Almost Optimal Distribution-Free Junta Testing

We consider the problem of testing whether an unknown n-variable Boolean function is a k-junta in the distribution-free property testing model, where the distance between functions is measured with respect to an arbitrary and unknown probability distribution over {0,1}^n. Chen, Liu, Servedio, Sheng and Xie [Zhengyang Liu et al., 2018] showed that the distribution-free k-junta testing can be performed, with one-sided error, by an adaptive algorithm that makes O~(k^2)/epsilon queries. In this paper, we give a simple two-sided error adaptive algorithm that makes O~(k/epsilon) queries

Lifting with Inner Functions of Polynomial Discrepancy

Lifting theorems are theorems that bound the communication complexity of a composed function f?g? in terms of the query complexity of f and the communication complexity of g. Such theorems constitute a powerful generalization of direct-sum theorems for g, and have seen numerous applications in recent years. We prove a new lifting theorem that works for every two functions f,g such that the discrepancy of g is at most inverse polynomial in the input length of f. Our result is a significant generalization of the known direct-sum theorem for discrepancy, and extends the range of inner functions g for which lifting theorems hold

Average-Case Hardness of NP and PH from Worst-Case Fine-Grained Assumptions

What is a minimal worst-case complexity assumption that implies non-trivial average-case hardness of NP or PH? This question is well motivated by the theory of fine-grained average-case complexity and fine-grained cryptography. In this paper, we show that several standard worst-case complexity assumptions are sufficient to imply non-trivial average-case hardness of NP or PH: - NTIME[n] cannot be solved in quasi-linear time on average if UP ? ? DTIME[2^{O?(?n)}]. - ??TIME[n] cannot be solved in quasi-linear time on average if ?_kSAT cannot be solved in time 2^{O?(?n)} for some constant k. Previously, it was not known if even average-case hardness of ??SAT implies the average-case hardness of ??TIME[n]. - Under the Exponential-Time Hypothesis (ETH), there is no average-case n^{1+?}-time algorithm for NTIME[n] whose running time can be estimated in time n^{1+?} for some constant ? > 0. Our results are given by generalizing the non-black-box worst-case-to-average-case connections presented by Hirahara (STOC 2021) to the settings of fine-grained complexity. To do so, we construct quite efficient complexity-theoretic pseudorandom generators under the assumption that the nondeterministic linear time is easy on average, which may be of independent interest

Almost Optimal Testers for Concise Representations

We give improved and almost optimal testers for several classes of Boolean functions on n variables that have concise representation in the uniform and distribution-free model. Classes, such as k-Junta, k-Linear, s-Term DNF, s-Term Monotone DNF, r-DNF, Decision List, r-Decision List, size-s Decision Tree, size-s Boolean Formula, size-s Branching Program, s-Sparse Polynomial over the binary field and functions with Fourier Degree at most d. The approach is new and combines ideas from Diakonikolas et al. [Ilias Diakonikolas et al., 2007], Bshouty [Nader H. Bshouty, 2018], Goldreich et al. [Oded Goldreich et al., 1998], and learning theory. The method can be extended to several other classes of functions over any domain that can be approximated by functions with a small number of relevant variables

Efficient Black-Box Identity Testing for Free Group Algebras

Hrubes and Wigderson [Pavel Hrubes and Avi Wigderson, 2014] initiated the study of noncommutative arithmetic circuits with division computing a noncommutative rational function in the free skew field, and raised the question of rational identity testing. For noncommutative formulas with inverses the problem can be solved in deterministic polynomial time in the white-box model [Ankit Garg et al., 2016; Ivanyos et al., 2018]. It can be solved in randomized polynomial time in the black-box model [Harm Derksen and Visu Makam, 2017], where the running time is polynomial in the size of the formula. The complexity of identity testing of noncommutative rational functions, in general, remains open for noncommutative circuits with inverses. We solve the problem for a natural special case. We consider expressions in the free group algebra F(X,X^{-1}) where X={x_1, x_2, ..., x_n}. Our main results are the following. 1) Given a degree d expression f in F(X,X^{-1}) as a black-box, we obtain a randomized poly(n,d) algorithm to check whether f is an identically zero expression or not. The technical contribution is an Amitsur-Levitzki type theorem [A. S. Amitsur and J. Levitzki, 1950] for F(X, X^{-1}). This also yields a deterministic identity testing algorithm (and even an expression reconstruction algorithm) that is polynomial time in the sparsity of the input expression. 2) Given an expression f in F(X,X^{-1}) of degree D and sparsity s, as black-box, we can check whether f is identically zero or not in randomized poly(n,log s, log D) time. This yields a randomized polynomial-time algorithm when D and s are exponential in n

