Electron. Colloquium Comput. Complex.最新文献_第8页

Electron. Colloquium Comput. Complex. Pub Date : 2022-09-12 DOI: 10.1090/bull/1777

Anup Rao

引用次数: 3

On the existence of strong proof complexity generators 论强证明复杂性生成器的存在性

Electron. Colloquium Comput. Complex. Pub Date : 2022-08-24 DOI: 10.48550/arXiv.2208.11642

J. Krajícek

{"title":"On the existence of strong proof complexity generators","authors":"J. Krajícek","doi":"10.48550/arXiv.2208.11642","DOIUrl":"https://doi.org/10.48550/arXiv.2208.11642","url":null,"abstract":"Cook and Reckhow 1979 pointed out that NP is not closed under complementation iff there is no propositional proof system that admits polynomial size proofs of all tautologies. Theory of proof complexity generators aims at constructing sets of tautologies hard for strong and possibly for all proof systems. We focus at a conjecture from K.2004 in foundations of the theory that there is a proof complexity generator hard for all proof systems. This can be equivalently formulated (for p-time generators) without a reference to proof complexity notions as follows: * There exist a p-time function $g$ stretching each input by one bit such that its range intersects all infinite NP sets. We consider several facets of this conjecture, including its links to bounded arithmetic (witnessing and independence results), to time-bounded Kolmogorov complexity, to feasible disjunction property of propositional proof systems and to complexity of proof search. We argue that a specific gadget generator from K.2009 is a good candidate for $g$. We define a new hardness property of generators, the $bigvee$-hardness, and shows that one specific gadget generator is the $bigvee$-hardest (w.r.t. any sufficiently strong proof system). We define the class of feasibly infinite NP sets and show, assuming a hypothesis from circuit complexity, that the conjecture holds for all feasibly infinite NP sets.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81515715","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}

引用次数: 3

Constant-Depth Sorting Networks 恒深度排序网络

Electron. Colloquium Comput. Complex. Pub Date : 2022-08-17 DOI: 10.48550/arXiv.2208.08394

Natalia Dobrokhotova-Maikova, A. Kozachinskiy, V. Podolskii

{"title":"Constant-Depth Sorting Networks","authors":"Natalia Dobrokhotova-Maikova, A. Kozachinskiy, V. Podolskii","doi":"10.48550/arXiv.2208.08394","DOIUrl":"https://doi.org/10.48550/arXiv.2208.08394","url":null,"abstract":"In this paper, we address sorting networks that are constructed from comparators of arity $k>2$. That is, in our setting the arity of the comparators -- or, in other words, the number of inputs that can be sorted at the unit cost -- is a parameter. We study its relationship with two other parameters -- $n$, the number of inputs, and $d$, the depth. This model received considerable attention. Partly, its motivation is to better understand the structure of sorting networks. In particular, sorting networks with large arity are related to recursive constructions of ordinary sorting networks. Additionally, studies of this model have natural correspondence with a recent line of work on constructing circuits for majority functions from majority gates of lower fan-in. Motivated by these questions, we obtain the first lower bounds on the arity of constant-depth sorting networks. More precisely, we consider sorting networks of depth $d$ up to 4, and determine the minimal $k$ for which there is such a network with comparators of arity $k$. For depths $d=1,2$ we observe that $k=n$. For $d=3$ we show that $k = lceil frac n2 rceil$. For $d=4$ the minimal arity becomes sublinear: $k = Theta(n^{2/3})$. This contrasts with the case of majority circuits, in which $k = O(n^{2/3})$ is achievable already for depth $d=3$.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85782977","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}

引用次数: 3

Direct Sum Theorems From Fortification 直接和定理

Electron. Colloquium Comput. Complex. Pub Date : 2022-08-16 DOI: 10.48550/arXiv.2208.07730

