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- research-articleJune 2024
Influences in Mixing Measures
STOC 2024: Proceedings of the 56th Annual ACM Symposium on Theory of ComputingJune 2024, Pages 527–536https://doi.org/10.1145/3618260.3649731The theory of influences in product measures has profound applications in theoretical computer science, combinatorics, and discrete probability. This deep theory is intimately connected to functional inequalities and to the Fourier analysis of discrete ...
- research-articleJune 2024
Characterizing Direct Product Testing via Coboundary Expansion
STOC 2024: Proceedings of the 56th Annual ACM Symposium on Theory of ComputingJune 2024, Pages 1978–1989https://doi.org/10.1145/3618260.3649714A d-dimensional simplicial complex X is said to support a direct product tester if any locally consistent function defined on its k-faces (where k≪ d) necessarily come from a function over its vertices. More precisely, a direct product tester has a ...
- research-articleJune 2024
Product Mixing in Compact Lie Groups
STOC 2024: Proceedings of the 56th Annual ACM Symposium on Theory of ComputingJune 2024, Pages 1415–1422https://doi.org/10.1145/3618260.3649626If G is a group, we say a subset S of G is product-free if the equation xy=z has no solutions with x,y,z ∈ S.In 1985, Babai and Sós [] asked, for a finite group G, how large a subset S⊆ G can be if it is product-free. The main tool (hitherto) for ...
- research-articleJune 2024
On Approximability of Satisfiable k-CSPs: IV
STOC 2024: Proceedings of the 56th Annual ACM Symposium on Theory of ComputingJune 2024, Pages 1423–1434https://doi.org/10.1145/3618260.3649610We prove a stability result for general 3-wise correlations over distributions satisfying mild connectivity properties. More concretely, we show that if Σ,Γ and Φ are alphabets of constant size, and µ is a distribution over Σ×Γ×Φ satisfying: (1) the ...
- research-articleJune 2024
Near Optimal Alphabet-Soundness Tradeoff PCPs
STOC 2024: Proceedings of the 56th Annual ACM Symposium on Theory of ComputingJune 2024, Pages 15–23https://doi.org/10.1145/3618260.3649606We show that for all є>0, for sufficiently large prime power q, for all δ>0, it is NP-hard to distinguish whether a 2-Prover-1-Round projection game with alphabet size q has value at least 1-δ, or value at most 1/q^(1-є). This establishes a nearly ...
- research-articleJune 2023
On Approximability of Satisfiable k-CSPs: III
STOC 2023: Proceedings of the 55th Annual ACM Symposium on Theory of ComputingJune 2023, Pages 643–655https://doi.org/10.1145/3564246.3585121In this paper we study functions on the Boolean hypercube that have the property that after applying certain random restrictions, the restricted function is correlated to a linear function with non-negligible probability. If the given function is ...
- research-articleJune 2023
On Approximability of Satisfiable k-CSPs: II
STOC 2023: Proceedings of the 55th Annual ACM Symposium on Theory of ComputingJune 2023, Pages 632–642https://doi.org/10.1145/3564246.3585120Let Σ be an alphabet and µ be a distribution on Σk for some k ≥ 2. Let α > 0 be the minimum probability of a tuple in the support of µ (denoted supp(µ)). Here, the support of µ is the set of all tuples in Σk that have a positive probability mass under ...
- bookDecember 2022
On Monotonicity Testing and the 2-to-2 Games Conjecture
This book discusses two questions in Complexity Theory: the Monotonicity Testing problem and the 2-to-2 Games Conjecture.
Monotonicity testing is a problem from the field of property testing, first considered by Goldreich et al. in 2000. The input of the ...
- research-articleJune 2022
On approximability of satisfiable k-CSPs: I
STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of ComputingJune 2022, Pages 976–988https://doi.org/10.1145/3519935.3520028We consider the P-CSP problem for 3-ary predicates P on satisfiable instances. We show that under certain conditions on P and a (1,s) integrality gap instance of the P-CSP problem, it can be translated into a dictatorship vs. quasirandomness test with ...
- research-articleJune 2022
Approximate polymorphisms
STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of ComputingJune 2022, Pages 195–202https://doi.org/10.1145/3519935.3519966For a function g∶{0,1}m→{0,1}, a function f∶ {0,1}n→{0,1} is called a g-polymorphism if their actions commute: f(g(row1(Z)),…,g(rown(Z))) = g(f(col1(Z)),…,f(colm(Z))) for all Z∈{0,1}n× m. The function f is called an approximate g-polymorphism if this ...
