A Neighborhood Balanced Coloring (NBC) of a graph is a red-blue coloring where each vertex has the same number of red and blue neighbors. This work proves that determining if a graph admits an NBC is NP-complete. We present a genetic algorithm to solve this problem, which we implemented and compared against exact and randomized algorithms.
{"title":"NP-Completeness of Neighborhood Balanced Colorings","authors":"Saeed Asaeedi","doi":"arxiv-2407.19793","DOIUrl":"https://doi.org/arxiv-2407.19793","url":null,"abstract":"A Neighborhood Balanced Coloring (NBC) of a graph is a red-blue coloring\u0000where each vertex has the same number of red and blue neighbors. This work\u0000proves that determining if a graph admits an NBC is NP-complete. We present a\u0000genetic algorithm to solve this problem, which we implemented and compared\u0000against exact and randomized algorithms.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"212 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141868273","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}
Yuting Fang, Mika Göös, Nathaniel Harms, Pooya Hatami
Every known communication problem whose randomized communication cost is constant (independent of the input size) can be reduced to $k$-Hamming Distance, that is, solved with a constant number of deterministic queries to some $k$-Hamming Distance oracle. We exhibit the first examples of constant-cost problems which cannot be reduced to $k$-Hamming Distance. To prove this separation, we relate it to a natural coding-theoretic question. For $f : {2, 4, 6} to mathbb{N}$, we say an encoding function $E : {0, 1}^n to {0, 1}^m$ is an $f$-code if it transforms Hamming distances according to $mathrm{dist}(E(x), E(y)) = f(mathrm{dist}(x, y))$ whenever $f$ is defined. We prove that, if there exist $f$-codes for infinitely many $n$, then $f$ must be affine: $f(4) = (f(2) + f(6))/2$.
{"title":"Constant-Cost Communication is not Reducible to k-Hamming Distance","authors":"Yuting Fang, Mika Göös, Nathaniel Harms, Pooya Hatami","doi":"arxiv-2407.20204","DOIUrl":"https://doi.org/arxiv-2407.20204","url":null,"abstract":"Every known communication problem whose randomized communication cost is\u0000constant (independent of the input size) can be reduced to $k$-Hamming\u0000Distance, that is, solved with a constant number of deterministic queries to\u0000some $k$-Hamming Distance oracle. We exhibit the first examples of\u0000constant-cost problems which cannot be reduced to $k$-Hamming Distance. To prove this separation, we relate it to a natural coding-theoretic\u0000question. For $f : {2, 4, 6} to mathbb{N}$, we say an encoding function $E\u0000: {0, 1}^n to {0, 1}^m$ is an $f$-code if it transforms Hamming distances\u0000according to $mathrm{dist}(E(x), E(y)) = f(mathrm{dist}(x, y))$ whenever $f$\u0000is defined. We prove that, if there exist $f$-codes for infinitely many $n$,\u0000then $f$ must be affine: $f(4) = (f(2) + f(6))/2$.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"113 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141868271","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}
Shreya Gupta, Boyang Huang, Russell Impagliazzo, Stanley Woo, Christopher Ye
Computational complexity is traditionally measured with respect to input size. For graphs, this is typically the number of vertices (or edges) of the graph. However, for large graphs even explicitly representing the graph could be prohibitively expensive. Instead, graphs with enough structure could admit more succinct representations. A number of previous works have considered various succinct representations of graphs, such as small circuits [Galperin, Wigderson '83]. We initiate the study of the computational complexity of problems on factored graphs: graphs that are given as a formula of products and union on smaller graphs. For any graph problem, we define a parameterized version by the number of operations used to construct the graph. For different graph problems, we show that the corresponding parameterized problems have a wide range of complexities that are also quite different from most parameterized problems. We give a natural example of a parameterized problem that is unconditionally not fixed parameter tractable (FPT). On the other hand, we show that subgraph counting is FPT. Finally, we show that reachability for factored graphs is FPT if and only if $mathbf{NL}$ is in some fixed polynomial time.
