Electron. Colloquium Comput. Complex.最新文献_第7页

On Disperser/Lifting Properties of the Index and Inner-Product Functions 指数函数和内积函数的分散/提升性质

Electron. Colloquium Comput. Complex. Pub Date : 2022-11-30 DOI: 10.48550/arXiv.2211.17211

P. Beame, Sajin Koroth

{"title":"On Disperser/Lifting Properties of the Index and Inner-Product Functions","authors":"P. Beame, Sajin Koroth","doi":"10.48550/arXiv.2211.17211","DOIUrl":"https://doi.org/10.48550/arXiv.2211.17211","url":null,"abstract":"Query-to-communication lifting theorems, which connect the query complexity of a Boolean function to the communication complexity of an associated `lifted' function obtained by composing the function with many copies of another function known as a gadget, have been instrumental in resolving many open questions in computational complexity. Several important complexity questions could be resolved if we could make substantial improvements in the input size required for lifting with the Index function, from its current near-linear size down to polylogarithmic in the number of inputs $N$ of the original function or, ideally, constant. The near-linear size bound was shown by Lovett, Meka, Mertz, Pitassi and Zhang using a recent breakthrough improvement on the Sunflower Lemma to show that a certain graph associated with the Index function of near-linear size is a disperser. They also stated a conjecture about the Index function that is essential for further improvements in the size required for lifting with Index using current techniques. In this paper we prove the following; 1) The conjecture of Lovett et al. is false when the size of the Index gadget is $log N-omega(1)$. 2) Also, the Inner-Product function, which satisfies the disperser property at size $O(log N)$, does not have this property when its size is $log N-omega(1)$. 3) Nonetheless, using Index gadgets of size at least 4, we prove a lifting theorem for a restricted class of communication protocols in which one of the players is limited to sending parities of its inputs. 4) Using the ideas from this lifting theorem, we derive a strong lifting theorem from decision tree size to parity decision tree size. We use this to derive a general lifting theorem in proof complexity from tree-resolution size to tree-like $Res(oplus)$ refutation size, which yields many new exponential lower bounds on such proofs.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72773772","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: 2

Lifting to Parity Decision Trees Via Stifling 通过窒息提升到奇偶性决策树

Electron. Colloquium Comput. Complex. Pub Date : 2022-11-30 DOI: 10.48550/arXiv.2211.17214

A. Chattopadhyay, Nikhil S. Mande, Swagato Sanyal, Suhail Sherif

{"title":"Lifting to Parity Decision Trees Via Stifling","authors":"A. Chattopadhyay, Nikhil S. Mande, Swagato Sanyal, Suhail Sherif","doi":"10.48550/arXiv.2211.17214","DOIUrl":"https://doi.org/10.48550/arXiv.2211.17214","url":null,"abstract":"We show that the deterministic decision tree complexity of a (partial) function or relation $f$ lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation $f circ g$ as long as the gadget $g$ satisfies a property that we call stifling. We observe that several simple gadgets of constant size, like Indexing on 3 input bits, Inner Product on 4 input bits, Majority on 3 input bits and random functions, satisfy this property. It can be shown that existing randomized communication lifting theorems ([G\"{o}\"{o}s, Pitassi, Watson. SICOMP'20], [Chattopadhyay et al. SICOMP'21]) imply PDT-size lifting. However there are two shortcomings of this approach: first they lift randomized decision tree complexity of $f$, which could be exponentially smaller than its deterministic counterpart when either $f$ is a partial function or even a total search problem. Second, the size of the gadgets in such lifting theorems are as large as logarithmic in the size of the input to $f$. Reducing the gadget size to a constant is an important open problem at the frontier of current research. Our result shows that even a random constant-size gadget does enable lifting to PDT size. Further, it also yields the first systematic way of turning lower bounds on the width of tree-like resolution proofs of the unsatisfiability of constant-width CNF formulas to lower bounds on the size of tree-like proofs in the resolution with parity system, i.e., $textit{Res}$($oplus$), of the unsatisfiability of closely related constant-width CNF formulas.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81641947","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

Derandomization under Different Resource Constraints 不同资源约束下的非随机化

Electron. Colloquium Comput. Complex. Pub Date : 2022-11-26 DOI: 10.48550/arXiv.2211.14640

Samuel Epstein

引用次数: 1

Parallel Repetition for the GHZ Game: Exponential Decay GHZ游戏的平行重复:指数衰减

Electron. Colloquium Comput. Complex. Pub Date : 2022-11-24 DOI: 10.48550/arXiv.2211.13741

