Siu On Chan

Approximation resistance from pairwise independent subgroups

We show optimal (up to constant factor) NP-hardness for Max-k-CSP over any domain, whenever k is larger than the domain size. This follows from our main result concerning predicates over abelian groups. We show that a predicate is approximation resistant if it contains a subgroup that is balanced pairwise independent. This gives an unconditional analogue of Austrin--Mossel hardness result, bypassing the Unique-Games Conjecture for predicates with an abelian subgroup structure. Our main ingredient is a new gap-amplification technique inspired by XOR-lemmas. Using this technique, we also improve the NP-hardness of approximating Independent-Set on bounded-degree graphs, Almost-Coloring, Two-Prover-One-Round-Game, and various other problems.

Approximate Constraint Satisfaction Requires Large LP Relaxations

We prove super-polynomial lower bounds on the size of linear programming relaxations for approximation versions of constraint satisfaction problems. We show that for these problems, polynomial-sized linear programs are exactly as powerful as programs arising from a constant number of rounds of the Sherali-Adams hierarchy. In particular, any polynomial-sized linear program for MAX CUT has an integrality gap of 1/2 and any such linear program for MAX 3-SAT has an integrality gap of 7/8.

Optimal algorithms for testing closeness of discrete distributions

We study the question of closeness testing for two discrete distributions. More precisely, given samples from two distributions p and q over an n-element set, we wish to distinguish whether p = q versus p is at least e-far from q, in either e1 or e2 distance. Batu et al [BFR+00, BFR+13] gave the first sub-linear time algorithms for these problems, which matched the lower bounds of [Val11] up to a logarithmic factor in n, and a polynomial factor of e. In this work, we present simple testers for both the e1 and e2 settings, with sample complexity that is information-theoretically optimal, to constant factors, both in the dependence on n, and the dependence on e for the e1 testing problem we establish that the sample complexity is Θ(max{n2/3/e4/3, n1/2/&epsilon2}).

Learning mixtures of structured distributions over discrete domains

Let

Efficient density estimation via piecewise polynomial approximation

\mathfrak{C}

Sum of Squares Lower Bounds from Pairwise Independence

be a class of probability distributions over the discrete domain

Quantifying unobserved protein-coding variants in human populations provides a roadmap for large-scale sequencing projects

[n] = \{1,...,n\}.

Approximation Resistance from Pairwise-Independent Subgroups

We show that if

(k+1)-cores have k-factors

\mathfrak{C}

On Extracting Common Random Bits From Correlated Sources on Large Alphabets

satisfies a rather general condition -- essentially, that each distribution in