{"title":"The Computational Complexity of Factored Graphs","authors":"Shreya Gupta, Boyang Huang, Russell Impagliazzo, Stanley Woo, Christopher Ye","doi":"arxiv-2407.19102","DOIUrl":"https://doi.org/arxiv-2407.19102","url":null,"abstract":"Computational complexity is traditionally measured with respect to input\u0000size. For graphs, this is typically the number of vertices (or edges) of the\u0000graph. However, for large graphs even explicitly representing the graph could\u0000be prohibitively expensive. Instead, graphs with enough structure could admit\u0000more succinct representations. A number of previous works have considered\u0000various succinct representations of graphs, such as small circuits [Galperin,\u0000Wigderson '83]. We initiate the study of the computational complexity of problems on factored\u0000graphs: graphs that are given as a formula of products and union on smaller\u0000graphs. For any graph problem, we define a parameterized version by the number\u0000of operations used to construct the graph. For different graph problems, we\u0000show that the corresponding parameterized problems have a wide range of\u0000complexities that are also quite different from most parameterized problems. We\u0000give a natural example of a parameterized problem that is unconditionally not\u0000fixed parameter tractable (FPT). On the other hand, we show that subgraph\u0000counting is FPT. Finally, we show that reachability for factored graphs is FPT\u0000if and only if $mathbf{NL}$ is in some fixed polynomial time.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"28 11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141868272","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}
Sebastian Ordyniak, Giacomo Paesani, Mateusz Rychlicki, Stefan Szeider
This paper presents a comprehensive theoretical investigation into the parameterized complexity of explanation problems in various machine learning (ML) models. Contrary to the prevalent black-box perception, our study focuses on models with transparent internal mechanisms. We address two principal types of explanation problems: abductive and contrastive, both in their local and global variants. Our analysis encompasses diverse ML models, including Decision Trees, Decision Sets, Decision Lists, Ordered Binary Decision Diagrams, Random Forests, and Boolean Circuits, and ensembles thereof, each offering unique explanatory challenges. This research fills a significant gap in explainable AI (XAI) by providing a foundational understanding of the complexities of generating explanations for these models. This work provides insights vital for further research in the domain of XAI, contributing to the broader discourse on the necessity of transparency and accountability in AI systems.
本文对各种机器学习（ML）模型中解释问题的参数化复杂性进行了全面的理论研究。与盛行的黑箱观念相反，我们的研究侧重于内部机制透明的模型。我们研究了两种主要类型的解释问题：归纳和对比，包括局部和全局变体。我们的分析涵盖多种 ML 模型，包括决策树、决策集、决策列表、有序二元决策图、随机森林和布尔电路，以及它们的集合，每种模型都提出了独特的解释性挑战。这项研究填补了可解释人工智能（XAI）领域的重大空白，提供了对这些模型生成解释的复杂性的基础性理解。这项工作为 XAI 领域的进一步研究提供了至关重要的见解，为更广泛地讨论人工智能系统中透明度和问责制的必要性做出了贡献。
{"title":"Explaining Decisions in ML Models: a Parameterized Complexity Analysis","authors":"Sebastian Ordyniak, Giacomo Paesani, Mateusz Rychlicki, Stefan Szeider","doi":"arxiv-2407.15780","DOIUrl":"https://doi.org/arxiv-2407.15780","url":null,"abstract":"This paper presents a comprehensive theoretical investigation into the\u0000parameterized complexity of explanation problems in various machine learning\u0000(ML) models. Contrary to the prevalent black-box perception, our study focuses\u0000on models with transparent internal mechanisms. We address two principal types\u0000of explanation problems: abductive and contrastive, both in their local and\u0000global variants. Our analysis encompasses diverse ML models, including Decision\u0000Trees, Decision Sets, Decision Lists, Ordered Binary Decision Diagrams, Random\u0000Forests, and Boolean Circuits, and ensembles thereof, each offering unique\u0000explanatory challenges. This research fills a significant gap in explainable AI\u0000(XAI) by providing a foundational understanding of the complexities of\u0000generating explanations for these models. This work provides insights vital for\u0000further research in the domain of XAI, contributing to the broader discourse on\u0000the necessity of transparency and accountability in AI systems.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141779574","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}
Often regarded as the study of how order emerges from randomness, Ramsey theory has played an important role in mathematics and computer science, giving rise to applications in numerous domains such as logic, parallel processing, and number theory. The core of graph Ramsey theory is arrowing: For fixed graphs $F$ and $H$, the $(F, H)$-Arrowing problem asks whether a given graph, $G$, has a red/blue coloring of the edges of $G$ such that there are no red copies of $F$ and no blue copies of $H$. For some cases, the problem has been shown to be coNP-complete, or solvable in polynomial time. However, a more systematic approach is needed to categorize the complexity of all cases. We focus on $(P_3, H)$-Arrowing as $F = P_3$ is the simplest meaningful case for which the complexity question remains open, and the hardness for this case likely extends to general $(F, H)$-Arrowing for nontrivial $F$. In this pursuit, we also gain insight into the complexity of a class of matching removal problems, since $(P_3, H)$-Arrowing is equivalent to $H$-free Matching Removal. We show that $(P_3, H)$-Arrowing is coNP-complete for all $2$-connected $H$ except when $H = K_3$, in which case the problem is in P. We introduce a new graph invariant to help us carefully combine graphs when constructing the gadgets for our reductions. Moreover, we show how $(P_3,H)$-Arrowing hardness results can be extended to other $(F,H)$-Arrowing problems. This allows for more intuitive and palatable hardness proofs instead of ad-hoc constructions of SAT gadgets, bringing us closer to categorizing the complexity of all $(F, H)$-Arrowing problems.
