Colloquium on Computational Complexity

Understanding the relationship between the worst-case and average-case complexities of $\mathrm{NP}$ and of other subclasses of $\mathrm{PH}$ is a long-standing problem in complexity theory. Over the last few years, much progress has been achieved in this front through the investigation of […]

We establish an $\epsilon$-sensitive hierarchy separation for monotone arithmetic computations. The notion of $\epsilon$-sensitive monotone lower bounds was recently introduced by Hrubes [Computational Complexity'20]. We show the following: (1) There exists a monotone polynomial over $n$ […]

In a recent result of Bhargava, Saraf and Volkovich [FOCS’18; JACM’20], the first sparsity bound for constant individual degree polynomials was shown. In particular, it was shown that any factor of a polynomial with at most $s$ terms and individual degree bounded by $d$ can itself have […]

In a breakthrough result, Ta-Shma described an explicit construction of an almost optimal binary code (STOC 2017). Ta-Shma's code has distance $\frac{1-\varepsilon}{2}$ and rate $\Omega\bigl(\varepsilon^{2+o(1)}\bigr)$ and thus it almost achieves the Gilbert-Varshamov bound, except for the $o(1)$ […]

We analyze the sketching approximability of constraint satisfaction problems on Boolean domains, where the constraints are balanced linear threshold functions applied to literals. In particular, we explore the approximability of monarchy-like functions where the value of the function is determined […]

Hrube\v{s} and Wigderson (2015) initiated the complexity-theoretic study of noncommutative formulas with inverse gates. They introduced the Rational Identity Testing (RIT) problem which is to decide whether a noncommutative rational formula computes zero in the free skew field. In the white-box […]

A Boolean maximum constraint satisfaction problem, Max-CSP\((f)\), is specified by a predicate \(f:\{-1,1\}^k\to\{0,1\}\). An \(n\)-variable instance of Max-CSP\((f)\) consists of a list of constraints, each of which applies \(f\) to \(k\) distinct literals drawn from the \(n\) variables. For […]

In this survey we describe progress over the last decade or so in understanding the complexity of solving constraint satisfaction problems (CSPs) approximately in the streaming and sketching models of computation. After surveying some of the results we give some sketches of the proofs and in […]

In this paper, we prove strengthened lower bounds for constant-depth set-multilinear formulas. More precisely, we show that over any field, there is an explicit polynomial $f$ in VNP defined over $n^2$ variables, and of degree $n$, such that any product-depth $\Delta$ set-multilinear formula […]

Multivariate multipoint evaluation is the problem of evaluating a multivariate polynomial, given as a coefficient vector, simultaneously at multiple evaluation points. In this work, we show that there exists a deterministic algorithm for multivariate multipoint evaluation over any finite field […]