We prove that any subset $A \subseteq [3]^n$ with $3^{-n}|A| \ge (\log\log\log\log n)^{-c}$ contains a combinatorial line of length $3$, i.e., $x, y, z \in A$, not all equal, with $x_i=y_i=z_i$ or $(x_i,y_i,z_i)=(0,1,2)$ for all $i = 1, 2, \dots, n$. This improves on the previous best bound of $3^{-n}|A| ... more >>>

Let $\Sigma_1,\ldots,\Sigma_k$ be finite alphabets, and let $\mu$ be a distribution over $\Sigma_1 \times \dots \times \Sigma_k$ in which the probability of each atom is at least $\alpha$. We prove that if $\mu$ does not admit Abelian embeddings, and $f_i: \Sigma_i \to \mathbb{C}$ are $1$-bounded functions (for $i=1,\ldots,k$) such that ... more >>>

We prove local and global inverse theorems for general $3$-wise correlations over pairwise-connected distributions. Let $\mu$ be a distribution over $\Sigma \times \Gamma \times \Phi$ such that the supports of $\mu_{xy}$, $\mu_{xz}$, and $\mu_{yz}$ are all connected, and let $f: \Sigma^n \to \mathbb{C}$, $g: \Gamma^n \to \mathbb{C}$, $h: \Phi^n \to ... more >>>

This paper studies the \emph{refuter} problems, a family of decision-tree $\mathrm{TFNP}$ problems capturing the metamathematical difficulty of proving proof complexity lower bounds. Suppose $\varphi$ is a hard tautology that does not admit any length-$s$ proof in some proof system $P$. In the corresponding refuter problem, we are given (query ... more >>>

We prove that for every odd $q\geq 3$, any $q$-query binary, possibly non-linear locally decodable code ($q$-LDC) $E:\{\pm 1\}^k \rightarrow \{\pm 1\}^n$ must satisfy $k \leq \tilde{O}(n^{1-2/q})$. For even $q$, this bound was established in a sequence of works (Katz and Trevisan (2000), Goldreich, Karloff, Schulman, and Trevisan (2002), and ... more >>>

The study of space-bounded computation has drawn extensively from ideas and results in the field of communication complexity. Catalytic Computation (Buhrman, Cleve, Koucký, Loff and Speelman, STOC 2013) studies the power of bounded space augmented with a pre-filled hard drive that can be used non-destructively during the computation. Presently, many ... more >>>

A code $C \colon \{0,1\}^k \to \{0,1\}^n$ is a $q$-query locally decodable code ($q$-LDC) if one can recover any chosen bit $b_i$ of the message $b \in \{0,1\}^k$ with good confidence by querying a corrupted string $\tilde{x}$ of the codeword $x = C(b)$ in at most $q$ coordinates. For $2$ ... more >>>

We exhibit a monotone function computable by a monotone circuit of quasipolynomial size such that any monotone circuit of polynomial depth requires exponential size. This is the first size-depth tradeoff result for monotone circuits in the so-called supercritical regime. Our proof is based on an analogous result in proof complexity: ... more >>>

We exhibit supercritical trade-off for monotone circuits, showing that there are functions computable by small circuits for which any circuit must have depth super-linear or even super-polynomial in the number of variables, far exceeding the linear worst-case upper bound. We obtain similar trade-offs in proof complexity, where we establish the ... more >>>

Understanding the power and limitations of classical and quantum information, and how they differ, is an important endeavor. On the classical side, property testing of distributions is a fundamental task: a tester, given samples of a distribution over a typically large domain such as $\{0,1\}^n$, is asked to verify properties ... more >>>

In this paper, we construct new t-server Private Information Retrieval (PIR) schemes with communication complexity subpolynomial in the previously best known, for all but finitely many t. Our results are
based on combining derivatives (in the spirit of Woodruff-Yekhanin) with the Matching Vector
based PIRs of Yekhanin and Efremenko. Previously ... more >>>

We show that the complexity class of exponential-time Arthur Merlin with sub-exponential advice ($AMEXP_{/2^{n^{\varepsilon}}}$) requires circuit complexity at least $2^n/n$. Previously, the best known such near-maximum lower bounds were for symmetric exponential time by Chen, Hirahara, and Ren (STOC'24) and Li (STOC'24), or randomized exponential time with MCSP oracle and ... more >>>

In a recent work, Cormode, Dall'Agnol, Gur and Hickey (CCC, 2024) introduced the model of Zero-Knowledge Streaming Interactive Proofs (zkSIPs).
Loosely speaking, such proof-systems enable a prover to convince astreaming verifier that the input $x$, to which it has read-once streaming access, satisfies some property, in such a way that ... more >>>

We characterize the power of constant-depth Boolean circuits in generating uniform symmetric distributions. Let $f\colon\{0,1\}^m\to\{0,1\}^n$ be a Boolean function where each output bit of $f$ depends only on $O(1)$ input bits. Assume the output distribution of $f$ on uniform input bits is close to a uniform distribution $\mathcal D$ with ... more >>>

First of all we give some reasons that “natural proofs” built not a barrier to prove P not= NP using Boolean complexity. Then we investigate the approximation method for its extension to prove super-polynomial lower bounds for the non-monotone complexity of suitable Boolean functions in NP or to understand why ... more >>>

Sensitivity measures how much the output of an algorithm changes, in terms of Hamming distance, when part of the input is modified. While approximation algorithms with low sensitivity have been developed for many problems, no sensitivity lower bounds were previously known for approximation algorithms. In this work, we establish the ... more >>>

We study the problem of partitioning the unit cube $[0,1]^n$ into $c$ parts so that each $d$-dimensional axis-parallel projection has small volume.

This natural combinatorial/geometric question was first studied by Kopparty and Nagargoje [KN23] as a reformulation of the problem of determining the achievable parameters for seedless multimergers -- which ... more >>>

Seeded extractors are fundamental objects in pseudorandomness and cryptography, and a deep line of work has designed polynomial-time seeded extractors with nearly-optimal parameters. However, existing constructions of seeded extractors with short seed length and large output length run in time $\Omega(n \log(1/\varepsilon))$ and often slower, where $n$ is the input ... more >>>

Estimating the second frequency moment of a stream up to $(1\pm\varepsilon)$ multiplicative error requires at most $O(\log n / \varepsilon^2)$ bits of space, due to a seminal result of Alon, Matias, and Szegedy. It is also known that at least $\Omega(\log n + 1/\varepsilon^{2})$ space is needed.
We prove an ... more >>>

The Schwartz-Zippel Lemma states that if a low-degree multivariate polynomial with coefficients in a field is not zero everywhere in the field, then it has few roots on every finite subcube of the field. This fundamental fact about multivariate polynomials has found many applications in algorithms, complexity theory, coding theory, ... more >>>