{"title":"The Complexity of (P3, H)-Arrowing and Beyond","authors":"Zohair Raza Hassan","doi":"arxiv-2407.15193","DOIUrl":"https://doi.org/arxiv-2407.15193","url":null,"abstract":"Often regarded as the study of how order emerges from randomness, Ramsey\u0000theory has played an important role in mathematics and computer science, giving\u0000rise to applications in numerous domains such as logic, parallel processing,\u0000and number theory. The core of graph Ramsey theory is arrowing: For fixed\u0000graphs $F$ and $H$, the $(F, H)$-Arrowing problem asks whether a given graph,\u0000$G$, has a red/blue coloring of the edges of $G$ such that there are no red\u0000copies of $F$ and no blue copies of $H$. For some cases, the problem has been\u0000shown to be coNP-complete, or solvable in polynomial time. However, a more\u0000systematic approach is needed to categorize the complexity of all cases. We focus on $(P_3, H)$-Arrowing as $F = P_3$ is the simplest meaningful case\u0000for which the complexity question remains open, and the hardness for this case\u0000likely extends to general $(F, H)$-Arrowing for nontrivial $F$. In this\u0000pursuit, we also gain insight into the complexity of a class of matching\u0000removal problems, since $(P_3, H)$-Arrowing is equivalent to $H$-free Matching\u0000Removal. We show that $(P_3, H)$-Arrowing is coNP-complete for all\u0000$2$-connected $H$ except when $H = K_3$, in which case the problem is in P. We\u0000introduce a new graph invariant to help us carefully combine graphs when\u0000constructing the gadgets for our reductions. Moreover, we show how\u0000$(P_3,H)$-Arrowing hardness results can be extended to other $(F,H)$-Arrowing\u0000problems. This allows for more intuitive and palatable hardness proofs instead\u0000of ad-hoc constructions of SAT gadgets, bringing us closer to categorizing the\u0000complexity of all $(F, H)$-Arrowing problems.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141779572","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}
Formal reasoning about the time complexity of algorithms and data structures is usually done in interactive theorem provers like Isabelle/HOL. This includes reasoning about amortized time complexity which looks at the worst case performance over a series of operations. However, most programs are not written within a theorem prover and thus use the data structures of the production language. To verify the correctness it is necessary to translate the data structures from the production language into the language of the prover. Such a translation step could introduce errors, for example due to a mismatch in features between the two languages. We show how to prove amortized complexity of data structures directly in Haskell using LiquidHaskell. Besides skipping the translation step, our approach can also provide a didactic advantage. Learners do not have to learn an additional language for proofs and can focus on the new concepts only. For this paper, we do not assume prior knowledge of amortized complexity as we explain the concepts and apply them in our first case study, a simple stack with multipop. Moving to more complicated (and useful) data structures, we show that the same technique works for binomial heaps which can be used to implement a priority queue. We also prove amortized complexity bounds for Claessen's version of the finger tree, a sequence-like data structure with constant-time cons/uncons on either end. Finally we discuss the current limitations of LiquidHaskell that made certain versions of the data structures not feasible.
