Pub Date : 2022-06-15DOI: 10.48550/arXiv.2206.07571
Anthony Leverrier, Gilles Z'emor
We introduce and analyse an efficient decoder for the quantum Tanner codes of that can correct adversarial errors of linear weight. Previous decoders for quantum low-density parity-check codes could only handle adversarial errors of weight $O(sqrt{n log n})$. We also work on the link between quantum Tanner codes and the Lifted Product codes of Panteleev and Kalachev, and show that our decoder can be adapted to the latter. The decoding algorithm alternates between sequential and parallel procedures and converges in linear time.
{"title":"Efficient decoding up to a constant fraction of the code length for asymptotically good quantum codes","authors":"Anthony Leverrier, Gilles Z'emor","doi":"10.48550/arXiv.2206.07571","DOIUrl":"https://doi.org/10.48550/arXiv.2206.07571","url":null,"abstract":"We introduce and analyse an efficient decoder for the quantum Tanner codes of that can correct adversarial errors of linear weight. Previous decoders for quantum low-density parity-check codes could only handle adversarial errors of weight $O(sqrt{n log n})$. We also work on the link between quantum Tanner codes and the Lifted Product codes of Panteleev and Kalachev, and show that our decoder can be adapted to the latter. The decoding algorithm alternates between sequential and parallel procedures and converges in linear time.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"109 1","pages":"1216-1244"},"PeriodicalIF":0.0,"publicationDate":"2022-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80858448","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 : 2022-05-26DOI: 10.48550/arXiv.2205.13428
M. Mahajan, Gaurav Sood
The Merge Resolution proof system (M-Res) for QBFs, proposed by Beyersdorff et al. in 2019, explicitly builds partial strategies inside refutations. The original motivation for this approach was to overcome the limitations encountered in long-distance Q-Resolution proof system (LD-Q-Res), where the syntactic side-conditions, while prohibiting all unsound resolutions, also end up prohibiting some sound resolutions. However, while the advantage of M-Res over many other resolution-based QBF proof systems was already demonstrated, a comparison with LD-Q-Res itself had remained open. In this paper, we settle this question. We show that M-Res has an exponential advantage over not only LD-Q-Res, but even over LQU$^+$-Res and IRM, the most powerful among currently known resolution-based QBF proof systems. Combining this with results from Beyersdorff et al. 2020, we conclude that M-Res is incomparable with LQU-Res and LQU$^+$-Res. Our proof method reveals two additional and curious features about M-Res: (i) MRes is not closed under restrictions, and is hence not a natural proof system, and (ii) weakening axiom clauses with existential variables provably yields an exponential advantage over M-Res without weakening. We further show that in the context of regular derivations, weakening axiom clauses with universal variables provably yields an exponential advantage over M-Res without weakening. These results suggest that MRes is better used with weakening, though whether M-Res with weakening is closed under restrictions remains open. We note that even with weakening, M-Res continues to be simulated by eFrege $+$ $forall$red (the simulation of ordinary M-Res was shown recently by Chew and Slivovsky).
{"title":"QBF Merge Resolution is powerful but unnatural","authors":"M. Mahajan, Gaurav Sood","doi":"10.48550/arXiv.2205.13428","DOIUrl":"https://doi.org/10.48550/arXiv.2205.13428","url":null,"abstract":"The Merge Resolution proof system (M-Res) for QBFs, proposed by Beyersdorff et al. in 2019, explicitly builds partial strategies inside refutations. The original motivation for this approach was to overcome the limitations encountered in long-distance Q-Resolution proof system (LD-Q-Res), where the syntactic side-conditions, while prohibiting all unsound resolutions, also end up prohibiting some sound resolutions. However, while the advantage of M-Res over many other resolution-based QBF proof systems was already demonstrated, a comparison with LD-Q-Res itself had remained open. In this paper, we settle this question. We show that M-Res has an exponential advantage over not only LD-Q-Res, but even over LQU$^+$-Res and IRM, the most powerful among currently known resolution-based QBF proof systems. Combining this with results from Beyersdorff et al. 2020, we conclude that M-Res is incomparable with LQU-Res and LQU$^+$-Res. Our proof method reveals two additional and curious features about M-Res: (i) MRes is not closed under restrictions, and is hence not a natural proof system, and (ii) weakening axiom clauses with existential variables provably yields an exponential advantage over M-Res without weakening. We further show that in the context of regular derivations, weakening axiom clauses with universal variables provably yields an exponential advantage over M-Res without weakening. These results suggest that MRes is better used with weakening, though whether M-Res with weakening is closed under restrictions remains open. We note that even with weakening, M-Res continues to be simulated by eFrege $+$ $forall$red (the simulation of ordinary M-Res was shown recently by Chew and Slivovsky).","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"34 1","pages":"22:1-22:19"},"PeriodicalIF":0.0,"publicationDate":"2022-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82282195","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 : 2022-05-24DOI: 10.48550/arXiv.2205.12377
