Quantified Derandomization of Linear Threshold Circuits

One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for $TC^0$, the class of constant-depth, polynomial-size circuits with majority gates. Relying on the results of Williams (2013), an appealing approach to prove such lower bounds is to construct a non-trivial derandomization algorithm for $TC^0$. In this work we take a first step towards the latter goal, by proving the first positive results regarding the derandomization of $TC^0$ circuits of depth $d>2$. Our first main result is a quantified derandomization algorithm for $TC^0$ circuits with a super-linear number of wires. Specifically, we construct an algorithm that gets as input a $TC^0$ circuit $C$ over $n$ input bits with depth $d$ and $n^{1+\exp(-d)}$ wires, runs in almost-polynomial-time, and distinguishes between the case that $C$ rejects at most $2^{n^{1-1/5d}}$ inputs and the case that $C$ accepts at most $2^{n^{1-1/5d}}$ inputs. In fact, our algorithm works even when the circuit $C$ is a linear threshold circuit, rather than just a $TC^0$ circuit (i.e., $C$ is a circuit with linear threshold gates, which are stronger than majority gates). Our second main result is that even a modest improvement of our quantified derandomization algorithm would yield a non-trivial algorithm for standard derandomization of all of $TC^0$, and would consequently imply that $NEXP\not\subseteq TC^0$. Specifically, if there exists a quantified derandomization algorithm that gets as input a $TC^0$ circuit with depth $d$ and $n^{1+O(1/d)}$ wires (rather than $n^{1+\exp(-d)}$ wires), runs in time at most $2^{n^{\exp(-d)}}$, and distinguishes between the case that $C$ rejects at most $2^{n^{1-1/5d}}$ inputs and the case that $C$ accepts at most $2^{n^{1-1/5d}}$ inputs, then there exists an algorithm with running time $2^{n^{1-\Omega(1)}}$ for standard derandomization of $TC^0$.Comment: Changes in this revision: An additional result (a PRG for quantified derandomization of depth-2 LTF circuits); rewrite of some of the exposition; minor correction

Derandomization in Game-theoretic Probability

We give a general method for constructing a deterministic strategy of Reality from a randomized strategy in game-theoretic probability. The construction can be seen as derandomization in game-theoretic probability.Comment: 19 page

Improved Approximation Algorithms for the Min-Max Selecting Items Problem

We give a simple deterministic $O(\log K / \log\log K)$ approximation algorithm for the Min-Max Selecting Items problem, where $K$ is the number of scenarios. While our main goal is simplicity, this result also improves over the previous best approximation ratio of $O(\log K)$ due to Kasperski, Kurpisz, and Zieli\'nski (Information Processing Letters (2013)). Despite using the method of pessimistic estimators, the algorithm has a polynomial runtime also in the RAM model of computation. We also show that the LP formulation for this problem by Kasperski and Zieli\'nski (Annals of Operations Research (2009)), which is the basis for the previous work and ours, has an integrality gap of at least $\Omega(\log K / \log\log K)$

Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas

We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: an $\varepsilon$-PRG for the class of size-$M$ depth-$d$ $\mathsf{AC}^0$ circuits with seed length $\log(M)^{d+O(1)}\cdot \log(1/\varepsilon)$, and an $\varepsilon$-PRG for the class of $S$-sparse $\mathbb{F}_2$ polynomials with seed length $2^{O(\sqrt{\log S})}\cdot \log(1/\varepsilon)$. These results bring the state of the art for unconditional derandomization of these classes into sharp alignment with the state of the art for computational hardness for all parameter settings: improving on the seed lengths of either PRG would require breakthrough progress on longstanding and notorious circuit lower bounds. The key enabling ingredient in our approach is a new \emph{pseudorandom multi-switching lemma}. We derandomize recently-developed \emph{multi}-switching lemmas, which are powerful generalizations of H{\aa}stad's switching lemma that deal with \emph{families} of depth-two circuits. Our pseudorandom multi-switching lemma---a randomness-efficient algorithm for sampling restrictions that simultaneously simplify all circuits in a family---achieves the parameters obtained by the (full randomness) multi-switching lemmas of Impagliazzo, Matthews, and Paturi [IMP12] and H{\aa}stad [H{\aa}s14]. This optimality of our derandomization translates into the optimality (given current circuit lower bounds) of our PRGs for $\mathsf{AC}^0$ and sparse $\mathbb{F}_2$ polynomials

Distributed local approximation algorithms for maximum matching in graphs and hypergraphs

We describe approximation algorithms in Linial's classic LOCAL model of distributed computing to find maximum-weight matchings in a hypergraph of rank $r$. Our main result is a deterministic algorithm to generate a matching which is an $O(r)$-approximation to the maximum weight matching, running in $\tilde O(r \log \Delta + \log^2 \Delta + \log^* n)$ rounds. (Here, the $\tilde O()$ notations hides $\text{polyloglog } \Delta$ and $\text{polylog } r$ factors). This is based on a number of new derandomization techniques extending methods of Ghaffari, Harris & Kuhn (2017). As a main application, we obtain nearly-optimal algorithms for the long-studied problem of maximum-weight graph matching. Specifically, we get a $(1+\epsilon)$ approximation algorithm using $\tilde O(\log \Delta / \epsilon^3 + \text{polylog}(1/\epsilon, \log \log n))$ randomized time and $\tilde O(\log^2 \Delta / \epsilon^4 + \log^*n / \epsilon)$ deterministic time. The second application is a faster algorithm for hypergraph maximal matching, a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of local graph algorithms. This gives an algorithm for $(2 \Delta - 1)$-edge-list coloring in $\tilde O(\log^2 \Delta \log n)$ rounds deterministically or $\tilde O( (\log \log n)^3 )$ rounds randomly. Another consequence (with additional optimizations) is an algorithm which generates an edge-orientation with out-degree at most $\lceil (1+\epsilon) \lambda \rceil$ for a graph of arboricity $\lambda$; for fixed $\epsilon$ this runs in $\tilde O(\log^6 n)$ rounds deterministically or $\tilde O(\log^3 n )$ rounds randomly