-3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132266227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We strengthen existing evidence for the so-called “algebrization barrier.” Algebrization—short for algebraic relativization—was introduced by Aaronson and Wigderson (AW) (STOC 2008) to characterize proofs involving arithmetization, simulation, and other “current techniques.” However, unlike relativization, eligible statements under this notion do not seem to have basic closure properties, making it conceivable to take two proofs, both with algebrizing conclusions, and combine them to get a proof without. Further, the notion is undefined for most types of statements and does not seem to yield a general criterion by which we can tell, given a proof, whether it algebrizes. In fact, the very notion of an algebrizing proof is never made explicit, and casual attempts to define it are problematic. All these issues raise the question of what evidence, if any, is obtained by knowing whether some statement does or does not algebrize. We give a reformulation of algebrization without these shortcomings. First, we define what it means for any statement/proof to hold relative to any language, with no need to refer to devices like a Turing machine with an oracle tape. Our approach dispels the widespread misconception that the notion of oracle access is inherently tied to a computational model. We also connect relativizing statements to proofs, by showing that every proof that some statement relativizes is essentially a relativizing proof of that statement. We then define a statement/proof as relativizing affinely if it holds relative to every affine oracle—here an affine oracle is the result of a particular error correcting code applied to the characteristic string of a language. We show that every statement that AW declare as algebrizing does relativize affinely, in fact, has a proof that relativizes affinely, and that no such proof exists for any of the statements shown not-algebrizing by AW in the classical computation model. Our work complements, and goes beyond, the subsequent work by Impagliazzo, Kabanets, and Kolokolova (STOC 2009), which also proposes a reformulation of algebrization, but falls short of recovering some key results of AW, most notably regarding the NEXP versus P/poly question. Using our definitions, we obtain new streamlined proofs of several classic results in complexity, including PSPACE ⊂ IP and NEXP ⊂ MIP. This may be of separate interest.
{"title":"Affine Relativization","authors":"Baris Aydinlioglu, E. Bach","doi":"10.1145/3170704","DOIUrl":"https://doi.org/10.1145/3170704","url":null,"abstract":"We strengthen existing evidence for the so-called “algebrization barrier.” Algebrization—short for algebraic relativization—was introduced by Aaronson and Wigderson (AW) (STOC 2008) to characterize proofs involving arithmetization, simulation, and other “current techniques.” However, unlike relativization, eligible statements under this notion do not seem to have basic closure properties, making it conceivable to take two proofs, both with algebrizing conclusions, and combine them to get a proof without. Further, the notion is undefined for most types of statements and does not seem to yield a general criterion by which we can tell, given a proof, whether it algebrizes. In fact, the very notion of an algebrizing proof is never made explicit, and casual attempts to define it are problematic. All these issues raise the question of what evidence, if any, is obtained by knowing whether some statement does or does not algebrize. We give a reformulation of algebrization without these shortcomings. First, we define what it means for any statement/proof to hold relative to any language, with no need to refer to devices like a Turing machine with an oracle tape. Our approach dispels the widespread misconception that the notion of oracle access is inherently tied to a computational model. We also connect relativizing statements to proofs, by showing that every proof that some statement relativizes is essentially a relativizing proof of that statement. We then define a statement/proof as relativizing affinely if it holds relative to every affine oracle—here an affine oracle is the result of a particular error correcting code applied to the characteristic string of a language. We show that every statement that AW declare as algebrizing does relativize affinely, in fact, has a proof that relativizes affinely, and that no such proof exists for any of the statements shown not-algebrizing by AW in the classical computation model. Our work complements, and goes beyond, the subsequent work by Impagliazzo, Kabanets, and Kolokolova (STOC 2009), which also proposes a reformulation of algebrization, but falls short of recovering some key results of AW, most notably regarding the NEXP versus P/poly question. Using our definitions, we obtain new streamlined proofs of several classic results in complexity, including PSPACE ⊂ IP and NEXP ⊂ MIP. This may be of separate interest.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131040539","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}
Mika Göös, T. S. Jayram, T. Pitassi, Thomas Watson
We show that randomized communication complexity can be superlogarithmic in the partition number of the associated communication matrix, and we obtain near-optimal randomized lower bounds for the Clique versus Independent Set problem. These results strengthen the deterministic lower bounds obtained in prior work (Göös, Pitassi, and Watson, FOCS’15). One of our main technical contributions states that information complexity when the cost is measured with respect to only 1-inputs (or only 0-inputs) is essentially equivalent to information complexity with respect to all inputs.