Elena Grigorescu, Brendan Juba, K. Wimmer, Ning Xie
Determinantal Point Processes (DPPs) are a widely used probabilistic model for negatively corre-lated sets. DPPs have been successfully employed in Machine Learning applications to select a diverse, yet representative subset of data. In these applications, the parameters of the DPP need to be fitted to match the data; typically, we seek a set of parameters that maximize the likelihood of the data. The algorithms used for this task to date either optimize over a limited family of DPPs, or use local improvement heuristics that do not provide theoretical guarantees of optimality. It is natural to ask if there exist efficient algorithms for finding a maximum likelihood DPP model for a given data set. In seminal work on DPPs in Machine Learning, Kulesza conjectured in his PhD Thesis (2012) that the problem is NP-complete. The lack of a formal proof prompted Brunel, Moitra, Rigollet and Urschel (2017a) to conjecture that, in opposition to Kulesza’s conjecture, there exists a polynomial-time algorithm for computing a maximum-likelihood DPP. They also presented some preliminary evidence supporting their conjecture. In this work we prove Kulesza’s conjecture. In fact, we prove the following stronger hardness of approximation result: even computing a 1 − 1 polylog N -approximation to the maximum log-likelihood of a DPP on a ground set of N elements is NP-complete. At the same time, we also obtain the first polynomial-time algorithm that achieves a nontrivial worst-case approximation to the optimal log-likelihood: the approximation factor is unconditionally (for data sets that consist of al., 2013b; et al., 2015; Affandi et al., 2013a), signal processing (Xu and Ou, Krause et al., Guestrin et al., 2005), clustering (Zou and 2012; Kang, 2013; and Ghahramani, 2013), recommendation systems (Zhou et al., 2010), revenue maximization (Dughmi et al., 2009), multi-agent reinforcement and al., 2020), modeling neural sketching for linear and low-rank
{"title":"Hardness of Maximum Likelihood Learning of DPPs","authors":"Elena Grigorescu, Brendan Juba, K. Wimmer, Ning Xie","doi":"10.48550/arXiv.2205.12377","DOIUrl":"https://doi.org/10.48550/arXiv.2205.12377","url":null,"abstract":"Determinantal Point Processes (DPPs) are a widely used probabilistic model for negatively corre-lated sets. DPPs have been successfully employed in Machine Learning applications to select a diverse, yet representative subset of data. In these applications, the parameters of the DPP need to be fitted to match the data; typically, we seek a set of parameters that maximize the likelihood of the data. The algorithms used for this task to date either optimize over a limited family of DPPs, or use local improvement heuristics that do not provide theoretical guarantees of optimality. It is natural to ask if there exist efficient algorithms for finding a maximum likelihood DPP model for a given data set. In seminal work on DPPs in Machine Learning, Kulesza conjectured in his PhD Thesis (2012) that the problem is NP-complete. The lack of a formal proof prompted Brunel, Moitra, Rigollet and Urschel (2017a) to conjecture that, in opposition to Kulesza’s conjecture, there exists a polynomial-time algorithm for computing a maximum-likelihood DPP. They also presented some preliminary evidence supporting their conjecture. In this work we prove Kulesza’s conjecture. In fact, we prove the following stronger hardness of approximation result: even computing a 1 − 1 polylog N -approximation to the maximum log-likelihood of a DPP on a ground set of N elements is NP-complete. At the same time, we also obtain the first polynomial-time algorithm that achieves a nontrivial worst-case approximation to the optimal log-likelihood: the approximation factor is unconditionally (for data sets that consist of al., 2013b; et al., 2015; Affandi et al., 2013a), signal processing (Xu and Ou, Krause et al., Guestrin et al., 2005), clustering (Zou and 2012; Kang, 2013; and Ghahramani, 2013), recommendation systems (Zhou et al., 2010), revenue maximization (Dughmi et al., 2009), multi-agent reinforcement and al., 2020), modeling neural sketching for linear and low-rank","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80645881","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 : 2022-05-16DOI: 10.48550/arXiv.2205.07803
Hunter Monroe