M. Braverman, Subhash Khot, Dor Minzer

引用次数: 3

Query Complexity of Inversion Minimization on Trees 树上逆最小化的查询复杂度

Electron. Colloquium Comput. Complex. Pub Date : 2022-11-22 DOI: 10.48550/arXiv.2211.12441

Ivan Hu, D. Melkebeek, Andrew Morgan

{"title":"Query Complexity of Inversion Minimization on Trees","authors":"Ivan Hu, D. Melkebeek, Andrew Morgan","doi":"10.48550/arXiv.2211.12441","DOIUrl":"https://doi.org/10.48550/arXiv.2211.12441","url":null,"abstract":"We consider the following computational problem: Given a rooted tree and a ranking of its leaves, what is the minimum number of inversions of the leaves that can be attained by ordering the tree? This variation of the problem of counting inversions in arrays originated in mathematical psychology, with the evaluation of the Mann--Whitney statistic for detecting differences between distributions as a special case. We study the complexity of the problem in the comparison-query model, used for problems like sorting and selection. For many types of trees with $n$ leaves, we establish lower bounds close to the strongest known in the model, namely the lower bound of $log_2(n!)$ for sorting $n$ items. We show: (a) $log_2((alpha(1-alpha)n)!) - O(log n)$ queries are needed whenever the tree has a subtree that contains a fraction $alpha$ of the leaves. This implies a lower bound of $log_2((frac{k}{(k+1)^2}n)!) - O(log n)$ for trees of degree $k$. (b) $log_2(n!) - O(log n)$ queries are needed in case the tree is binary. (c) $log_2(n!) - O(k log k)$ queries are needed for certain classes of trees of degree $k$, including perfect trees with even $k$. The lower bounds are obtained by developing two novel techniques for a generic problem $Pi$ in the comparison-query model and applying them to inversion minimization on trees. Both techniques can be described in terms of the Cayley graph of the symmetric group with adjacent-rank transpositions as the generating set. Consider the subgraph consisting of the edges between vertices with the same value under $Pi$. We show that the size of any decision tree for $Pi$ must be at least: (i) the number of connected components of this subgraph, and (ii) the factorial of the average degree of the complementary subgraph, divided by $n$. Lower bounds on query complexity then follow by taking the base-2 logarithm.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"23 1","pages":"2836-2866"},"PeriodicalIF":0.0,"publicationDate":"2022-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78976391","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

Low-depth arithmetic circuit lower bounds via shifted partials 低深度算术电路下界通过移位偏

Electron. Colloquium Comput. Complex. Pub Date : 2022-11-14 DOI: 10.48550/arXiv.2211.07691

Prashanth Amireddy, A. Garg, N. Kayal, Chandan Saha, Bhargav Thankey

引用次数: 1

Geometry of Rounding 舍入几何

Electron. Colloquium Comput. Complex. Pub Date : 2022-11-04 DOI: 10.48550/arXiv.2211.02694

Jason Vander Woude, P. Dixon, A. Pavan, Jamie Radcliffe, N. V. Vinodchandran

{"title":"Geometry of Rounding","authors":"Jason Vander Woude, P. Dixon, A. Pavan, Jamie Radcliffe, N. V. Vinodchandran","doi":"10.48550/arXiv.2211.02694","DOIUrl":"https://doi.org/10.48550/arXiv.2211.02694","url":null,"abstract":"Rounding has proven to be a fundamental tool in theoretical computer science. By observing that rounding and partitioning of $mathbb{R}^d$ are equivalent, we introduce the following natural partition problem which we call the {em secluded hypercube partition problem}: Given $kin mathbb{N}$ (ideally small) and $epsilon>0$ (ideally large), is there a partition of $mathbb{R}^d$ with unit hypercubes such that for every point $p in mathbb{R}^d$, its closed $epsilon$-neighborhood (in the $ell_{infty}$ norm) intersects at most $k$ hypercubes? We undertake a comprehensive study of this partition problem. We prove that for every $din mathbb{N}$, there is an explicit (and efficiently computable) hypercube partition of $mathbb{R}^d$ with $k = d+1$ and $epsilon = frac{1}{2d}$. We complement this construction by proving that the value of $k=d+1$ is the best possible (for any $epsilon$) for a broad class of ``reasonable'' partitions including hypercube partitions. We also investigate the optimality of the parameter $epsilon$ and prove that any partition in this broad class that has $k=d+1$, must have $epsilonleqfrac{1}{2sqrt{d}}$. These bounds imply limitations of certain deterministic rounding schemes existing in the literature. Furthermore, this general bound is based on the currently known lower bounds for the dissection number of the cube, and improvements to this bound will yield improvements to our bounds. While our work is motivated by the desire to understand rounding algorithms, one of our main conceptual contributions is the introduction of the {em secluded hypercube partition problem}, which fits well with a long history of investigations by mathematicians on various hypercube partitions/tilings of Euclidean space.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"76 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86094891","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