This work makes two distinct yet related contributions. The first contribution is a new information-theoretic model, the query-with-sketch model, and tools to show lower bounds within it. The second contribution is conceptual, technically builds on the first contribution, and is a barrier in the derandomization of randomized logarithmic space (BPL). ... more >>>

We exhibit a total search problem whose communication complexity in the quantum SMP (simultaneous message passing) model is exponentially smaller than in the classical two-way randomized model. Moreover, the quantum protocol is computationally efficient and its solutions are classically verifiable, that is, the problem lies in the communication analogue of ... more >>>

We investigate the task of deterministically condensing randomness from Online Non-Oblivious Symbol Fixing (oNOSF) sources, a natural model of defective random sources for which it is known that extraction is impossible [AORSV, EUROCRYPT'20]. A $(g,\ell)$-oNOSF source is a sequence of $\ell$ blocks $\mathbf{X} = (\mathbf{X}_1, \dots, \mathbf{X}_{\ell})\sim (\{0, 1\}^{n})^{\ell}$, where ... more >>>

We improve the best known upper bounds on the density of corner-free sets over quasirandom groups from inverse poly-logarithmic to quasi-polynomial. We make similarly substantial improvements to the best known lower bounds on the communication complexity of a large class of permutation functions in the 3-player Number-on-Forehead model. Underpinning both ... more >>>

We initiate the study of the *randomness complexity* of differential privacy, i.e., how many random bits an algorithm needs in order to generate accurate differentially private releases. As a test case, we focus on the task of releasing the results of $d$ counting queries, or equivalently all one-way marginals on ... more >>>

Trevisan and Vadhan (FOCS 2000) introduced the notion of (seedless) extractors for samplable distributions as a possible solution to the problem of extracting random keys for cryptographic protocols from weak sources of randomness.
They showed that under a very strong complexity theoretic assumption, there exists a constant $\alpha>0$ such that ... more >>>

We introduce two models of space-bounded quantum interactive proof systems, $\mathbf{QIPL}$ and $\mathbf{QIP_\mathrm{U}L}$. The $\mathbf{QIP_\mathrm{U}L}$ model, a space-bounded variant of quantum interactive proofs ($\mathbf{QIP}$) introduced by Watrous (CC 2003) and Kitaev and Watrous (STOC 2000), restricts verifier actions to unitary circuits. In contrast, $\mathbf{QIPL}$ allows logarithmically many intermediate measurements per ... more >>>

Commuting local Hamiltonians provide a testing ground for studying many of the most interesting open questions in quantum information theory, including the quantum PCP conjecture and the existence of area laws. Although they are a simplified model of quantum computation, the status of the commuting local Hamiltonian problem remains largely ... more >>>

A natural model of a source of randomness consists of a long stream of symbols $X = X_1\circ\ldots\circ X_t$, with some guarantee on the entropy of $X_i$ conditioned on the outcome of the prefix $x_1,\dots,x_{i-1}$. We study unpredictable sources, a generalization of the almost Chor--Goldreich (CG) sources considered in [DMOZ23]. ... more >>>

In this work, we show that the class of multivariate degree-$d$ polynomials mapping $\{0,1\}^{n}$ to any Abelian group $G$ is locally correctable with $\widetilde{O}_{d}((\log n)^{d})$ queries for up to a fraction of errors approaching half the minimum distance of the underlying code. In particular, this result holds even for polynomials ... more >>>

This note revisits the study of search problems that are solvable in probabilistic polynomial time. Previously, Goldreich (2011) introduced a class called ``$\mathcal{BPP}$-search'', and studied search-to-decision reductions for problems in this class.

In Goldreich's original formulation, the definition of what counts as ``successfully solving'' a $\mathcal{BPP}$-search problem is implicit, and ... more >>>

There has been tremendous progress in the past decade in the field of quantified Boolean formulas (QBF), both in practical solving as well as in creating a theory of corresponding proof systems and their proof complexity analysis. Both for solving and for proof complexity, it is important to have interesting ... more >>>

We propose a new definition of the class of search problems that correspond to BPP.
Specifically, a problem in this class is specified by a polynomial-time approximable function $q:\{0,1\}^*\times\{0,1\}^*\to[0,1]$ that associates, with each possible solution $y$ to an instance $x$, a quality $q(x,y)$.
Intuitively, quality 1 corresponds to perfectly ... more >>>

In this paper we study syntactic branching programs of bounded repetition
representing CNFs of bounded treewidth.
For this purpose we introduce two new structural graph
parameters $d$-pathwidth and clique preserving $d$-pathwidth denoted
by $pw_d(G)$ and $cpw_d(G)$ where $G$ is a graph.
We show that $cpw_2(G) \leq O(tw(G) \Delta(G))$ ... more >>>

We survey recent and classical results and techniques concerning binary codes in the large distance (or, high-noise) regime, and the closely related notion of $\varepsilon$-balanced codes. Our (hopefully small-biased) column will mainly discuss encoding, and decoding from adversarial errors.

A previous version of this text originally appeared as an ACM ... more >>>

We study uniquely decodable codes and list decodable codes in the high-noise regime, specifically codes that are uniquely decodable from $\frac{1-\varepsilon}{2}$ fraction of errors and list decodable from $1-\varepsilon$ fraction of errors. We present several improved explicit constructions that achieve near-optimal rates, as well as efficient or even linear-time decoding ... more >>>

We investigate the computational complexity of estimating the trace of quantum state powers $\text{tr}(\rho^q)$ for an $n$-qubit mixed quantum state $\rho$, given its state-preparation circuit of size $\text{poly}(n)$. This quantity is closely related to and often interchangeable with the Tsallis entropy $\text{S}_q(\rho) = \frac{1-\text{tr}(\rho^q)}{q-1}$, where $q = 1$ corresponds to ... more >>>

We prove the first meta-complexity characterization of a quantum cryptographic primitive. We show that one-way puzzles exist if and only if there is some quantum samplable distribution of binary strings over which it is hard to approximate Kolmogorov complexity. Therefore, we characterize one-way puzzles by the average-case hardness of a ... more >>>

The coding theorem for Kolmogorov complexity states that any string sampled from a computable distribution has a description length close to its information content. A coding theorem for resource-bounded Kolmogorov complexity is the key to obtaining fundamental results in average-case complexity, yet whether any samplable distribution admits a coding theorem ... more >>>

One of the earliest models of weak randomness is the Chor-Goldreich (CG) source. A $(t,n,k)$-CG source is a sequence of random variables $\mathbf{X}=(\mathbf{X}_1,\dots,\mathbf{X}_t) \sim (\{0,1\}^n)^t$, where each $\mathbf{X}_i$ has min-entropy $k$ conditioned on any fixing of $\mathbf{X}_1,\dots,\mathbf{X}_{i-1}$. Chor and Goldreich proved that there is no deterministic way to extract randomness ... more >>>