我们证明了随机通信复杂度在相关通信矩阵的分区数上可以是超对数的，并且我们得到了团对独立集问题的近最优随机下界。这些结果加强了先前工作中获得的确定性下限(Göös, Pitassi, and Watson, FOCS ' 15)。我们的主要技术贡献之一是，当仅以1个输入(或仅0个输入)衡量成本时，信息复杂性本质上等同于与所有输入相关的信息复杂性。
{"title":"Randomized Communication versus Partition Number","authors":"Mika Göös, T. S. Jayram, T. Pitassi, Thomas Watson","doi":"10.1145/3170711","DOIUrl":"https://doi.org/10.1145/3170711","url":null,"abstract":"We show that randomized communication complexity can be superlogarithmic in the partition number of the associated communication matrix, and we obtain near-optimal randomized lower bounds for the Clique versus Independent Set problem. These results strengthen the deterministic lower bounds obtained in prior work (Göös, Pitassi, and Watson, FOCS’15). One of our main technical contributions states that information complexity when the cost is measured with respect to only 1-inputs (or only 0-inputs) is essentially equivalent to information complexity with respect to all inputs.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128003650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove nearly matching upper and lower bounds on the randomized communication complexity of the following problem: Alice and Bob are each given a probability distribution over n elements, and they wish to estimate within ±ε the statistical (total variation) distance between their distributions. For some range of parameters, there is up to a log n factor gap between the upper and lower bounds, and we identify a barrier to using information complexity techniques to improve the lower bound in this case. We also prove a side result that we discovered along the way: the randomized communication complexity of n-bit Majority composed with n-bit Greater Than is Θ (n log n).
{"title":"Communication Complexity of Statistical Distance","authors":"Thomas Watson","doi":"10.1145/3170708","DOIUrl":"https://doi.org/10.1145/3170708","url":null,"abstract":"We prove nearly matching upper and lower bounds on the randomized communication complexity of the following problem: Alice and Bob are each given a probability distribution over n elements, and they wish to estimate within ±ε the statistical (total variation) distance between their distributions. For some range of parameters, there is up to a log n factor gap between the upper and lower bounds, and we identify a barrier to using information complexity techniques to improve the lower bound in this case. We also prove a side result that we discovered along the way: the randomized communication complexity of n-bit Majority composed with n-bit Greater Than is Θ (n log n).","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130765918","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}
Bennett and Gill [1981] showed that PA ≠ NPA ≠ coNPA for a random oracle A, with probability 1. We investigate whether this result extends to individual polynomial-time random oracles. We consider two notions of random oracles: p-random oracles in the sense of martingales and resource-bounded measure [Lutz 1992; Ambos-Spies et al. 1997], and p-betting-game random oracles using the betting games generalization of resource-bounded measure [Buhrman et al. 2000]. Every p-betting-game random oracle is also p-random; whether the two notions are equivalent is an open problem. (1) We first show that PA ≠ NPA for every oracle A that is p-betting-game random. Ideally, we would extend (1) to p-random oracles. We show that answering this either way would imply an unrelativized complexity class separation: (2) If PA ≠ NPA relative to every p-random oracle A, then BPP ≠ EXP. (3) If PA ≠ NPA relative to some p-random oracle A, then P ≠ PSPACE. Rossman, Servedio, and Tan [2015] showed that the polynomial-time hierarchy is infinite relative to a random oracle, solving a longstanding open problem. We consider whether we can extend (1) to show that PHA is infinite relative to oracles A that are p-betting-game random. Showing that PHA separates at even its first level would also imply an unrelativized complexity class separation: (4) If NPA ≠ coNPA for a p-betting-game measure 1 class of oracles A, then NP ≠ EXP. (5) If PHA is infinite relative to every p-random oracle A, then PH ≠ EXP. We also consider random oracles for time versus space, for example: (6) LA ≠ PA relative to every oracle A that is p-betting-game random.