We consolidate two widely believed conjectures about tautologies -- no optimal proof system exists, and most require superpolynomial size proofs in any system -- into a $p$-isomorphism-invariant condition satisfied by all paddable $textbf{coNP}$-complete languages or none. The condition is: for any Turing machine (TM) $M$ accepting the language, $textbf{P}$-uniform input families requiring superpolynomial time by $M$ exist (equivalent to the first conjecture) and appear with positive upper density in an enumeration of input families (implies the second). In that case, no such language is easy on average (in $textbf{AvgP}$) for a distribution applying non-negligible weight to the hard families. The hardness of proving tautologies and theorems is likely related. Motivated by the fact that arithmetic sentences encoding"string $x$ is Kolmogorov random"are true but unprovable with positive density in a finitely axiomatized theory $mathcal{T}$ (Calude and J{"u}rgensen), we conjecture that any propositional proof system requires superpolynomial size proofs for a dense set of $textbf{P}$-uniform families of tautologies encoding"there is no $mathcal{T}$ proof of size $leq t$ showing that string $x$ is Kolmogorov random". This implies the above condition. The conjecture suggests that there is no optimal proof system because undecidable theories help prove tautologies and do so more efficiently as axioms are added, and that constructing hard tautologies seems difficult because it is impossible to construct Kolmogorov random strings. Similar conjectures that computational blind spots are manifestations of noncomputability would resolve other open problems.
{"title":"Average-Case Hardness of Proving Tautologies and Theorems","authors":"Hunter Monroe","doi":"10.48550/arXiv.2205.07803","DOIUrl":"https://doi.org/10.48550/arXiv.2205.07803","url":null,"abstract":"We consolidate two widely believed conjectures about tautologies -- no optimal proof system exists, and most require superpolynomial size proofs in any system -- into a $p$-isomorphism-invariant condition satisfied by all paddable $textbf{coNP}$-complete languages or none. The condition is: for any Turing machine (TM) $M$ accepting the language, $textbf{P}$-uniform input families requiring superpolynomial time by $M$ exist (equivalent to the first conjecture) and appear with positive upper density in an enumeration of input families (implies the second). In that case, no such language is easy on average (in $textbf{AvgP}$) for a distribution applying non-negligible weight to the hard families. The hardness of proving tautologies and theorems is likely related. Motivated by the fact that arithmetic sentences encoding\"string $x$ is Kolmogorov random\"are true but unprovable with positive density in a finitely axiomatized theory $mathcal{T}$ (Calude and J{\"u}rgensen), we conjecture that any propositional proof system requires superpolynomial size proofs for a dense set of $textbf{P}$-uniform families of tautologies encoding\"there is no $mathcal{T}$ proof of size $leq t$ showing that string $x$ is Kolmogorov random\". This implies the above condition. The conjecture suggests that there is no optimal proof system because undecidable theories help prove tautologies and do so more efficiently as axioms are added, and that constructing hard tautologies seems difficult because it is impossible to construct Kolmogorov random strings. Similar conjectures that computational blind spots are manifestations of noncomputability would resolve other open problems.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"135 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77529249","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 : 2022-05-04DOI: 10.48550/arXiv.2205.02345
Chi-Ning Chou, Alexander Golovnev, Amirbehshad Shahrasbi, M. Sudan, Santhoshini Velusamy
We analyze the sketching approximability of constraint satisfaction problems on Boolean domains, where the constraints are balanced linear threshold functions applied to literals. In~particular, we explore the approximability of monarchy-like functions where the value of the function is determined by a weighted combination of the vote of the first variable (the president) and the sum of the votes of all remaining variables. The pure version of this function is when the president can only be overruled by when all remaining variables agree. For every $k geq 5$, we show that CSPs where the underlying predicate is a pure monarchy function on $k$ variables have no non-trivial sketching approximation algorithm in $o(sqrt{n})$ space. We also show infinitely many weaker monarchy functions for which CSPs using such constraints are non-trivially approximable by $O(log(n))$ space sketching algorithms. Moreover, we give the first example of sketching approximable asymmetric Boolean CSPs. Our results work within the framework of Chou, Golovnev, Sudan, and Velusamy (FOCS 2021) that characterizes the sketching approximability of all CSPs. Their framework can be applied naturally to get a computer-aided analysis of the approximability of any specific constraint satisfaction problem. The novelty of our work is in using their work to get an analysis that applies to infinitely many problems simultaneously.