- research-articleSeptember 2021
Optimal tiling of the euclidean space using permutation-symmetric bodies
CCC '21: Proceedings of the 36th Computational Complexity ConferenceJuly 2021, Article No.: lipics, Page 1https://doi.org/10.4230/LIPIcs.CCC.2021.5What is the least surface area of a permutation-symmetric body B whose Zn translations tile Rn? Since any such body must have volume 1, the isoperimetric inequality implies that its surface area must be at least [EQUATION]. Remarkably, Kindler et al. ...
- research-articleJune 2021
New separations results for external information
STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of ComputingJune 2021, Pages 248–258https://doi.org/10.1145/3406325.3451044We obtain new separation results for the two-party external information complexity of Boolean functions. The external information complexity of a function f(x,y) is the minimum amount of information a two-party protocol computing f must reveal to an ...
- research-articleJune 2020
AND testing and robust judgement aggregation
STOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of ComputingJune 2020, Pages 222–233https://doi.org/10.1145/3357713.3384254A function f∶{0,1} n → {0,1} is called an approximate AND-homomorphism if choosing x,y∈n uniformly at random, we have that f(x∧ y) = f(x)∧ f(y) with probability at least 1−ε, where x∧ y = (x 1∧ y 1,…,x n ∧ y n ). We prove that if f∶ {0,1} n → {0,1} ...
- research-articleJuly 2019
Can Distributed Uniformity Testing Be Local?
PODC '19: Proceedings of the 2019 ACM Symposium on Principles of Distributed ComputingJuly 2019, Pages 228–237https://doi.org/10.1145/3293611.3331613In the distributed uniformity testing problem, k servers draw samples from some unknown distribution, and the goal is to determine whether the unknown distribution is uniform or whether it is ε-far from uniform, where ε is a proximity parameter. Each ...
- research-articleJune 2018
On non-optimally expanding sets in Grassmann graphs
STOC 2018: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of ComputingJune 2018, Pages 940–951https://doi.org/10.1145/3188745.3188806We study the structure of non-expanding sets in the Grassmann graph. We put forth a hypothesis stating that every small set whose expansion is smaller than 1−δ must be correlated with one of a specified list of sets which are isomorphic to smaller ...
- research-articleJune 2018
Towards a proof of the 2-to-1 games conjecture?
STOC 2018: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of ComputingJune 2018, Pages 376–389https://doi.org/10.1145/3188745.3188804We present a polynomial time reduction from gap-3LIN to label cover with 2-to-1 constraints. In the “yes” case the fraction of satisfied constraints is at least 1 −ε, and in the “no” case we show that this fraction is at most ε, assuming a certain (new) ...
- research-articleJanuary 2018
On Monotonicity Testing and Boolean Isoperimetric-type Theorems
SIAM Journal on Computing (SICOMP), Volume 47, Issue 62018, Pages 2238–2276https://doi.org/10.1137/16M1065872We show a directed and robust analogue of a boolean isoperimetric-type theorem of Talagrand [Geom. Funct. Anal., 3 (1993), pp. 295--314]. As an application, we give a monotonicity testing algorithm that makes $\tilde{O}(\sqrt{n}/\varepsilon^2)$ nonadaptive ...
- research-articleJune 2017
On independent sets, 2-to-2 games, and Grassmann graphs
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of ComputingJune 2017, Pages 576–589https://doi.org/10.1145/3055399.3055432We present a candidate reduction from the 3-Lin problem to the 2-to-2 Games problem and present a combinatorial hypothesis about Grassmann graphs which, if correct, is sufficient to show the soundness of the reduction in a certain non-standard sense. A ...
- ArticleOctober 2015
On Monotonicity Testing and Boolean Isoperimetric Type Theorems
FOCS '15: Proceedings of the 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS)October 2015, Pages 52–58https://doi.org/10.1109/FOCS.2015.13We show a directed and robust analogue of a boolean isoperimetric type theorem of Talagrand. As an application, we give a monotonicity testing algorithm that makes, up to polylog factors, (square root of n)/ε2 non-adaptive queries to a ...