{"title":"Liquid Amortization: Proving Amortized Complexity with LiquidHaskell (Functional Pearl)","authors":"Jan van Brügge","doi":"arxiv-2407.13671","DOIUrl":"https://doi.org/arxiv-2407.13671","url":null,"abstract":"Formal reasoning about the time complexity of algorithms and data structures\u0000is usually done in interactive theorem provers like Isabelle/HOL. This includes\u0000reasoning about amortized time complexity which looks at the worst case\u0000performance over a series of operations. However, most programs are not written\u0000within a theorem prover and thus use the data structures of the production\u0000language. To verify the correctness it is necessary to translate the data\u0000structures from the production language into the language of the prover. Such a\u0000translation step could introduce errors, for example due to a mismatch in\u0000features between the two languages. We show how to prove amortized complexity\u0000of data structures directly in Haskell using LiquidHaskell. Besides skipping\u0000the translation step, our approach can also provide a didactic advantage.\u0000Learners do not have to learn an additional language for proofs and can focus\u0000on the new concepts only. For this paper, we do not assume prior knowledge of\u0000amortized complexity as we explain the concepts and apply them in our first\u0000case study, a simple stack with multipop. Moving to more complicated (and\u0000useful) data structures, we show that the same technique works for binomial\u0000heaps which can be used to implement a priority queue. We also prove amortized\u0000complexity bounds for Claessen's version of the finger tree, a sequence-like\u0000data structure with constant-time cons/uncons on either end. Finally we discuss\u0000the current limitations of LiquidHaskell that made certain versions of the data\u0000structures not feasible.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"95 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141746191","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}
Julian Dörfler, Benito van der Zander, Markus Bläser, Maciej Liskiewicz
Learning the unknown causal parameters of a linear structural causal model is a fundamental task in causal analysis. The task, known as the problem of identification, asks to estimate the parameters of the model from a combination of assumptions on the graphical structure of the model and observational data, represented as a non-causal covariance matrix. In this paper, we give a new sound and complete algorithm for generic identification which runs in polynomial space. By standard simulation results, this algorithm has exponential running time which vastly improves the state-of-the-art double exponential time method using a Gr"obner basis approach. The paper also presents evidence that parameter identification is computationally hard in general. In particular, we prove, that the task asking whether, for a given feasible correlation matrix, there are exactly one or two or more parameter sets explaining the observed matrix, is hard for $forall R$, the co-class of the existential theory of the reals. In particular, this problem is $coNP$-hard. To our best knowledge, this is the first hardness result for some notion of identifiability.
{"title":"On the Complexity of Identification in Linear Structural Causal Models","authors":"Julian Dörfler, Benito van der Zander, Markus Bläser, Maciej Liskiewicz","doi":"arxiv-2407.12528","DOIUrl":"https://doi.org/arxiv-2407.12528","url":null,"abstract":"Learning the unknown causal parameters of a linear structural causal model is\u0000a fundamental task in causal analysis. The task, known as the problem of\u0000identification, asks to estimate the parameters of the model from a combination\u0000of assumptions on the graphical structure of the model and observational data,\u0000represented as a non-causal covariance matrix. In this paper, we give a new\u0000sound and complete algorithm for generic identification which runs in\u0000polynomial space. By standard simulation results, this algorithm has\u0000exponential running time which vastly improves the state-of-the-art double\u0000exponential time method using a Gr\"obner basis approach. The paper also\u0000presents evidence that parameter identification is computationally hard in\u0000general. In particular, we prove, that the task asking whether, for a given\u0000feasible correlation matrix, there are exactly one or two or more parameter\u0000sets explaining the observed matrix, is hard for $forall R$, the co-class of\u0000the existential theory of the reals. In particular, this problem is\u0000$coNP$-hard. To our best knowledge, this is the first hardness result for some\u0000notion of identifiability.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"95 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141746471","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 construct 2-query, quasi-linear sized probabilistically checkable proofs (PCPs) with arbitrarily small constant soundness, improving upon Dinur's 2-query quasi-linear size PCPs with soundness $1-Omega(1)$. As an immediate corollary, we get that under the exponential time hypothesis, for all $epsilon >0$ no approximation algorithm for $3$-SAT can obtain an approximation ratio of $7/8+epsilon$ in time $2^{n/log^C n}$, where $C$ is a constant depending on $epsilon$. Our result builds on a recent line of works showing the existence of linear sized direct product testers with small soundness by independent works of Bafna, Lifshitz, and Minzer, and of Dikstein, Dinur, and Lubotzky. The main new ingredient in our proof is a technique that embeds a given PCP construction into a PCP on a prescribed graph, provided that the latter is a graph underlying a sufficiently good high-dimensional expander. Towards this end, we use ideas from fault-tolerant distributed computing, and more precisely from the literature of the almost everywhere agreement problem starting with the work of Dwork, Peleg, Pippenger, and Upfal (1986). We show that graphs underlying HDXs admit routing protocols that are tolerant to adversarial edge corruptions, and in doing so we also improve the state of the art in this line of work. Our PCP construction requires variants of the aforementioned direct product testers with poly-logarithmic degree. The existence and constructability of these variants is shown in an appendix by Zhiwei Yun.