Bennett和Gill[1981]证明了随机oracle a的PA≠NPA≠coNPA，概率为1。我们研究了这个结果是否可以推广到单个多项式时间随机预言。我们考虑了随机预言的两个概念:鞅意义上的p-随机预言和资源有界测度[Lutz 1992;Ambos-Spies et al. 1997]，以及使用资源有限测度的投注游戏泛化的p-betting-game随机预言[Buhrman et al. 2000]。每个p-赌局随机神谕也是p-随机的;这两个概念是否等同是一个悬而未决的问题。(1)我们首先证明PA≠NPA对于每一个p-bet -game随机的oracle A。理想情况下，我们可以将(1)扩展到p-随机预言机。我们证明，无论以哪种方式回答这个问题，都意味着一个非相对化的复杂性类分离:(2)如果PA相对于每个P -随机oracle A≠NPA，则BPP≠EXP。(3)如果PA相对于某个P -随机oracle A≠NPA，则P≠PSPACE。Rossman, Servedio和Tan[2015]表明，相对于随机预言，多项式时间层次结构是无限的，解决了一个长期存在的开放问题。我们考虑是否可以推广(1)来证明PHA相对于p-投注-博弈随机的神谕A是无限的。表明PHA即使在其第一层也分离，也意味着非相对化的复杂性类分离:(4)如果对于p-投注-游戏测度1类神谕a的NPA≠coNPA，则NP≠EXP。(5)如果PHA相对于每个p-随机神谕a是无限的，则PH≠EXP。我们还考虑时间与空间的随机神谕，例如:(6)相对于每个p-投注-游戏随机的神谕a, LA≠PA。
{"title":"Polynomial-Time Random Oracles and Separating Complexity Classes","authors":"J. M. Hitchcock, Adewale Sekoni, Hadi Shafei","doi":"10.1145/3434389","DOIUrl":"https://doi.org/10.1145/3434389","url":null,"abstract":"Bennett and Gill [1981] showed that PA ≠ NPA ≠ coNPA for a random oracle A, with probability 1. We investigate whether this result extends to individual polynomial-time random oracles. We consider two notions of random oracles: p-random oracles in the sense of martingales and resource-bounded measure [Lutz 1992; Ambos-Spies et al. 1997], and p-betting-game random oracles using the betting games generalization of resource-bounded measure [Buhrman et al. 2000]. Every p-betting-game random oracle is also p-random; whether the two notions are equivalent is an open problem. (1) We first show that PA ≠ NPA for every oracle A that is p-betting-game random. Ideally, we would extend (1) to p-random oracles. We show that answering this either way would imply an unrelativized complexity class separation: (2) If PA ≠ NPA relative to every p-random oracle A, then BPP ≠ EXP. (3) If PA ≠ NPA relative to some p-random oracle A, then P ≠ PSPACE. Rossman, Servedio, and Tan [2015] showed that the polynomial-time hierarchy is infinite relative to a random oracle, solving a longstanding open problem. We consider whether we can extend (1) to show that PHA is infinite relative to oracles A that are p-betting-game random. Showing that PHA separates at even its first level would also imply an unrelativized complexity class separation: (4) If NPA ≠ coNPA for a p-betting-game measure 1 class of oracles A, then NP ≠ EXP. (5) If PHA is infinite relative to every p-random oracle A, then PH ≠ EXP. We also consider random oracles for time versus space, for example: (6) LA ≠ PA relative to every oracle A that is p-betting-game random.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129248003","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}
Given metric spaces (X, DX) and (Y, DY), an embedding F: X → Y is an injective mapping from X to Y. Expansion eF and contraction cF of an embedding F: X → Y are defined as eF = maxx;1, x2 (≠ x1) ∈ X DY(F(x1), F(x2))/ DX(x1, x2) and cF = maxx1, x2 (≠ x1) ∈ X DX(x1, x2)/DY(F(x1), F(x2)), respectively, and distortion dF is defined as dF= eF ⋅ cF. Observe that dF ≥ 1. An embedding F : X → Y is noncontracting if cF ≤ 1. When d=1, then F is isometry. The Metric Embedding problem takes as input two metric spaces (X,DX) and (Y,DY), and a positive integer d. The objective is to determine whether there is an embedding F: X → Y such that dF ≤ d. Such an embedding is called a distortion d embedding. The bijective Metric Embedding problem is a special case of