We fully determine the communication complexity of approximating matrix rank, over any finite field $\mathbb{F}$. We study the most general version of this problem, where $0\leq r < R\leq n$ are given integers, Alice and Bob's inputs are matrices $A,B\in\mathbb{F}^{n\times n}$, respectively, and they need to distinguish between the cases ... more >>>

We establish strong inapproximability for finding the sparsest nonzero vector in a real subspace (where sparsity refers to the number of nonzero entries). Formally we show that it is NP-Hard (under randomized reductions) to approximate the sparsest vector in a subspace within any constant factor. By simple tensoring the inapproximability ... more >>>

Two matrices are said to be principal minor equivalent if they have equal
corresponding principal minors of all orders. We give a characterization of
principal minor equivalence and a deterministic polynomial time algorithm to
check if two given matrices are principal minor equivalent. Earlier such
results were known for ... more >>>

We show that every NP relation that can be verified by a bounded-depth polynomial-sized circuit, or a bounded-space polynomial-time algorithm, has a computational zero-knowledge proof (with statistical soundness) with communication that is only additively larger than the witness length. Our construction relies only on the minimal assumption that one-way functions ... more >>>

The classical Reed-Muller codes over a finite field $\mathbb{F}_q$ are based on evaluations of $m$-variate polynomials of degree at most $d$ over a product set $U^m$, for some $d$ less than $|U|$. Because of their good distance properties, as well as the ubiquity and expressive power of polynomials, these codes ... more >>>

We present a polynomial-time pseudo-deterministic algorithm for constructing irreducible polynomial of degree $d$ over finite field $\mathbb{F}_q$. A pseudo-deterministic algorithm is allowed to use randomness, but with high probability it must output a canonical irreducible polynomial. Our construction runs in time $\tilde{O}(d^4 \log^4{q})$.

Our construction extends Shoup's deterministic algorithm ... more >>>

We consider indistinguishability obfuscation (iO) for multi-output circuits $C:\{0,1\}^n\to\{0,1\}^n$ of size s, where s is the number of AND/OR/NOT gates in C. Under the worst-case assumption that NP $\nsubseteq$ BPP, we establish that there is no efficient indistinguishability obfuscation scheme that outputs circuits of size $s + o(s/ \log s)$. ... more >>>

The Circuit Size Hierarchy CSH$^a_b$ states that if $a > b \geq 1$ then the set of functions on $n$ variables computed by Boolean circuits of size $n^a$ is strictly larger than the set of functions computed by circuits of size $n^b$. This result, which is a cornerstone of circuit ... more >>>

Classical data can be copied and re-used for computation, with adverse consequences economically and in terms of data privacy. Motivated by this, we formulate problems in one-way communication complexity where Alice holds some data and Bob holds $m$ inputs, and he wants to compute $m$ instances of a bipartite relation ... more >>>

We initiate a study of doubly-efficient interactive proofs of proximity, while focusing on properties that can be tested within query-complexity that is significantly sub-linear, and seeking interactive proofs of proximity in which

1. The query-complexity of verification is significantly smaller than the query-complexity of testing.

2. The query-complexity of the ... more >>>

A line of work has shown how nontrivial uniform algorithms for analyzing circuits can be used to derive non-uniform circuit lower bounds. We show how the non-existence of nontrivial circuit-analysis algorithms can also imply non-uniform circuit lower bounds. Our connections yield new win-win circuit lower bounds, and suggest a potential ... more >>>

Assume we are given sample access to an unknown distribution $D$ over a large domain $[N]$. An emerging line of work has demonstrated that many basic quantities relating to the distribution, such as its distance from uniform and its Shannon entropy, despite being hard to approximate through the samples only, ... more >>>

Designing algorithms for space bounded models with restoration requirements on (most of) the space used by the algorithm is an important challenge posed about the catalytic computation model introduced by Buhrman et al (2014). Motivated by the scenarios where we do not need to restore unless $w$ is "useful", we ... more >>>

This paper revisits the study of two classical technical tools in theoretical computer science: Yao's transformation of distinguishers to next-bit predictors (FOCS 1982), and the ``reconstruction paradigm'' in pseudorandomness (e.g., as in Nisan and Wigderson, JCSS 1994). Recent works of Pyne, Raz, and Zhan (FOCS 2023) and Doron, Pyne, and ... more >>>

A catalytic machine is a model of computation where a traditional space-bounded machine is augmented with an additional, significantly larger, "catalytic" tape, which, while being available as a work tape, has the caveat of being initialized with an arbitrary string, which must be preserved at the end of the computation. ... more >>>

We study the challenge of derandomizing constant-width standard-order read-once branching programs (ROBPs). Let $c \in [1, 2)$ be any constant. We prove that if there are explicit pseudorandom generators (PRGs) for width-$6$ length-$n$ permutation ROBPs with error $1/n$ and seed length $\widetilde{O}(\log^c n)$, then there are explicit hitting set generators ... more >>>

We introduce $\mathrm{pKt}$ complexity, a new notion of time-bounded Kolmogorov complexity that can be seen as a probabilistic analogue of Levin's $\mathrm{Kt}$ complexity. Using $\mathrm{pKt}$ complexity, we upgrade two recent frameworks that characterize one-way functions ($\mathrm{OWFs}$) via symmetry of information and meta-complexity, respectively. Among other contributions, we establish the following ... more >>>

We present the first explicit construction of two-sided lossless expanders in the unbalanced setting (bipartite graphs that have many more nodes on the left than on the right). Prior to our work, all known explicit constructions in the unbalanced setting achieved only one-sided lossless expansion.

Specifically, we show ... more >>>

Razborov [J. ACM, 2016] exhibited the following surprisingly strong trade-off phenomenon in propositional proof complexity: for a parameter $k = k(n)$, there exists $k$-CNF formulas over $n$ variables, having resolution refutations of $O(k)$ width, but every tree-like refutation of width $n^{1-\epsilon}/k$ needs size $\text{exp}\big(n^{\Omega(k)}\big)$. We extend this result to tree-like ... more >>>

Interactive proofs of proximity (IPPs) for a property are relaxed proof systems analogous to property testers in which the goal is for the verifier to be convinced to accept inputs that are in the property, and to not be fooled into accepting inputs that are far from the property.

Kumar (CCC, 2023) used a novel switching lemma to prove exponential-size lower bounds for a circuit class $GC^0$ that not only contains $AC^0$ but can---with a single gate---compute functions that require exponential-size $TC^0$ circuits. Their main result was that switching-lemma lower bounds for $AC^0$ lift to $GC^0$ with no loss ... more >>>

We propose a framework of algorithm vs. hardness for all Max-CSPs and demonstrate it for a large class of predicates. This framework extends the work of Raghavendra [STOC, 2008], who showed a similar result for almost satisfiable Max-CSPs.