{"title":"Quasi-Linear Size PCPs with Small Soundness from HDX","authors":"Mitali Bafna, Dor Minzer, Nikhil Vyas","doi":"arxiv-2407.12762","DOIUrl":"https://doi.org/arxiv-2407.12762","url":null,"abstract":"We construct 2-query, quasi-linear sized probabilistically checkable proofs\u0000(PCPs) with arbitrarily small constant soundness, improving upon Dinur's\u00002-query quasi-linear size PCPs with soundness $1-Omega(1)$. As an immediate\u0000corollary, we get that under the exponential time hypothesis, for all $epsilon\u0000>0$ no approximation algorithm for $3$-SAT can obtain an approximation ratio of\u0000$7/8+epsilon$ in time $2^{n/log^C n}$, where $C$ is a constant depending on\u0000$epsilon$. Our result builds on a recent line of works showing the existence\u0000of linear sized direct product testers with small soundness by independent\u0000works of Bafna, Lifshitz, and Minzer, and of Dikstein, Dinur, and Lubotzky. The main new ingredient in our proof is a technique that embeds a given PCP\u0000construction into a PCP on a prescribed graph, provided that the latter is a\u0000graph underlying a sufficiently good high-dimensional expander. Towards this\u0000end, we use ideas from fault-tolerant distributed computing, and more precisely\u0000from the literature of the almost everywhere agreement problem starting with\u0000the work of Dwork, Peleg, Pippenger, and Upfal (1986). We show that graphs\u0000underlying HDXs admit routing protocols that are tolerant to adversarial edge\u0000corruptions, and in doing so we also improve the state of the art in this line\u0000of work. Our PCP construction requires variants of the aforementioned direct product\u0000testers with poly-logarithmic degree. The existence and constructability of\u0000these variants is shown in an appendix by Zhiwei Yun.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141746470","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}
Harm Derksen, Peter Ivanov, Chin Ho Lee, Emanuele Viola
We prove several new results on the Hamming weight of bounded uniform and small-bias distributions. We exhibit bounded-uniform distributions whose weight is anti-concentrated, matching existing concentration inequalities. This construction relies on a recent result in approximation theory due to Erd'eyi (Acta Arithmetica 2016). In particular, we match the classical tail bounds, generalizing a result by Bun and Steinke (RANDOM 2015). Also, we improve on a construction by Benjamini, Gurel-Gurevich, and Peled (2012). We give a generic transformation that converts any bounded uniform distribution to a small-bias distribution that almost preserves its weight distribution. Applying this transformation in conjunction with the above results and others, we construct small-bias distributions with various weight restrictions. In particular, we match the concentration that follows from that of bounded uniformity and the generic closeness of small-bias and bounded-uniform distributions, answering a question by Bun and Steinke (RANDOM 2015). Moreover, these distributions are supported on only a constant number of Hamming weights. We further extend the anti-concentration constructions to small-bias distributions perturbed with noise, a class that has received much attention recently in derandomization. Our results imply (but are not implied by) a recent result of the authors (CCC 2024), and are based on different techniques. In particular, we prove that the standard Gaussian distribution is far from any mixture of Gaussians with bounded variance.
