Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > 2020:
All reports in year 2020:
TR20-022 | 19th February 2020
Klim Efremenko, Gillat Kol, Raghuvansh Saxena

#### Interactive Error Resilience Beyond $\frac{2}{7}$

Interactive error correcting codes can protect interactive communication protocols against a constant fraction of adversarial errors, while incurring only a constant multiplicative overhead in the total communication. What is the maximum fraction of errors that such codes can protect against?

For the non-adaptive channel, where the parties must agree ... more >>>

TR20-021 | 21st February 2020
Rahul Ilango, Bruno Loff, Igor Carboni Oliveira

#### NP-Hardness of Circuit Minimization for Multi-Output Functions

Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an ... more >>>

TR20-020 | 21st February 2020
Nikhil Mande, Justin Thaler, Shuchen Zhu

#### Improved Approximate Degree Bounds For $k$-distinctness

An open problem that is widely regarded as one of the most important in quantum query complexity is to resolve the quantum query complexity of the $k$-distinctness function on inputs of size $N$. While the case of $k=2$ (also called Element Distinctness) is well-understood, there is a polynomial gap between ... more >>>

TR20-019 | 19th February 2020
Siddharth Bhandari, Prahladh Harsha

#### A note on the explicit constructions of tree codes over polylogarithmic-sized alphabet

Recently, Cohen, Haeupler and Schulman gave an explicit construction of binary tree codes over polylogarithmic-sized output alphabet based on Pudl\'{a}k's construction of maximum-distance-separable (MDS) tree codes using totally-non-singular triangular matrices. In this short note, we give a unified and simpler presentation of Pudl\'{a}k and Cohen-Haeupler-Schulman's constructions.

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TR20-018 | 18th February 2020
Valentine Kabanets, Sajin Koroth, Zhenjian Lu, Dimitrios Myrisiotis, Igor Oliveira

#### Algorithms and Lower Bounds for de Morgan Formulas of Low-Communication Leaf Gates

The class $FORMULA[s] \circ \mathcal{G}$ consists of Boolean functions computable by size-$s$ de Morgan formulas whose leaves are any Boolean functions from a class $\mathcal{G}$. We give lower bounds and (SAT, Learning, and PRG) algorithms for $FORMULA[n^{1.99}]\circ \mathcal{G}$, for classes $\mathcal{G}$ of functions with low communication complexity. Let $R^{(k)}(\mathcal{G})$ be ... more >>>

TR20-017 | 18th February 2020
Alexander Kozachinskiy, Vladimir Podolskii

#### Multiparty Karchmer-Wigderson Games and Threshold Circuits

We suggest a generalization of Karchmer-Wigderson communication games to the multiparty setting. Our generalization turns out to be tightly connected to circuits consisting of threshold gates. This allows us to obtain new explicit constructions of such circuits for several functions. In particular, we provide an explicit (polynomial-time computable) log-depth monotone ... more >>>

TR20-016 | 17th February 2020
Kuan Cheng, William Hoza

#### Hitting Sets Give Two-Sided Derandomization of Small Space

A hitting set is a "one-sided" variant of a pseudorandom generator (PRG), naturally suited to derandomizing algorithms that have one-sided error. We study the problem of using a given hitting set to derandomize algorithms that have two-sided error, focusing on space-bounded algorithms. For our first result, we show that if ... more >>>

TR20-015 | 18th February 2020
Emanuele Viola

#### New lower bounds for probabilistic degree and AC0 with parity gates

We prove new lower bounds for computing some functions $f:\{0,1\}^{n}\to\{0,1\}$ in $E^{NP}$ by polynomials modulo $2$, constant-depth circuits with parity gates ($AC^{0}[\oplus]$), and related classes. Results include:

(1) $\Omega(n/\log^{2}n)$ lower bounds probabilistic degree. This is optimal up to a factor $O(\log^{2}n)$. The previous best lower bound was $\Omega(\sqrt{n})$ proved in ... more >>>

TR20-014 | 16th February 2020
Gil Cohen, Shahar Samocha

#### Palette-Alternating Tree Codes

A tree code is an edge-coloring of the complete infinite binary tree such that every two nodes of equal depth have a fraction--bounded away from $0$--of mismatched colors between the corresponding paths to their least common ancestor. Tree codes were introduced in a seminal work by Schulman (STOC 1993) and ... more >>>