Our framework is based on a new hybrid approximation algorithm, which uses ... more >>>

The propositional proof system resolution over parities (Res($\oplus$)) combines resolution and the linear algebra over GF(2). It is a challenging open question to prove a superpolynomial lower bound on the proof size in this system. For many years, superpolynomial lower bounds were known only in tree-like cases. Recently, Efremenko, Garlik, ... more >>>

We prove a Carbery-Wright style anti-concentration inequality for the unitary Haar measure, by showing that the probability of a polynomial in the entries of a random unitary falling into an $\varepsilon$ range is at most a polynomial in $\varepsilon$. Using it, we show that the scrambling speed of a random ... more >>>

One of the most famous TFNP subclasses is PPP, which is the set of all search problems whose totality is guaranteed by the pigeonhole principle. The author's recent preprint [ECCC TR24-002 2024] has introduced a TFNP problem related to the pigeonhole principle over a quotient set, called Quotient Pigeon, and ... more >>>

We consider read-$k$ determinantal representations of polynomials and prove some non-expressibility results. A square matrix $M$ whose entries are variables or field elements will be called \emph{read-$k$}, if every variable occurs at most $k$ times in $M$. It will be called a \emph{determinantal representation} of a polynomial $f$ if $f=\det(M)$. ... more >>>

The input to the Tree Evaluation problem is a binary tree of height $h$ in which each internal vertex is associated with a function mapping pairs of $\ell$-bit strings to $\ell$-bit strings,and each leaf is assigned an $\ell$-bit string.
The desired output is the value of the root, where ... more >>>

We present a deterministic $2^{k^{\mathcal{O}(1)}} \text{poly}(n,d)$ time algorithm for decomposing $d$-dimensional, width-$n$ tensors of rank at most $k$ over $\mathbb{R}$ and $\mathbb{C}$. This improves upon the previous randomized algorithm of Peleg, Shpilka, and Volk (ITCS '24) that takes $2^{k^{k^{\mathcal{O}(k)}}} \text{poly}(n,d)$ time and the deterministic $n^{k^k}$ time algorithms of Bhargava, Saraf, ... more >>>

We present a new algorithm for solving homogeneous multilinear equations, which are high dimensional generalisations of solving homogeneous linear equations. First, we present a linear time reduction from solving generic homogeneous multilinear equations to computing hyperdeterminants, via a high dimensional Cramer's rule. Hyperdeterminants are generalisations of determinants, associated with tensors ... more >>>

Consider the model where we can access a parity function through random uniform labeled examples in the presence of random classification noise. In this paper, we show that approximating the number of relevant variables in the parity function is as hard as properly learning parities.

More specifically, let $\gamma:{\mathbb R}^+\to ... more >>>

A central open question within meta-complexity is that of NP-hardness of problems such as MCSP and MK$^t$P. Despite a large body of work giving consequences of and barriers for NP-hardness of these problems under (restricted) deterministic reductions, very little is known in the setting of randomized reductions. In this work, ... more >>>

The dimension of partial derivatives (Nisan and Wigderson, 1997) is a popular measure for proving lower bounds in algebraic complexity. It is used to give strong lower bounds on the Waring decomposition of polynomials (called Waring rank). This naturally leads to an interesting open question: does this measure essentially characterize ... more >>>

Numerous works have studied the probability that a length $t-1$ random walk on an expander is confined to a given rectangle $S_1 \times \ldots \times S_t$, providing both upper and lower bounds for this probability.
However, when the densities of the sets $S_i$ may depend on the walk length (e.g., ... more >>>

We show that assuming the Exponential Time Hypothesis, the Partial Minimum Branching Program Size Problem (MBPSP*) requires superpolynomial time. This result also applies to the partial minimization problems for many interesting subclasses of branching programs, such as read-$k$ branching programs and OBDDs.

Combining these results with our recent result (Glinskih ... more >>>

A classical challenge in complexity theory and cryptography is to simulate interactive proof systems by non-interactive proof systems. In this work we leverage approaches from recent works in derandomization to address this challenge, focusing on non-interactive simulations that are sound against uniform adversarial algorithms.

Our results concern fundamental questions in ... more >>>

We introduce and study the following natural total search problem, which we call the {\it heavy element avoidance} (Heavy Avoid) problem: for a distribution on $N$ bits specified by a Boolean circuit sampling it, and for some parameter $\delta(N) \ge 1/\poly(N)$ fixed in advance, output an $N$-bit string that has ... more >>>

A dot-product proof (DPP) is a simple probabilistic proof system in which the input statement $x$ and the proof ${\pi}$ are vectors over a finite field $\mathbb{F}$, and the proof is verified by making a single dot-product query $\langle {q},({x} \| {\pi})\rangle$ jointly to ${x}$ and ${\pi}$. A DPP can ... more >>>

This paper studies the interaction of oracles with algorithmic approaches to proving circuit complexity lower bounds, establishing new results on two different kinds of questions.

1. We revisit some prominent open questions in circuit lower bounds, and provide a clean way of viewing them as circuit upper bound questions. Let ... more >>>

Kol and Raz [STOC 2013] showed how to simulate any alternating two-party communication protocol designed to work over the noiseless channel, by a protocol that works over a stochastic channel that corrupts each sent symbol with probability $\epsilon>0$ independently, with only a $1+\mathcal{O}(\sqrt{\H(\epsilon)})$ blowup to the communication. In particular, this ... more >>>

We prove a lower bound on the communication complexity of computing the $n$-fold xor of an arbitrary function $f$, in terms of the communication complexity and rank of $f$. We prove that $D(f^{\oplus n}) \geq n \cdot \Big(\frac{\Omega(D(f))}{\log rk(f)} -\log rk(f)\Big )$, where here $D(f), D(f^{\oplus n})$ represent the ... more >>>

Time efficient decoding algorithms for error correcting codes often require linear space. However, locally decodable codes yield more efficient randomized decoders that run in time $n^{1+o(1)}$ and space $n^{o(1)}$. In this work we focus on deterministic decoding.
Gronemeier showed that any non-adaptive deterministic decoder for a good code running ... more >>>

The input to the Tree Evaluation problem is a binary tree of height $h$ in which each internal vertex is associated with a function mapping pairs of $\ell$-bit strings to $\ell$-bit strings, and each leaf is assigned an $\ell$-bit string.
The desired output is the value of the root, ... more >>>

The problem of minimizing the share size of threshold secret-sharing schemes is a basic research question that has been extensively studied. Ideally, one strives for schemes in which the share size equals the secret size. While this is achievable for large secrets (Shamir, CACM '79), no similar solutions are known ... more >>>

We study the formula complexity of Iterated Sub-Permutation Matrix Multiplication, the logspace-complete problem of computing the product of $k$ $n$-by-$n$ Boolean matrices with at most a single $1$ in each row and column. For all $d \le \log k$, this problem is solvable by $n^{O(dk^{1/d})}$ size monotone formulas of two ... more >>>