{"title":"Pseudorandomness, symmetry, smoothing: II","authors":"Harm Derksen, Peter Ivanov, Chin Ho Lee, Emanuele Viola","doi":"arxiv-2407.12110","DOIUrl":"https://doi.org/arxiv-2407.12110","url":null,"abstract":"We prove several new results on the Hamming weight of bounded uniform and\u0000small-bias distributions. We exhibit bounded-uniform distributions whose weight is anti-concentrated,\u0000matching existing concentration inequalities. This construction relies on a\u0000recent result in approximation theory due to Erd'eyi (Acta Arithmetica 2016).\u0000In particular, we match the classical tail bounds, generalizing a result by Bun\u0000and Steinke (RANDOM 2015). Also, we improve on a construction by Benjamini,\u0000Gurel-Gurevich, and Peled (2012). We give a generic transformation that converts any bounded uniform\u0000distribution to a small-bias distribution that almost preserves its weight\u0000distribution. Applying this transformation in conjunction with the above\u0000results and others, we construct small-bias distributions with various weight\u0000restrictions. In particular, we match the concentration that follows from that\u0000of bounded uniformity and the generic closeness of small-bias and\u0000bounded-uniform distributions, answering a question by Bun and Steinke (RANDOM\u00002015). Moreover, these distributions are supported on only a constant number of\u0000Hamming weights. We further extend the anti-concentration constructions to small-bias\u0000distributions perturbed with noise, a class that has received much attention\u0000recently in derandomization. Our results imply (but are not implied by) a\u0000recent result of the authors (CCC 2024), and are based on different techniques.\u0000In particular, we prove that the standard Gaussian distribution is far from any\u0000mixture of Gaussians with bounded variance.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141746192","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 dimension of partial derivatives (Nisan and Wigderson, 1997) is a popular measure for proving lower bounds in algebraic complexity. It is used to give strong lower bounds on the Waring decomposition of polynomials (called Waring rank). This naturally leads to an interesting open question: does this measure essentially characterize the Waring rank of any polynomial? The well-studied model of Read-once Oblivious ABPs (ROABPs for short) lends itself to an interesting hierarchy of 'sub-models': Any-Order-ROABPs (ARO), Commutative ROABPs, and Diagonal ROABPs. It follows from previous works that for any polynomial, a bound on its Waring rank implies an analogous bound on its Diagonal ROABP complexity (called the duality trick), and a bound on its dimension of partial derivatives implies an analogous bound on its 'ARO complexity': ROABP complexity in any order (Nisan, 1991). Our work strengthens the latter connection by showing that a bound on the dimension of partial derivatives in fact implies a bound on the commutative ROABP complexity. Thus, we improve our understanding of partial derivatives and move a step closer towards answering the above question. Our proof builds on the work of Ramya and Tengse (2022) to show that the commutative-ROABP-width of any homogeneous polynomial is at most the dimension of its partial derivatives. The technique itself is a generalization of the proof of the duality trick due to Saxena (2008).
{"title":"Explicit Commutative ROABPs from Partial Derivatives","authors":"Vishwas Bhargava, Anamay Tengse","doi":"arxiv-2407.10143","DOIUrl":"https://doi.org/arxiv-2407.10143","url":null,"abstract":"The dimension of partial derivatives (Nisan and Wigderson, 1997) is a popular\u0000measure for proving lower bounds in algebraic complexity. It is used to give\u0000strong lower bounds on the Waring decomposition of polynomials (called Waring\u0000rank). This naturally leads to an interesting open question: does this measure\u0000essentially characterize the Waring rank of any polynomial? The well-studied model of Read-once Oblivious ABPs (ROABPs for short) lends\u0000itself to an interesting hierarchy of 'sub-models': Any-Order-ROABPs (ARO),\u0000Commutative ROABPs, and Diagonal ROABPs. It follows from previous works that\u0000for any polynomial, a bound on its Waring rank implies an analogous bound on\u0000its Diagonal ROABP complexity (called the duality trick), and a bound on its\u0000dimension of partial derivatives implies an analogous bound on its 'ARO\u0000complexity': ROABP complexity in any order (Nisan, 1991). Our work strengthens\u0000the latter connection by showing that a bound on the dimension of partial\u0000derivatives in fact implies a bound on the commutative ROABP complexity. Thus,\u0000we improve our understanding of partial derivatives and move a step closer\u0000towards answering the above question. Our proof builds on the work of Ramya and Tengse (2022) to show that the\u0000commutative-ROABP-width of any homogeneous polynomial is at most the dimension\u0000of its partial derivatives. The technique itself is a generalization of the\u0000proof of the duality trick due to Saxena (2008).","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719611","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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