Hao Wu

{"title":"Direct Sum Theorems From Fortification","authors":"Hao Wu","doi":"10.48550/arXiv.2208.07730","DOIUrl":"https://doi.org/10.48550/arXiv.2208.07730","url":null,"abstract":"We revisit the direct sum questions in communication complexity which asks whether the resource needed to solve $n$ communication problems together is (approximately) the sum of resources needed to solve these problems separately. Our work starts with the observation that Dinur and Meir's fortification lemma can be generalized to a general fortification lemma for a sub-additive measure over set. By applying this lemma to the case of cover number, we obtain a dual form of cover number, called\"$delta$-fooling set\"which is a generalized fooling set. Any rectangle which contains enough number of elements from a $delta$-fooling set can not be monochromatic. With this fact, we are able to reprove the classic direct sum theorem of cover number with a simple double counting argument. Formally, let $S subseteq (Atimes B) times O$ and $T subseteq (Ptimes Q) times Z$ be two communication problems, $ log mathsf{Cov}left(Stimes Tright) geq log mathsf{Cov}left(Sright) + logmathsf{Cov}(T) -loglog|P||Q|-4.$ where $mathsf{Cov}$ denotes the cover number. One issue of current deterministic direct sum theorems about communication complexity is that they provide no information when $n$ is small, especially when $n=2$. In this work, we prove a new direct sum theorem about protocol size which imply a better direct sum theorem for two functions in terms of protocol size. Formally, let $mathsf{L}$ denotes complexity of the protocol size of a communication problem, given a communication problem $F:A times B rightarrow {0,1}$, $ logmathsf{L}left(Ftimes Fright)geq log mathsf{L}left(Fright) +Omegaleft(sqrt{logmathsf{L}left(Fright)}right)-loglog|A||B| -4$. All our results are obtained in a similar way using the $delta$-fooling set to construct a hardcore for the direct sum problem.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83396482","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}

引用次数: 0

Communication Complexity of Collision 碰撞的通信复杂度

Electron. Colloquium Comput. Complex. Pub Date : 2022-07-29 DOI: 10.48550/arXiv.2208.00029

Mika Göös, Siddhartha Jain

引用次数: 1

Testing of Index-Invariant Properties in the Huge Object Model 大型对象模型中索引不变性的检验

Electron. Colloquium Comput. Complex. Pub Date : 2022-07-25 DOI: 10.48550/arXiv.2207.12514

Sourav Chakraborty, E. Fischer, Arijit Ghosh, Gopinath Mishra, Sayantan Sen

{"title":"Testing of Index-Invariant Properties in the Huge Object Model","authors":"Sourav Chakraborty, E. Fischer, Arijit Ghosh, Gopinath Mishra, Sayantan Sen","doi":"10.48550/arXiv.2207.12514","DOIUrl":"https://doi.org/10.48550/arXiv.2207.12514","url":null,"abstract":"The study of distribution testing has become ubiquitous in the area of property testing, both for its theoretical appeal, as well as for its applications in other fields of Computer Science. The original distribution testing model relies on samples drawn independently from the distribution to be tested. However, when testing distributions over the $n$-dimensional Hamming cube $left{0,1right}^{n}$ for a large $n$, even reading a few samples is infeasible. To address this, Goldreich and Ron [ITCS 2022] have defined a model called the huge object model, in which the samples may only be queried in a few places. In this work, we initiate a study of a general class of properties in the huge object model, those that are invariant under a permutation of the indices of the vectors in $left{0,1right}^{n}$, while still not being necessarily fully symmetric as per the definition used in traditional distribution testing. We prove that every index-invariant property satisfying a bounded VC-dimension restriction admits a property tester with a number of queries independent of n. To complement this result, we argue that satisfying only index-invariance or only a VC-dimension bound is insufficient to guarantee a tester whose query complexity is independent of n. Moreover, we prove that the dependency of sample and query complexities of our tester on the VC-dimension is tight. As a second part of this work, we address the question of the number of queries required for non-adaptive testing. We show that it can be at most quadratic in the number of queries required for an adaptive tester of index-invariant properties. This is in contrast with the tight exponential gap for general non-index-invariant properties. Finally, we provide an index-invariant property for which the quadratic gap between adaptive and non-adaptive query complexities for testing is almost tight.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"6 1","pages":"3065-3136"},"PeriodicalIF":0.0,"publicationDate":"2022-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78728189","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}

引用次数: 3

On Hardness of Testing Equivalence to Sparse Polynomials Under Shifts 位移下稀疏多项式等价检验的硬度

Electron. Colloquium Comput. Complex. Pub Date : 2022-07-21 DOI: 10.48550/arXiv.2207.10588