In the catalytic logspace ($CL$) model of (Buhrman et.~al.~STOC 2013), we are given a small work tape, and a larger catalytic tape that has an arbitrary initial configuration. We may edit this tape, but it must be exactly restored to its initial configuration at the completion of the computation. This ... more >>>

In this note, we study the interplay between the communication from a verifier in a general private-coin interactive protocol and the number of random bits it uses in the protocol. Under worst-case derandomization assumptions, we show that it is possible to transform any $I$-round interactive protocol that uses $\rho$ random ... more >>>

An $s$-sparse polynomial has at most $s$ monomials with nonzero coefficients. The Equivalence Testing problem for sparse polynomials (ETsparse) asks to decide if a given polynomial $f$ is equivalent to (i.e., in the orbit of) some $s$-sparse polynomial. In other words, given $f \in \mathbb{F}[\mathbf{x}]$ and $s \in \mathbb{N}$, ETsparse ... more >>>

In a recent result, Knop, Lovett, McGuire and Yuan (STOC 2021) proved the log-rank conjecture for communication complexity, up to $\log n$ factor, for any Boolean function composed with $AND$ function as the inner gadget. One of the main tools in this result was the relationship between monotone analogues of ... more >>>

High dimensional expanders (HDXs) are a hypergraph generalization of expander graphs. They are extensively studied in the math and TCS communities due to their many applications. Like expander graphs, HDXs are especially interesting for applications when they are bounded degree, namely, if the number of edges adjacent to every vertex ... more >>>

A sequence of recent works, concluding with Mu et al. (Eurocrypt, 2024) has shown that every problem $\Pi$ admitting a non-interactive statistical zero-knowledge proof (NISZK) has an efficient zero-knowledge batch verification protocol. Namely, an NISZK protocol for proving that $x_1,\dots,x_k \in \Pi$ with communication that only scales poly-logarithmically with $k$. ... more >>>

In an Instance-Hiding Interactive Proof (IHIP) [Beaver et al. CRYPTO 90], an efficient verifier with a _private_ input x interacts with an unbounded prover to determine whether x is contained in a language L. In addition to completeness and soundness, the instance-hiding property requires that the prover should not learn ... more >>>

For every $n$, we construct a sum-of-squares identity
$ (\sum_{i=1}^n x_i^2) (\sum_{j=1}^n y_j^2)= \sum_{k=1}^s f_k^2$,
where $f_k$ are bilinear forms with complex coefficients and $s= O(n^{1.62})$. Previously, such a construction was known with $s=O(n^2/\log n)$.
The same bound holds over any field of positive characteristic.

As an application to ... more >>>

How can we trust the correctness of a learned model on a particular input of interest? Model accuracy is typically measured $on\ average$ over a distribution of inputs, giving no guarantee for any fixed input. This paper proposes a theoretically-founded solution to this problem: to train $Self$-$Proving\ models$ that prove ... more >>>

We present the first efficient averaging sampler that achieves asymptotically optimal randomness complexity and near-optimal sample complexity for natural parameter choices. Specifically, for any constant $\alpha > 0$, for $\delta > 2^{-\mathrm{poly}(1 / \varepsilon)}$, it uses $m + O(\log (1 / \delta))$ random bits to output $t = O(\log(1 ... more >>>

What are the minimal cryptographic assumptions that suffice for constructing efficient argument systems, and for which tasks? Recently, Amit and Rothblum [STOC 2023] showed that one-way functions suffice for constructing constant-round arguments for bounded-depth computations. In this work we ask: what other tasks have efficient argument systems based only on ... more >>>

We present a simple alternative exposition of the the recent result of Hirahara and Nanashima (STOC’24) showing that one-way functions exist if (1) every language in NP has a zero-knowledge proof/argument and (2) ZKA contains non-trivial languages. Our presentation does not rely on meta-complexity and we hope it may be ... more >>>

Suppose Alice has collected a small number of samples from an unknown distribution, and would like to learn about the distribution. Bob, an untrusted data analyst, claims that he ran a sophisticated data analysis on the distribution, and makes assertions about its properties. Can Alice efficiently verify Bob's claims using ... more >>>

We prove that random low-degree polynomials (over $\mathbb{F}_2$) are unbiased, in an extremely general sense. That is, we show that random low-degree polynomials are good randomness extractors for a wide class of distributions. Prior to our work, such results were only known for the small families of (1) uniform sources, ... more >>>

For $S\subseteq \mathbb{F}^n$, consider the linear space of restrictions of degree-$d$ polynomials to $S$. The Hilbert function of $S$, denoted $\mathrm{h}_S(d,\mathbb{F})$, is the dimension of this space. We obtain a tight lower bound on the smallest value of the Hilbert function of subsets $S$ of arbitrary finite grids in $\mathbb{F}^n$ ... more >>>

The Gilbert--Varshamov (GV) bound is a classical existential result in coding theory. It implies that a random linear binary code of rate $\varepsilon^2$ has relative distance at least $\frac{1}{2} - O(\varepsilon)$ with high probability. However, it is a major challenge to construct explicit codes with similar parameters.

One hope to ... more >>>

Error reduction procedures play a crucial role in constructing weighted PRGs [PV'21, CDRST'21], which are central to many recent advances in space-bounded derandomization. The fundamental method driving error reduction procedures is the Richardson iteration, which is adapted from the literature on fast Laplacian solvers. In the context of space-bounded derandomization, ... more >>>

The Zig-Zag product of two graphs, $Z = G \circ H$, was introduced in the seminal work of Reingold, Vadhan, and Wigderson (Ann. of Math. 2002) and has since become a pivotal tool in theoretical computer science. The classical bound, which is used throughout, states that the spectral expansion of ... more >>>

Tree codes, introduced in the seminal works of Schulman (STOC 93', IEEE Transactions on Information Theory 96') are codes designed for interactive communication. Encoding in a tree code is done in an online manner: the $i$-th codeword symbol depends only on the first $i$ message symbols. Codewords should have good ... more >>>

This paper explores the connection between classical isoperimetric inequalities, their directed analogues, and monotonicity testing. We study the setting of real-valued functions $f : [0,1]^d \to \mathbb{R}$ on the solid unit cube, where the goal is to test with respect to the $L^p$ distance. Our goals are twofold: to further ... more >>>

One of the major open problems in complexity theory is to demonstrate an explicit function which requires super logarithmic depth, a.k.a, the $\mathbf{P}$ versus $\mathbf{NC^1}$ problem. The current best depth lower bound is $(3-o(1))\cdot \log n$, and it is widely open how to prove a super-$3\log n$ depth lower bound. ... more >>>

We develop new characterizations of Impagliazzo's worlds Algorithmica, Heuristica and Pessiland by the intractability of conditional Kolmogorov complexity $\mathrm{K}$ and conditional probabilistic time-bounded Kolmogorov complexity $\mathrm{pK}^t$.