{"title":"Sketching Approximability of (Weak) Monarchy Predicates","authors":"Chi-Ning Chou, Alexander Golovnev, Amirbehshad Shahrasbi, M. Sudan, Santhoshini Velusamy","doi":"10.48550/arXiv.2205.02345","DOIUrl":"https://doi.org/10.48550/arXiv.2205.02345","url":null,"abstract":"We analyze the sketching approximability of constraint satisfaction problems on Boolean domains, where the constraints are balanced linear threshold functions applied to literals. In~particular, we explore the approximability of monarchy-like functions where the value of the function is determined by a weighted combination of the vote of the first variable (the president) and the sum of the votes of all remaining variables. The pure version of this function is when the president can only be overruled by when all remaining variables agree. For every $k geq 5$, we show that CSPs where the underlying predicate is a pure monarchy function on $k$ variables have no non-trivial sketching approximation algorithm in $o(sqrt{n})$ space. We also show infinitely many weaker monarchy functions for which CSPs using such constraints are non-trivially approximable by $O(log(n))$ space sketching algorithms. Moreover, we give the first example of sketching approximable asymmetric Boolean CSPs. Our results work within the framework of Chou, Golovnev, Sudan, and Velusamy (FOCS 2021) that characterizes the sketching approximability of all CSPs. Their framework can be applied naturally to get a computer-aided analysis of the approximability of any specific constraint satisfaction problem. The novelty of our work is in using their work to get an analysis that applies to infinitely many problems simultaneously.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"10 1","pages":"35:1-35:17"},"PeriodicalIF":0.0,"publicationDate":"2022-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85160630","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 : 2022-05-02DOI: 10.48550/arXiv.2205.00762
P. Liberatore
A superredundant clause is a clause that is redundant in the resolution closure of a formula. The converse concept of superirredundancy ensures membership of the clause in all minimal CNF formulae that are equivalent to the given one. This allows for building formulae where some clauses are fixed when minimizing size. An example are proofs of complexity hardness of the problems of minimal formula size. Others are proofs of size when forgetting variables or revising a formula. Most clauses can be made superirredundant by splitting them over a new variable.
{"title":"Superredundancy: A tool for Boolean formula minimization complexity analysis","authors":"P. Liberatore","doi":"10.48550/arXiv.2205.00762","DOIUrl":"https://doi.org/10.48550/arXiv.2205.00762","url":null,"abstract":"A superredundant clause is a clause that is redundant in the resolution closure of a formula. The converse concept of superirredundancy ensures membership of the clause in all minimal CNF formulae that are equivalent to the given one. This allows for building formulae where some clauses are fixed when minimizing size. An example are proofs of complexity hardness of the problems of minimal formula size. Others are proofs of size when forgetting variables or revising a formula. Most clauses can be made superirredundant by splitting them over a new variable.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77279044","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 : 2022-04-18DOI: 10.48550/arXiv.2204.08404
Vipul Arora, Arnab Bhattacharyya, Noah Fleming, E. Kelman, Yuichi Yoshida
We study the problem of testing whether a function $f: mathbb{R}^n to mathbb{R}$ is a polynomial of degree at most $d$ in the emph{distribution-free} testing model. Here, the distance between functions is measured with respect to an unknown distribution $mathcal{D}$ over $mathbb{R}^n$ from which we can draw samples. In contrast to previous work, we do not assume that $mathcal{D}$ has finite support. We design a tester that given query access to $f$, and sample access to $mathcal{D}$, makes $(d/varepsilon)^{O(1)}$ many queries to $f$, accepts with probability $1$ if $f$ is a polynomial of degree $d$, and rejects with probability at least $2/3$ if every degree-$d$ polynomial $P$ disagrees with $f$ on a set of mass at least $varepsilon$ with respect to $mathcal{D}$. Our result also holds under mild assumptions when we receive only a polynomial number of bits of precision for each query to $f$, or when $f$ can only be queried on rational points representable using a logarithmic number of bits. Along the way, we prove a new stability theorem for multivariate polynomials that may be of independent interest.
