19 research outputs found

    Random local algorithms

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    Consider the problem when we want to construct some structure on a bounded degree graph, e.g. an almost maximum matching, and we want to decide about each edge depending only on its constant radius neighbourhood. We show that the information about the local statistics of the graph does not help here. Namely, if there exists a random local algorithm which can use any local statistics about the graph, and produces an almost optimal structure, then the same can be achieved by a random local algorithm using no statistics.Comment: 9 page

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    Testing bounded arboricity

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    In this paper we consider the problem of testing whether a graph has bounded arboricity. The family of graphs with bounded arboricity includes, among others, bounded-degree graphs, all minor-closed graph classes (e.g. planar graphs, graphs with bounded treewidth) and randomly generated preferential attachment graphs. Graphs with bounded arboricity have been studied extensively in the past, in particular since for many problems they allow for much more efficient algorithms and/or better approximation ratios. We present a tolerant tester in the sparse-graphs model. The sparse-graphs model allows access to degree queries and neighbor queries, and the distance is defined with respect to the actual number of edges. More specifically, our algorithm distinguishes between graphs that are ϵ\epsilon-close to having arboricity α\alpha and graphs that cϵc \cdot \epsilon-far from having arboricity 3α3\alpha, where cc is an absolute small constant. The query complexity and running time of the algorithm are O~(nmlog(1/ϵ)ϵ+nαm(1ϵ)O(log(1/ϵ)))\tilde{O}\left(\frac{n}{\sqrt{m}}\cdot \frac{\log(1/\epsilon)}{\epsilon} + \frac{n\cdot \alpha}{m} \cdot \left(\frac{1}{\epsilon}\right)^{O(\log(1/\epsilon))}\right) where nn denotes the number of vertices and mm denotes the number of edges. In terms of the dependence on nn and mm this bound is optimal up to poly-logarithmic factors since Ω(n/m)\Omega(n/\sqrt{m}) queries are necessary (and α=O(m))\alpha = O(\sqrt{m})). We leave it as an open question whether the dependence on 1/ϵ1/\epsilon can be improved from quasi-polynomial to polynomial. Our techniques include an efficient local simulation for approximating the outcome of a global (almost) forest-decomposition algorithm as well as a tailored procedure of edge sampling

    Property Testing via Set-Theoretic Operations

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    Given two testable properties P1\mathcal{P}_{1} and P2\mathcal{P}_{2}, under what conditions are the union, intersection or set-difference of these two properties also testable? We initiate a systematic study of these basic set-theoretic operations in the context of property testing. As an application, we give a conceptually different proof that linearity is testable, albeit with much worse query complexity. Furthermore, for the problem of testing disjunction of linear functions, which was previously known to be one-sided testable with a super-polynomial query complexity, we give an improved analysis and show it has query complexity O(1/\eps^2), where \eps is the distance parameter.Comment: Appears in ICS 201

    Testing probability distributions underlying aggregated data

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    In this paper, we analyze and study a hybrid model for testing and learning probability distributions. Here, in addition to samples, the testing algorithm is provided with one of two different types of oracles to the unknown distribution DD over [n][n]. More precisely, we define both the dual and cumulative dual access models, in which the algorithm AA can both sample from DD and respectively, for any i[n]i\in[n], - query the probability mass D(i)D(i) (query access); or - get the total mass of {1,,i}\{1,\dots,i\}, i.e. j=1iD(j)\sum_{j=1}^i D(j) (cumulative access) These two models, by generalizing the previously studied sampling and query oracle models, allow us to bypass the strong lower bounds established for a number of problems in these settings, while capturing several interesting aspects of these problems -- and providing new insight on the limitations of the models. Finally, we show that while the testing algorithms can be in most cases strictly more efficient, some tasks remain hard even with this additional power

    Lower Bounds on Query Complexity for Testing Bounded-Degree CSPs

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    In this paper, we consider lower bounds on the query complexity for testing CSPs in the bounded-degree model. First, for any ``symmetric'' predicate P:0,1k0,1P:{0,1}^{k} \to {0,1} except \equ where k3k\geq 3, we show that every (randomized) algorithm that distinguishes satisfiable instances of CSP(P) from instances (P1(0)/2kϵ)(|P^{-1}(0)|/2^k-\epsilon)-far from satisfiability requires Ω(n1/2+δ)\Omega(n^{1/2+\delta}) queries where nn is the number of variables and δ>0\delta>0 is a constant that depends on PP and ϵ\epsilon. This breaks a natural lower bound Ω(n1/2)\Omega(n^{1/2}), which is obtained by the birthday paradox. We also show that every one-sided error tester requires Ω(n)\Omega(n) queries for such PP. These results are hereditary in the sense that the same results hold for any predicate QQ such that P1(1)Q1(1)P^{-1}(1) \subseteq Q^{-1}(1). For EQU, we give a one-sided error tester whose query complexity is O~(n1/2)\tilde{O}(n^{1/2}). Also, for 2-XOR (or, equivalently E2LIN2), we show an Ω(n1/2+δ)\Omega(n^{1/2+\delta}) lower bound for distinguishing instances between ϵ\epsilon-close to and (1/2ϵ)(1/2-\epsilon)-far from satisfiability. Next, for the general k-CSP over the binary domain, we show that every algorithm that distinguishes satisfiable instances from instances (12k/2kϵ)(1-2k/2^k-\epsilon)-far from satisfiability requires Ω(n)\Omega(n) queries. The matching NP-hardness is not known, even assuming the Unique Games Conjecture or the dd-to-11 Conjecture. As a corollary, for Maximum Independent Set on graphs with nn vertices and a degree bound dd, we show that every approximation algorithm within a factor d/\poly\log d and an additive error of ϵn\epsilon n requires Ω(n)\Omega(n) queries. Previously, only super-constant lower bounds were known

    Correlation Testing for Affine Invariant Properties on Fpn\mathbb{F}_p^n in the High Error Regime

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    Recently there has been much interest in Gowers uniformity norms from the perspective of theoretical computer science. This is mainly due to the fact that these norms provide a method for testing whether the maximum correlation of a function f:FpnFpf:\mathbb{F}_p^n \rightarrow \mathbb{F}_p with polynomials of degree at most dpd \le p is non-negligible, while making only a constant number of queries to the function. This is an instance of {\em correlation testing}. In this framework, a fixed test is applied to a function, and the acceptance probability of the test is dependent on the correlation of the function from the property. This is an analog of {\em proximity oblivious testing}, a notion coined by Goldreich and Ron, in the high error regime. In this work, we study general properties which are affine invariant and which are correlation testable using a constant number of queries. We show that any such property (as long as the field size is not too small) can in fact be tested by Gowers uniformity tests, and hence having correlation with the property is equivalent to having correlation with degree dd polynomials for some fixed dd. We stress that our result holds also for non-linear properties which are affine invariant. This completely classifies affine invariant properties which are correlation testable. The proof is based on higher-order Fourier analysis. Another ingredient is a nontrivial extension of a graph theoretical theorem of Erd\"os, Lov\'asz and Spencer to the context of additive number theory.Comment: 43 pages. A preliminary version of this work appeared in STOC' 201
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