Coresets Meet EDCS: Algorithms for Matching and Vertex Cover on Massive Graphs

As massive graphs become more prevalent, there is a rapidly growing need for scalable algorithms that solve classical graph problems, such as maximum matching and minimum vertex cover, on large datasets. For massive inputs, several different computational models have been introduced, including the streaming model, the distributed communication model, and the massively parallel computation (MPC) model that is a common abstraction of MapReduce-style computation. In each model, algorithms are analyzed in terms of resources such as space used or rounds of communication needed, in addition to the more traditional approximation ratio. In this paper, we give a single unified approach that yields better approximation algorithms for matching and vertex cover in all these models. The highlights include: * The first one pass, significantly-better-than-2-approximation for matching in random arrival streams that uses subquadratic space, namely a $(1.5+\epsilon)$-approximation streaming algorithm that uses $O(n^{1.5})$ space for constant $\epsilon > 0$. * The first 2-round, better-than-2-approximation for matching in the MPC model that uses subquadratic space per machine, namely a $(1.5+\epsilon)$-approximation algorithm with $O(\sqrt{mn} + n)$ memory per machine for constant $\epsilon > 0$. By building on our unified approach, we further develop parallel algorithms in the MPC model that give a $(1 + \epsilon)$-approximation to matching and an $O(1)$-approximation to vertex cover in only $O(\log\log{n})$ MPC rounds and $O(n/poly\log{(n)})$ memory per machine. These results settle multiple open questions posed in the recent paper of Czumaj~et.al. [STOC 2018]

Exponential Time Paradigms Through the Polynomial Time Lens

We propose a general approach to modelling algorithmic paradigms for the exact solution of NP-hard problems. Our approach is based on polynomial time reductions to succinct versions of problems solvable in polynomial time. We use this viewpoint to explore and compare the power of paradigms such as branching and dynamic programming, and to shed light on the true complexity of various problems. As one instantiation, we model branching using the notion of witness compression, i.e., reducibility to the circuit satisfiability problem parameterized by the number of variables of the circuit. We show this is equivalent to the previously studied notion of `OPP-algorithms\u27, and provide a technique for proving conditional lower bounds for witness compressions via a constructive variant of AND-composition, which is a notion previously studied in theory of preprocessing. In the context of parameterized complexity we use this to show that problems such as Pathwidth and Treewidth and Independent Set parameterized by pathwidth do not have witness compression, assuming NP subseteq coNP/poly. Since these problems admit fast fixed parameter tractable algorithms via dynamic programming, this shows that dynamic programming can be stronger than branching, under a standard complexity hypothesis. Our approach has applications outside parameterized complexity as well: for example, we show if a polynomial time algorithm outputs a maximum independent set of a given planar graph on n vertices with probability exp(-n^{1-epsilon}) for some epsilon>0, then NP subseteq coNP/poly. This negative result dims the prospects for one very natural approach to sub-exponential time algorithms for problems on planar graphs. As two other illustrations (more exploratory) of our approach, we model algorithms based on inclusion-exclusion or group algebras via the notion of "parity compression", and we model a subclass of dynamic programming algorithms with the notion of "disjunctive dynamic programming". These models give us a way to naturally classify various parameterized problems with FPT algorithms. In the case of the dynamic programming model, we show that Independent Set parameterized by pathwidth is complete for this model