In our first set of results, we show that $\mathrm{NP} \subseteq \mathrm{BPP}$ iff $\mathrm{pK}^t(x \mid y)$ can be computed efficiently in the worst case ... more >>>

We study the \emph{noncommutative rank} problem, $\NCRANK$, of computing the rank of matrices with linear entries in $n$ noncommuting variables and the problem of \emph{noncommutative Rational Identity Testing}, $\RIT$, which is to decide if a given rational formula in $n$ noncommuting variables is zero on its domain of definition.

... more >>>

We show that there is a constant $k$ such that Buss's intuitionistic theory $\mathbf{IS}^1_2$ does not prove that SAT requires co-nondeterministic circuits of size at least $n^k$. To our knowledge, this is the first unconditional unprovability result in bounded arithmetic in the context of worst-case fixed-polynomial size circuit lower bounds. ... more >>>

We prove optimal concentration of measure for lifted functions on high dimensional expanders (HDX). Let $X$ be a $k$-dimensional HDX. We show for any $i \leq k$ and function $f: X(i) \to [0,1]$:
\[
\Pr_{s \in X(k)}\left[\left|\underset{{t \subseteq s}}{\mathbb{E}}[f(t)] - \mu \right| \geq \varepsilon \right] \leq \exp\left(-\varepsilon^2 \frac{k}{i}\right).
\]
Using ... more >>>

In this paper we present a new proof system framework CLIP (Cumulation Linear Induction Proposition) for propositional model counting. A CLIP proof firstly involves a circuit, calculating the cumulative function (or running count) of models counted up to a point, and secondly a propositional proof arguing for the correctness of ... more >>>

We design polynomial size, constant depth (namely, $AC^0$) arithmetic formulae for the greatest common divisor (GCD) of two polynomials, as well as the related problems of the discriminant, resultant, Bézout coefficients, squarefree decomposition, and the inversion of structured matrices like Sylvester and Bézout matrices. Our GCD algorithm extends to any ... more >>>

Strong algebraic proof systems such as IPS (Ideal Proof System; Grochow-Pitassi JACM 2018) offer a general model for
deriving polynomials in an ideal and refuting unsatisfiable propositional formulas, subsuming most standard propositional proof systems. A major approach for lower bounding the size of IPS refutations is the Functional Lower Bound ... more >>>

A locally decodable code (LDC) is an error correcting code that allows for recovery of any desired bit in the message based on a constant number of randomly selected bits in the possibly corrupted codeword.
A relaxed LDC requires correct recovery only in case of actual codewords, while requiring that ... more >>>

In this work, we study the worst-case to average-case hardness of the Learning with Errors problem (LWE) under an alternative measure of hardness - the maximum success probability achievable by a probabilistic polynomial-time (PPT) algorithm. Previous works by Regev (STOC 2005), Peikert (STOC 2009), and Brakerski, Peikert, Langlois, Regev, Stehle ... more >>>

In a pair of recent breakthroughs \cite{CHR,Li} it was shown that the classes $S_2^E, ZPE^{NP}$, and $\Sigma_2^E$ require exponential circuit complexity, giving the first unconditional improvements to a classical result of Kannan. These results were obtained by designing a surprising new algorithm for the total search problem Range Avoidance: given ... more >>>

The Parameterized Inapproximability Hypothesis (PIH), which is an analog of the PCP theorem in parameterized complexity, asserts that, there is a constant $\varepsilon> 0$ such that for any computable function $f:\mathbb{N}\to\mathbb{N}$, no $f(k)\cdot n^{O(1)}$-time algorithm can, on input a $k$-variable CSP instance with domain size $n$, find an assignment satisfying ... more >>>

The class $ACC$ consists of Boolean functions that can be computed by constant-depth circuits of polynomial size with $AND, NOT$ and $MOD_m$ gates, where $m$ is a natural number. At the frontier of our understanding lies a widely believed conjecture asserting that $MAJORITY$ does not belong to $ACC$. The Boolean ... more >>>

Let $X=X_1\sqcup X_2\sqcup\ldots\sqcup X_k$ be a partitioned set of variables such that the variables in each part $X_i$ are noncommuting but for any $i\neq j$, the variables $x \in X_i$ commute with the variables $x' \in X_j$. Given as input a square matrix $T$ whose entries are linear forms over ... more >>>

Propositional proof complexity deals with the lengths of polynomial-time verifiable proofs for Boolean tautologies. An abundance of proof systems is known, including algebraic and semialgebraic systems, which work with polynomial equations and inequalities, respectively. The most basic algebraic proof system is based on Hilbert's Nullstellensatz (Beame et al., 1996). Tropical ... more >>>

Lifting theorems are theorems that bound the communication complexity
of a composed function $f\circ g^{n}$ in terms of the query complexity
of $f$ and the communication complexity of $g$. Such theorems
constitute a powerful generalization of direct-sum theorems for $g$,
and have seen numerous applications in recent years.

We prove ... more >>>

The notion of query-to-communication lifting theorems is a generic framework to convert query lower bounds into two-party communication lower bounds. Though this framework is very generic and beautiful, it has some inherent limitations such as it only applies to lifted functions. In order to address this issue, we propose gadgetless ... more >>>

The Stepanov-Bombieri proof of the Hasse-Weil bound also gives non-trivial bounds on the bias of character sums over curves with small genus, for any low-degree function $f$ that is not completely biased. For high genus curves, and in particular for curves used in AG codes over constant size fields, the ... more >>>

We give improved lower bounds for binary $3$-query locally correctable codes (3-LCCs) $C \colon \{0,1\}^k \rightarrow \{0,1\}^n$. Specifically, we prove:

(1) If $C$ is a linear design 3-LCC, then $n \geq 2^{(1 - o(1))\sqrt{k} }$. A design 3-LCC has the additional property that the correcting sets for every ... more >>>

Information complexity is one of the most powerful tools to prove information-theoretical lower bounds, with broad applications in communication complexity and streaming algorithms. A core notion in information complexity analysis is the Shannon entropy. Though it has some convenient properties, such as chain rules, Shannon entropy still has inherent limitations. ... more >>>

Relaxations for the constraint satisfaction problem (CSP) include bounded width, linear program (LP), semidefinite program (SDP), afinfe integer program (AIP), and the combined LP+AIP of Brakensiek, Guruswami, Wrochna, and Živný (SICOMP 2020). Tightening relaxations systematically leads to hierarchies and stronger algorithms. For the LP+AIP hierarchy, a constant level lower bound ... more >>>

Estimating quantiles is one of the foundational problems of data sketching. Given $n$ elements $x_1, x_2, \dots, x_n$ from some universe of size $U$ arriving in a data stream, a quantile sketch estimates the rank of any element with additive error at most $\varepsilon n$. A low-space algorithm solving this ... more >>>

We prove that the class of communication problems with public-coin randomized constant-cost protocols, called $BPP^0$, does not contain a complete problem. In other words, there is no randomized constant-cost problem $Q \in BPP^0$, such that all other problems $P \in BPP^0$ can be computed by a constant-cost deterministic protocol with ... more >>>