在emph{无分布}检验模型中，研究了一个函数$f: mathbb{R}^n to mathbb{R}$是否为次多项式(最多$d$)的检验问题。在这里，函数之间的距离是相对于一个未知分布$mathcal{D}$ / $mathbb{R}^n$来测量的，我们可以从中绘制样本。与以前的工作相反，我们不假设$mathcal{D}$具有有限的支持。我们设计了一个测试器，给定对$f$的查询访问权和对$mathcal{D}$的样本访问权，对$f$进行$(d/varepsilon)^{O(1)}$多次查询，如果$f$是次$d$的多项式，则以概率$1$接受。并且拒绝的概率至少为$2/3$如果每个度- $d$多项式$P$在一组质量上与$f$不一致至少$varepsilon$相对于$mathcal{D}$。我们的结果在温和的假设下也成立，当我们对$f$的每个查询只接收到多项式位数的精度，或者$f$只能在使用对数位数表示的有理点上查询时。在此过程中，我们证明了一个新的多元多项式的稳定性定理，这可能是独立的兴趣。
{"title":"Low Degree Testing over the Reals","authors":"Vipul Arora, Arnab Bhattacharyya, Noah Fleming, E. Kelman, Yuichi Yoshida","doi":"10.48550/arXiv.2204.08404","DOIUrl":"https://doi.org/10.48550/arXiv.2204.08404","url":null,"abstract":"We study the problem of testing whether a function $f: mathbb{R}^n to mathbb{R}$ is a polynomial of degree at most $d$ in the emph{distribution-free} testing model. Here, the distance between functions is measured with respect to an unknown distribution $mathcal{D}$ over $mathbb{R}^n$ from which we can draw samples. In contrast to previous work, we do not assume that $mathcal{D}$ has finite support. We design a tester that given query access to $f$, and sample access to $mathcal{D}$, makes $(d/varepsilon)^{O(1)}$ many queries to $f$, accepts with probability $1$ if $f$ is a polynomial of degree $d$, and rejects with probability at least $2/3$ if every degree-$d$ polynomial $P$ disagrees with $f$ on a set of mass at least $varepsilon$ with respect to $mathcal{D}$. Our result also holds under mild assumptions when we receive only a polynomial number of bits of precision for each query to $f$, or when $f$ can only be queried on rational points representable using a logarithmic number of bits. Along the way, we prove a new stability theorem for multivariate polynomials that may be of independent interest.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"26 1","pages":"738-792"},"PeriodicalIF":0.0,"publicationDate":"2022-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81745948","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 : 2022-04-04DOI: 10.48550/arXiv.2204.01224
Meghal Gupta, N. Manoj
Given query access to a monotone function $fcolon{0,1}^nto{0,1}$ with certificate complexity $C(f)$ and an input $x^{star}$, we design an algorithm that outputs a size-$C(f)$ subset of $x^{star}$ certifying the value of $f(x^{star})$. Our algorithm makes $O(C(f) cdot log n)$ queries to $f$, which matches the information-theoretic lower bound for this problem and resolves the concrete open question posed in the STOC '22 paper of Blanc, Koch, Lange, and Tan [BKLT22]. We extend this result to an algorithm that finds a size-$2C(f)$ certificate for a real-valued monotone function with $O(C(f) cdot log n)$ queries. We also complement our algorithms with a hardness result, in which we show that finding the shortest possible certificate in $x^{star}$ may require $Omegaleft(binom{n}{C(f)}right)$ queries in the worst case.