S. Chillara, Coral Grichener, Amir Shpilka

{"title":"On Hardness of Testing Equivalence to Sparse Polynomials Under Shifts","authors":"S. Chillara, Coral Grichener, Amir Shpilka","doi":"10.48550/arXiv.2207.10588","DOIUrl":"https://doi.org/10.48550/arXiv.2207.10588","url":null,"abstract":"We say that two given polynomials $f, g in R[X]$, over a ring $R$, are equivalent under shifts if there exists a vector $a in R^n$ such that $f(X+a) = g(X)$. Grigoriev and Karpinski (FOCS 1990), Lakshman and Saunders (SICOMP, 1995), and Grigoriev and Lakshman (ISSAC 1995) studied the problem of testing polynomial equivalence of a given polynomial to any $t$-sparse polynomial, over the rational numbers, and gave exponential time algorithms. In this paper, we provide hardness results for this problem. Formally, for a ring $R$, let $mathrm{SparseShift}_R$ be the following decision problem. Given a polynomial $P(X)$, is there a vector $a$ such that $P(X+a)$ contains fewer monomials than $P(X)$. We show that $mathrm{SparseShift}_R$ is at least as hard as checking if a given system of polynomial equations over $R[x_1,ldots, x_n]$ has a solution (Hilbert's Nullstellensatz). As a consequence of this reduction, we get the following results. 1. $mathrm{SparseShift}_mathbb{Z}$ is undecidable. 2. For any ring $R$ (which is not a field) such that $mathrm{HN}_R$ is $mathrm{NP}_R$-complete over the Blum-Shub-Smale model of computation, $mathrm{SparseShift}_{R}$ is also $mathrm{NP}_{R}$-complete. In particular, $mathrm{SparseShift}_{mathbb{Z}}$ is also $mathrm{NP}_{mathbb{Z}}$-complete. We also study the gap version of the $mathrm{SparseShift}_R$ and show the following. 1. For every function $beta: mathbb{N}tomathbb{R}_+$ such that $betain o(1)$, $N^beta$-gap-$mathrm{SparseShift}_mathbb{Z}$ is also undecidable (where $N$ is the input length). 2. For $R=mathbb{F}_p, mathbb{Q}, mathbb{R}$ or $mathbb{Z}_q$ and for every $beta>1$ the $beta$-gap-$mathrm{SparseShift}_R$ problem is $mathrm{NP}$-hard.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"77 1","pages":"22:1-22:20"},"PeriodicalIF":0.0,"publicationDate":"2022-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80380528","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}

引用次数: 1

Streaming complexity of CSPs with randomly ordered constraints 随机有序约束下csp的流复杂度

Electron. Colloquium Comput. Complex. Pub Date : 2022-07-14 DOI: 10.1137/1.9781611977554.ch156

Raghuvansh R. Saxena, Noah G. Singer, M. Sudan, Santhoshini Velusamy

{"title":"Streaming complexity of CSPs with randomly ordered constraints","authors":"Raghuvansh R. Saxena, Noah G. Singer, M. Sudan, Santhoshini Velusamy","doi":"10.1137/1.9781611977554.ch156","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch156","url":null,"abstract":"We initiate a study of the streaming complexity of constraint satisfaction problems (CSPs) when the constraints arrive in a random order. We show that there exists a CSP, namely $textsf{Max-DICUT}$, for which random ordering makes a provable difference. Whereas a $4/9 approx 0.445$ approximation of $textsf{DICUT}$ requires $Omega(sqrt{n})$ space with adversarial ordering, we show that with random ordering of constraints there exists a $0.48$-approximation algorithm that only needs $O(log n)$ space. We also give new algorithms for $textsf{Max-DICUT}$ in variants of the adversarial ordering setting. Specifically, we give a two-pass $O(log n)$ space $0.48$-approximation algorithm for general graphs and a single-pass $tilde{O}(sqrt{n})$ space $0.48$-approximation algorithm for bounded degree graphs. On the negative side, we prove that CSPs where the satisfying assignments of the constraints support a one-wise independent distribution require $Omega(sqrt{n})$-space for any non-trivial approximation, even when the constraints are randomly ordered. This was previously known only for adversarially ordered constraints. Extending the results to randomly ordered constraints requires switching the hard instances from a union of random matchings to simple Erd\"os-Renyi random (hyper)graphs and extending tools that can perform Fourier analysis on such instances. The only CSP to have been considered previously with random ordering is $textsf{Max-CUT}$ where the ordering is not known to change the approximability. Specifically it is known to be as hard to approximate with random ordering as with adversarial ordering, for $o(sqrt{n})$ space algorithms. Our results show a richer variety of possibilities and motivate further study of CSPs with randomly ordered constraints.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"51 1","pages":"4083-4103"},"PeriodicalIF":0.0,"publicationDate":"2022-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77239653","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}

引用次数: 5

QCDCL with Cube Learning or Pure Literal Elimination - What is best? QCDCL与立方体学习或纯文字消除-哪个是最好的?

Electron. Colloquium Comput. Complex. Pub Date : 2022-07-01 DOI: 10.24963/ijcai.2022/248

Olaf Beyersdorff, Benjamin Böhm

引用次数: 3

Improved bounds on the AN-complexity of O(1)-linear functions 改进了O(1)-线性函数的复杂度界

Electron. Colloquium Comput. Complex. Pub Date : 2022-06-23 DOI: 10.1007/s00037-022-00224-7

O. Goldreich

引用次数: 0