The fundamental theorem of Goldreich, Micali, and Wigderson (J. ACM 1991) shows that the existence of a one-way function is sufficient for constructing computational zero knowledge ($\mathrm{CZK}$) proofs for all languages in $\mathrm{NP}$. We prove its converse, thereby establishing characterizations of one-way functions based on the worst-case complexities of ... more >>>

We prove that a binary linear code of block length $n$ that is locally correctable with $3$ queries against a fraction $\delta > 0$ of adversarial errors must have dimension at most $O_{\delta}(\log^2 n \cdot \log \log n)$. This is almost tight in view of quadratic Reed-Muller codes being a ... more >>>

A Matching Vector ($\mathbf{MV}$) family modulo a positive integer $m \ge 2$ is a pair of ordered lists $\mathcal{U} = (\mathbf{u}_1, \cdots, \mathbf{u}_K)$ and $\mathcal{V} = (\mathbf{v}_1, \cdots, \mathbf{v}_K)$ where $\mathbf{u}_i, \mathbf{v}_j \in \mathbb{Z}_m^n$ with the following property: for any $i \in [K]$, the inner product $\langle \mathbf{u}_i, \mathbf{v}_i \rangle ... more >>>

Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are necessary to prove a given theorem. In this work, we systematically explore the reverse mathematics of complexity lower bounds. We explore reversals in the setting of bounded arithmetic, with Cook's theory $\mathbf{PV}_1$ as the base ... more >>>

A search-to-decision reduction is a procedure that allows one to find a solution to a problem from the mere ability to decide when a solution exists. The existence of a search-to-decision reduction for time-bounded Kolmogorov complexity, i.e., the problem of checking if a string $x$ can be generated by a ... more >>>

The planted clique conjecture states that no polynomial-time algorithm can find a hidden clique of size $k \ll \sqrt{n}$ in an $n$-vertex Erd\H{o}s--R\'enyi random graph with a $k$-clique planted. In this paper, we prove the equivalence among many (in fact, \emph{most}) variants of planted clique conjectures, such as search ... more >>>

We give an algorithm for finding an $\epsilon$-fixed point of a contraction map $f:[0,1]^k\rightarrow [0,1]^k$ under the $\ell_\infty$-norm with query complexity $O (k^2\log (1/\epsilon ) )$.

We consider the task of locally correcting, and locally list-correcting, multivariate linear functions over the domain $\{0,1\}^n$ over arbitrary fields and more generally Abelian groups. Such functions form error-correcting codes of relative distance $1/2$ and we give local-correction algorithms correcting up to nearly $1/4$-fraction errors making $\widetilde{\mathcal{O}}(\log n)$ queries. This ... more >>>

Only a handful candidates for computational assumptions that imply secure key-agreement protocols (KA) are known, and even fewer are believed to be quantum safe. In this paper, we present a new hardness assumption---the worst-case hardness of a promise problem related to an interactive version of Kolmogorov Complexity.
Roughly speaking, the ... more >>>

We study the problem of function inversion with preprocessing where, given a function $f : [N] \to [N]$ and a point $y$ in its image, the goal is to find an $x$ such that $f(x) = y$ using at most $T$ oracle queries to $f$ and $S$ bits of preprocessed ... more >>>

We demonstrate that under believable cryptographic hardness assumptions, Gap versions of standard meta-complexity problems, such as the Minimum Circuit Size problem (MCSP) and the Minimum Time-Bounded Kolmogorov Complexity problem (MKTP) are not NP-complete w.r.t. Levin (i.e., witness-preserving many-to-one) reductions.

In more detail:
- Assuming the existence of indistinguishability obfuscation, and ... more >>>

We give a corrected proof that if PP $\subseteq$ BQP/qpoly, then the Counting Hierarchy collapses, as originally claimed by [Aaronson, CCC 2006]. This recovers the related unconditional claim that PP does not have circuits of any fixed size $n^k$ even with quantum advice. We do so by proving that YQP*, ... more >>>

While classic result in the 1980s establish that one-way functions (OWFs) imply the existence of pseudorandom generators (PRGs) which in turn imply pseudorandom functions (PRFs), the constructions (most notably the one from OWFs to PRGs) is complicated and inefficient.

Consequently, researchers have developed alternative \emph{direct} constructions of PRFs from various ... more >>>

Quantum computers are expected to revolutionize our ability to process information. The advancement from classical to quantum computing is a product of our advancement from classical to quantum physics -- the more our understanding of the universe grows, so does our ability to use it for computation. A natural question ... more >>>

In this work we study oblivious complexity classes. Among our results:
1) For each $k \in \mathbb{N}$, we construct an explicit language $L_k \in O_2P$ that cannot be computed by circuits of size $n^k$.
2) We prove a hierarchy theorem for $O_2TIME$. In particular, for any function $t:\mathbb{N} \rightarrow \mathbb{N}$ ... more >>>

Whether $BPL=L$ (which is conjectured to be equal), or even whether $BPL\subseteq NL$, is a big open problem in theoretical computer science. It is well known that $L-NC^1\subseteq L\subseteq NL\subseteq L-AC^1$. In this work we will show that $BPL\subseteq L-AC^1$, which was not known before. Our proof is based on ... more >>>

We consider the query complexity of testing local graph properties in the bounded-degree graph model.
A local property is defined in terms of forbidden subgraphs that are augmented by degree information, where the latter account also for neighbors that are not in the subgraph.
Indeed, this formulation yields a generalized ... more >>>

For every $n >0$, we show the existence of a CNF tautology over $O(n^2)$ variables of width $O(\log n)$ such that it has a Polynomial Calculus Resolution refutation over $\{0,1\}$ variables of size $O(n^3polylog(n))$ but any Polynomial Calculus refutation over $\{+1,-1\}$ variables requires size $2^{\Omega(n)}$. This shows that Polynomial Calculus ... more >>>

The concept of redundancy in SAT lead to more expressive and powerful proof search techniques, e.g. able to express various inprocessing techniques, and to interesting hierarchies of proof systems [Heule et.al’20, Buss-Thapen’19].
We propose a general way to integrate redundancy rules in MaxSAT, that is we define MaxSAT variants of ... more >>>

The matching and linear matroid intersection problems are solvable in quasi-NC, meaning that there exist deterministic algorithms that run in polylogarithmic time and use quasi-polynomially many parallel processors. However, such a parallel algorithm is unknown for linear matroid matching, which generalizes both of these problems. In this work, we propose ... more >>>

We design a deterministic subexponential time algorithm that takes as input a multivariate polynomial $f$ computed by a constant-depth circuit over rational numbers, and outputs a list $L$ of circuits (of unbounded depth and possibly with division gates) that contains all irreducible factors of $f$ computable by constant-depth circuits. This ... more >>>

Analyzing refutations of the well known
pebbling formulas we prove some new strong connections between pebble games and algebraic proof system, showing that
there is a parallelism between the reversible, black and black-white pebbling games on one side, and
the three algebraic proof systems NS, MC and ... more >>>

In this work, we initiate the study of the space complexity of the Polynomial Identity Testing problem (PIT).
First, we observe that the majority of the existing (time-)efficient ``blackbox'' PIT algorithms already give rise to space-efficient ``whitebox'' algorithms for the respective classes of arithmetic formulas via a space-efficient ... more >>>

We study extractors computable in uniform $\mathrm{AC}^0$ and uniform $\mathrm{NC}^1$.