给定对具有证书复杂度$C(f)$和输入$x^{star}$的单调函数$fcolon{0,1}^nto{0,1}$的查询访问权，我们设计了一个算法，该算法输出验证$f(x^{star})$值的$x^{star}$的一个大小为$C(f)$的子集。我们的算法对$f$进行$O(C(f) cdot log n)$查询，该查询匹配该问题的信息论下界，并解决了Blanc, Koch, Lange, and Tan [BKLT22]的STOC '22论文中提出的具体开放问题。我们将这个结果扩展到一个算法，该算法通过$O(C(f) cdot log n)$查询为一个实值单调函数找到一个size- $2C(f)$证书。我们还用硬度结果补充了我们的算法，其中我们表明，在最坏的情况下，在$x^{star}$中找到最短的可能证书可能需要查询$Omegaleft(binom{n}{C(f)}right)$。
{"title":"An Optimal Algorithm for Certifying Monotone Functions","authors":"Meghal Gupta, N. Manoj","doi":"10.48550/arXiv.2204.01224","DOIUrl":"https://doi.org/10.48550/arXiv.2204.01224","url":null,"abstract":"Given query access to a monotone function $fcolon{0,1}^nto{0,1}$ with certificate complexity $C(f)$ and an input $x^{star}$, we design an algorithm that outputs a size-$C(f)$ subset of $x^{star}$ certifying the value of $f(x^{star})$. Our algorithm makes $O(C(f) cdot log n)$ queries to $f$, which matches the information-theoretic lower bound for this problem and resolves the concrete open question posed in the STOC '22 paper of Blanc, Koch, Lange, and Tan [BKLT22]. We extend this result to an algorithm that finds a size-$2C(f)$ certificate for a real-valued monotone function with $O(C(f) cdot log n)$ queries. We also complement our algorithms with a hardness result, in which we show that finding the shortest possible certificate in $x^{star}$ may require $Omegaleft(binom{n}{C(f)}right)$ queries in the worst case.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"25 1","pages":"207-212"},"PeriodicalIF":0.0,"publicationDate":"2022-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83531762","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 : 2022-04-02DOI: 10.48550/arXiv.2204.00858
Uma Girish, Kunal Mittal, R. Raz, Wei Zhan
We prove that for every 3-player (3-prover) game $mathcal G$ with value less than one, whose query distribution has the support $mathcal S = {(1,0,0), (0,1,0), (0,0,1)}$ of hamming weight one vectors, the value of the $n$-fold parallel repetition $mathcal G^{otimes n}$ decays polynomially fast to zero; that is, there is a constant $c = c(mathcal G)>0$ such that the value of the game $mathcal G^{otimes n}$ is at most $n^{-c}$. Following the recent work of Girish, Holmgren, Mittal, Raz and Zhan (STOC 2022), our result is the missing piece that implies a similar bound for a much more general class of multiplayer games: For $textbf{every}$ 3-player game $mathcal G$ over $textit{binary questions}$ and $textit{arbitrary answer lengths}$, with value less than 1, there is a constant $c = c(mathcal G)>0$ such that the value of the game $mathcal G^{otimes n}$ is at most $n^{-c}$. Our proof technique is new and requires many new ideas. For example, we make use of the Level-$k$ inequalities from Boolean Fourier Analysis, which, to the best of our knowledge, have not been explored in this context prior to our work.
{"title":"Polynomial Bounds On Parallel Repetition For All 3-Player Games With Binary Inputs","authors":"Uma Girish, Kunal Mittal, R. Raz, Wei Zhan","doi":"10.48550/arXiv.2204.00858","DOIUrl":"https://doi.org/10.48550/arXiv.2204.00858","url":null,"abstract":"We prove that for every 3-player (3-prover) game $mathcal G$ with value less than one, whose query distribution has the support $mathcal S = {(1,0,0), (0,1,0), (0,0,1)}$ of hamming weight one vectors, the value of the $n$-fold parallel repetition $mathcal G^{otimes n}$ decays polynomially fast to zero; that is, there is a constant $c = c(mathcal G)>0$ such that the value of the game $mathcal G^{otimes n}$ is at most $n^{-c}$. Following the recent work of Girish, Holmgren, Mittal, Raz and Zhan (STOC 2022), our result is the missing piece that implies a similar bound for a much more general class of multiplayer games: For $textbf{every}$ 3-player game $mathcal G$ over $textit{binary questions}$ and $textit{arbitrary answer lengths}$, with value less than 1, there is a constant $c = c(mathcal G)>0$ such that the value of the game $mathcal G^{otimes n}$ is at most $n^{-c}$. Our proof technique is new and requires many new ideas. For example, we make use of the Level-$k$ inequalities from Boolean Fourier Analysis, which, to the best of our knowledge, have not been explored in this context prior to our work.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"9 1","pages":"6:1-6:17"},"PeriodicalIF":0.0,"publicationDate":"2022-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89046023","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 : 2022-03-31DOI: 10.48550/arXiv.2203.16978
V. Arvind, Pushkar S. Joglekar
In continuation to our recent work [AJ22] on noncommutative polynomial factorization, we consider the factorization problem for matrices of polynomials and show the following results.
{"title":"Matrix Polynomial Factorization via Higman Linearization","authors":"V. Arvind, Pushkar S. Joglekar","doi":"10.48550/arXiv.2203.16978","DOIUrl":"https://doi.org/10.48550/arXiv.2203.16978","url":null,"abstract":"In continuation to our recent work [AJ22] on noncommutative polynomial factorization, we consider the factorization problem for matrices of polynomials and show the following results.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"25 1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79714667","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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