TR20-013 | 17th February 2020
Noga Ron-Zewi, Mary Wootters, Gilles Z\'{e}mor

We give a linear-time erasure list-decoding algorithm for expander codes. More precisely, let $r > 0$ be any integer. Given an inner code $\cC_0$ of length $d$, and a $d$-regular bipartite expander graph $G$ with $n$ vertices on each side, we give an algorithm to list-decode the expander code $\cC ... more >>> TR20-012 | 14th February 2020 Dmitry Sokolov #### (Semi)Algebraic Proofs over$\{\pm 1\}$Variables One of the major open problems in proof complexity is to prove lower bounds on$AC_0[p]$-Frege proof systems. As a step toward this goal Impagliazzo, Mouli and Pitassi in a recent paper suggested to prove lower bounds on the size for Polynomial Calculus over the$\{\pm 1\}$basis. In this ... more >>> TR20-011 | 9th February 2020 Dominik Scheder, Navid Talebanfard #### Super Strong ETH is true for strong PPSZ We construct$k$-CNFs with$m$variables on which the strong version of PPSZ$k$-SAT algorithm, which uses bounded width resolution, has success probability at most$2^{-(1 - (1 + \epsilon)2/k)m}$for every$\epsilon > 0$. Previously such a bound was known only for the weak PPSZ algorithm which exhaustively searches ... more >>> TR20-010 | 12th February 2020 Lijie Chen, Hanlin Ren #### Strong Average-Case Circuit Lower Bounds from Non-trivial Derandomization We prove that for all constants a, NQP = NTIME[n^{polylog(n)}] cannot be (1/2 + 2^{-log^a n})-approximated by 2^{log^a n}-size ACC^0 of THR circuits (ACC^0 circuits with a bottom layer of THR gates). Previously, it was even open whether E^NP can be (1/2+1/sqrt{n})-approximated by AC^0[2] circuits. As a straightforward application, ... more >>> TR20-009 | 6th February 2020 Esty Kelman, Subhash Khot, Guy Kindler, Dor Minzer, Muli Safra #### Theorems of KKL, Friedgut, and Talagrand via Random Restrictions and Log-Sobolev Inequality We give alternate proofs for three related results in analysis of Boolean functions, namely the KKL Theorem, Friedgut’s Junta Theorem, and Talagrand’s strengthening of the KKL Theorem. We follow a new approach: looking at the first Fourier level of the function after a suitable random restriction and applying the Log-Sobolev ... more >>> TR20-008 | 26th January 2020 Benny Applebaum, Amos Beimel, Oded Nir, Naty Peter #### Better Secret-Sharing via Robust Conditional Disclosure of Secrets A secret-sharing scheme allows to distribute a secret$s$among$n$parties such that only some predefined authorized'' sets of parties can reconstruct the secret, and all other unauthorized'' sets learn nothing about$s$. The collection of authorized sets is called the access structure. For over 30 years, it was ... more >>> TR20-007 | 19th December 2019 Claude Crépeau, Arnaud Massenet, Louis Salvail, Lucas Stinchcombe, Nan Yang #### Practical Relativistic Zero-Knowledge for NP In this work we consider the following problem: in a Multi-Prover environment, how close can we get to prove the validity of an NP statement in Zero-Knowledge ? We exhibit a set of two novel Zero-Knowledge protocols for the 3-COLorability problem that use two (local) provers or three (entangled) provers ... more >>> TR20-006 | 22nd January 2020 Anup Rao, Amir Yehudayoff #### The Communication Complexity of the Exact Gap-Hamming Problem We prove a sharp lower bound on the distributional communication complexity of the exact gap-hamming problem. more >>> TR20-005 | 17th January 2020 Olaf Beyersdorff, Joshua Blinkhorn, Meena Mahajan #### Hardness Characterisations and Size-Width Lower Bounds for QBF Resolution We provide a tight characterisation of proof size in resolution for quantified Boolean formulas (QBF) by circuit complexity. Such a characterisation was previously obtained for a hierarchy of QBF Frege systems (Beyersdorff & Pich, LICS 2016), but leaving open the most important case of QBF resolution. Different from the Frege ... more >>> TR20-004 | 17th January 2020 Joshua Brakensiek, Venkatesan Guruswami, Marcin Wrochna, Stanislav Zivny #### The Power of the Combined Basic LP and Affine Relaxation for Promise CSPs In the field of constraint satisfaction problems (CSP), promise CSPs are an exciting new direction of study. In a promise CSP, each constraint comes in two forms: "strict" and "weak," and in the associated decision problem one must distinguish between being able to satisfy all the strict constraints versus not ... more >>> TR20-003 | 15th January 2020 Giuseppe Persiano, Kevin Yeo #### Tight Static Lower Bounds for Non-Adaptive Data Structures In this paper, we study the static cell probe complexity of non-adaptive data structures that maintain a subset of$n$points from a universe consisting of$m=n^{1+\Omega(1)}$points. A data structure is defined to be non-adaptive when the memory locations that are chosen to be accessed during a query depend ... more >>> TR20-002 | 6th January 2020 Sophie Laplante, Reza Naserasr, Anupa Sunny #### Sensitivity lower bounds from linear dependencies Recently, using spectral techniques, H. Huang proved that every subgraph of the hypercube of dimension n induced on more than half the vertices has maximum degree at least the square root of n. Combined with some earlier work, this completed a proof of the sensitivity conjecture. In this work we ... more >>> TR20-001 | 31st December 2019 Or Meir, Jakob Nordström, Robert Robere, Susanna de Rezende #### Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling Revisions: 2 We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatz refutations of pebbling formulas, showing that a graph$G$can be reversibly pebbled in time$t$and space$s$if and only if there is a Nullstellensatz refutation of the pebbling formula over$G$in size$t+1\$ ... more >>>

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