the Metric Embedding problem where ∣X∣ = ∣Y∣. In parameterized complexity, the Metric Embedding problem, in full generality, is known to be W-hard and, therefore, not expected to have an FPT algorithm. In this article, we consider the Gen-Graph Metric Embedding problem, where the two metric spaces are graph metrics. We explore the extent of tractability of the problem in the parameterized complexity setting. We determine whether an unweighted graph metric (G,DG) can be embedded, or bijectively embedded, into another unweighted graph metric (H,DH), where the graph H has low structural complexity. For example, H is a cycle, or H has bounded treewidth or bounded connected treewidth. The parameters for the algorithms are chosen from the upper bound d on distortion, bound Δ on the maximum degree of H, treewidth α of H, and connected treewidth αc of H. Our general approach to these problems can be summarized as trying to understand the behavior of the shortest paths in G under a low-distortion embedding into H, and the structural relation the mapping of these paths has to shortest paths in H.
{"title":"FPT Algorithms for Embedding into Low-Complexity Graphic Metrics","authors":"Arijit Ghosh, Sudeshna Kolay, Gopinath Mishra","doi":"10.1145/3369933","DOIUrl":"https://doi.org/10.1145/3369933","url":null,"abstract":"Given metric spaces (X, DX) and (Y, DY), an embedding F: X → Y is an injective mapping from X to Y. Expansion eF and contraction cF of an embedding F: X → Y are defined as eF = maxx;1, x2 (≠ x1) ∈ X DY(F(x1), F(x2))/ DX(x1, x2) and cF = maxx1, x2 (≠ x1) ∈ X DX(x1, x2)/DY(F(x1), F(x2)), respectively, and distortion dF is defined as dF= eF ⋅ cF. Observe that dF ≥ 1. An embedding F : X → Y is noncontracting if cF ≤ 1. When d=1, then F is isometry. The Metric Embedding problem takes as input two metric spaces (X,DX) and (Y,DY), and a positive integer d. The objective is to determine whether there is an embedding F: X → Y such that dF ≤ d. Such an embedding is called a distortion d embedding. The bijective Metric Embedding problem is a special case of the Metric Embedding problem where ∣X∣ = ∣Y∣. In parameterized complexity, the Metric Embedding problem, in full generality, is known to be W-hard and, therefore, not expected to have an FPT algorithm. In this article, we consider the Gen-Graph Metric Embedding problem, where the two metric spaces are graph metrics. We explore the extent of tractability of the problem in the parameterized complexity setting. We determine whether an unweighted graph metric (G,DG) can be embedded, or bijectively embedded, into another unweighted graph metric (H,DH), where the graph H has low structural complexity. For example, H is a cycle, or H has bounded treewidth or bounded connected treewidth. The parameters for the algorithms are chosen from the upper bound d on distortion, bound Δ on the maximum degree of H, treewidth α of H, and connected treewidth αc of H. Our general approach to these problems can be summarized as trying to understand the behavior of the shortest paths in G under a low-distortion embedding into H, and the structural relation the mapping of these paths has to shortest paths in H.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"116 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125583031","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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