The power of the Binary Value Principle 二进制值原理的力量

Electron. Colloquium Comput. Complex. Pub Date : 2022-10-31 DOI: 10.48550/arXiv.2210.17429

Yaroslav Alekseev, E. Hirsch

{"title":"The power of the Binary Value Principle","authors":"Yaroslav Alekseev, E. Hirsch","doi":"10.48550/arXiv.2210.17429","DOIUrl":"https://doi.org/10.48550/arXiv.2210.17429","url":null,"abstract":"The (extended) Binary Value Principle (eBVP: $sum_{i=1}^n x_i2^{i-1} = -k$ for $k>0$ and $x^2_i=x_i$) has received a lot of attention recently, several lower bounds have been proved for it (Alekseev et al 2020, Alekseev 2021, Part and Tzameret 2021). Also it has been shown (Alekseev et al 2020) that the probabilistically verifiable Ideal Proof System (IPS) (Grochow and Pitassi 2018) together with eBVP polynomially simulates a similar semialgebraic proof system. In this paper we consider Polynomial Calculus with the algebraic version of Tseitin's extension rule (Ext-PC). Contrary to IPS, this is a Cook--Reckhow proof system. We show that in this context eBVP still allows to simulate similar semialgebraic systems. We also prove that it allows to simulate the Square Root Rule (Grigoriev and Hirsch 2003), which is in sharp contrast with the result of (Alekseev 2021) that shows an exponential lower bound on the size of Ext-PC derivations of the Binary Value Principle from its square. On the other hand, we demonstrate that eBVP probably does not help in proving exponential lower bounds for Boolean formulas: we show that an Ext-PC (even with the Square Root Rule) derivation of any unsatisfiable Boolean formula in CNF from eBVP must be of exponential size.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"69 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75657885","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

Downward Self-Reducibility in TFNP TFNP的向下自约性

Electron. Colloquium Comput. Complex. Pub Date : 2022-09-21 DOI: 10.48550/arXiv.2209.10509

P. Harsha, Daniel Mitropolsky, Alon Rosen

引用次数: 0

Decision Tree Complexity versus Block Sensitivity and Degree 决策树复杂性与块灵敏度和度

Electron. Colloquium Comput. Complex. Pub Date : 2022-09-16 DOI: 10.48550/arXiv.2209.08042

Rahul Chugh, Supartha Podder, Swagato Sanyal

{"title":"Decision Tree Complexity versus Block Sensitivity and Degree","authors":"Rahul Chugh, Supartha Podder, Swagato Sanyal","doi":"10.48550/arXiv.2209.08042","DOIUrl":"https://doi.org/10.48550/arXiv.2209.08042","url":null,"abstract":"Relations between the decision tree complexity and various other complexity measures of Boolean functions is a thriving topic of research in computational complexity. It is known that decision tree complexity is bounded above by the cube of block sensitivity, and the cube of polynomial degree. However, the widest separation between decision tree complexity and each of block sensitivity and degree that is witnessed by known Boolean functions is quadratic. In this work, we investigate the tightness of the existing cubic upper bounds. We improve the cubic upper bounds for many interesting classes of Boolean functions. We show that for graph properties and for functions with a constant number of alternations, both of the cubic upper bounds can be improved to quadratic. We define a class of Boolean functions, which we call the zebra functions, that comprises Boolean functions where each monotone path from 0^n to 1^n has an equal number of alternations. This class contains the symmetric and monotone functions as its subclasses. We show that for any zebra function, decision tree complexity is at most the square of block sensitivity, and certificate complexity is at most the square of degree. Finally, we show using a lifting theorem of communication complexity by G{\"{o}}{\"{o}}s, Pitassi and Watson that the task of proving an improved upper bound on the decision tree complexity for all functions is in a sense equivalent to the potentially easier task of proving a similar upper bound on communication complexity for each bi-partition of the input variables, for all functions. In particular, this implies that to bound the decision tree complexity it suffices to bound smaller measures like parity decision tree complexity, subcube decision tree complexity and decision tree rank, that are defined in terms of models that can be efficiently simulated by communication protocols.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72780024","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