For the $\mathrm{AC}^0$ setting, we give a construction such that for every $k \ge n/ \mathrm{poly} \log n, \eps \ge 2^{-\mathrm{poly} \log n}$, it can extract $(1-\gamma)k$ randomness from an $(n, k)$ source for an arbitrary constant ... more >>>

In the Minmax Set Cover Reconfiguration problem, given a set system $\mathcal{F}$ over a universe and its two covers $\mathcal{C}^\mathrm{start}$ and $\mathcal{C}^\mathrm{goal}$ of size $k$, we wish to transform $\mathcal{C}^\mathrm{start}$ into $\mathcal{C}^\mathrm{goal}$ by repeatedly adding or removing a single set of $\mathcal{F}$ while covering the universe in any intermediate state. ... more >>>

We initiate an in-depth proof-complexity analysis of polynomial calculus (Q-PC) for Quantified Boolean Formulas (QBF). In the course of this we establish a tight proof-size characterisation of Q-PC in terms of a suitable circuit model (polynomial decision lists). Using this correspondence we show a size-degree relation for Q-PC, similar in ... more >>>

Lifting theorems are used for transferring lower bounds between Boolean function complexity measures. Given a lower bound on a complexity measure $A$ for some function $f$, we compose $f$ with a carefully chosen gadget function $g$ and get essentially the same lower bound on a complexity measure $B$ for the ... more >>>

A $q$-locally correctable code (LCC) $C:\{0,1\}^k \to \{0,1\}^n$ is a code in which it is possible to correct every bit of a (not too) corrupted codeword by making at most $q$ queries to the word. The cases in which $q$ is constant are of special interest, and so are the ... more >>>

In an attempt to show that the acceptance probability of a quantum query algorithm making $q$ queries can be well-approximated almost everywhere by a classical decision tree of depth $\leq \text{poly}(q)$, Aaronson and Ambainis proposed the following conjecture: let $f: \{ \pm 1\}^n \rightarrow [0,1]$ be a degree $d$ polynomial ... more >>>

Let $f$ be a Boolean function given as either a truth table or a circuit. How difficult is it to find the decision tree complexity, also known as deterministic query complexity, of $f$ in both cases? We prove that this problem is $NC$-hard and PSPACE-hard, respectively. The second bound is ... more >>>

Regular resolution is a refinement of the resolution proof system requiring that no variable be resolved on more than once along any path in the proof. It is known that there exist sequences of formulas that require exponential-size proofs in regular resolution while admitting polynomial-size proofs in resolution. Thus, with ... more >>>

We give the first explicit constant rate, constant relative distance, linear codes with an encoder that runs in time $n^{1 + o(1)}$ and space $\mathop{polylog}(n)$ provided random access to the message. Prior to this work, the only such codes were non-explicit, for instance repeat accumulate codes [DJM98] and the codes ... more >>>

Spurred by the influential work of Viola (Journal of Computing 2012), the past decade has witnessed an active line of research into the complexity of (approximately) sampling distributions, in contrast to the traditional focus on the complexity of computing functions.

We build upon and make explicit earlier implicit results of ... more >>>

Recently, the proof system MICE for the model counting problem #SAT was introduced by Fichte, Hecher and Roland (SAT’22). As demonstrated by Fichte et al., the system MICE can be used for proof logging for state-of-the-art #SAT solvers.
We perform a proof-complexity study of MICE. For this we first simplify ... more >>>

We prove the first hardness results against efficient proof search by quantum algorithms. We show that under Learning with Errors (LWE), the standard lattice-based cryptographic assumption, no quantum algorithm can weakly automate $\mathbf{TC}^0$-Frege. This extends the line of results of Kraí?ek and Pudlák (Information and Computation, 1998), Bonet, Pitassi, and ... more >>>

We construct explicit pseudorandom generators that fool $n$-variate polynomials of degree at most $d$ over a finite field $\mathbb{F}_q$. The seed length of our generators is $O(d \log n + \log q)$, over fields of size exponential in $d$ and characteristic at least $d(d-1)+1$. Previous constructions such as Bogdanov's (STOC ... more >>>

We show that for all $\varepsilon>0$, for sufficiently large prime power $q\in\mathbb{N}$, for all $\delta>0$, it is NP-hard to distinguish whether a $2$-Prover-$1$-Round projection game with alphabet size $q$ has value at least $1-\delta$, or value at most $1/q^{1-\varepsilon}$. This establishes a nearly optimal alphabet-to-soundness tradeoff for $2$-query PCPs ... more >>>

For every $n$, we construct a sum-of-squares identitity
\[ (\sum_{i=1}^n x_i^2) (\sum_{j=1}^n y_j^2)= \sum_{k=1}^s f_k^2\,,\]
where $f_k$ are bilinear forms with complex coefficients and $s= O(n^{1.62})$. Previously, such a construction was known with $s=O(n^2/\log n)$.
The same bound holds over any field of positive characteristic.

A nearest neighbor representation of a Boolean function $f$ is a set of vectors (anchors) labeled by $0$ or $1$ such that $f(x) = 1$ if and only if the closest anchor to $x$ is labeled by $1$. This model was introduced by Hajnal, Liu, and Turán (2022), who studied ... more >>>

A zero-knowledge proof enables a prover to convince a verifier that $x \in S$, without revealing anything beyond this fact. By running a zero-knowledge proof $k$ times, it is possible to prove (still in zero-knowledge) that $k$ separate instances $x_1,\dots,x_k$ are all in $S$. However, this increases the communication by ... more >>>

Motivated by the inapproximability of reconfiguration problems, we present a new PCP-type characterization of PSPACE, which we call a probabilistically checkable reconfiguration proof (PCRP): Any PSPACE computation can be encoded into an exponentially long sequence of polynomially long proofs such that every adjacent pair of the proofs differs in at ... more >>>

Proving super-polynomial lower bounds on the size of proofs of unsatisfiability of Boolean formulas using resolution over parities, is an outstanding problem that has received a lot of attention after its introduction by Raz and Tzamaret (2008). Very recently, Efremenko, Garlik and Itsykson (2023) proved the first exponential lower bounds ... more >>>

In this paper we prove the following two results.
* We show that for any $C \in {mVF, mVP, mVNP}$, $C = \overline{C}$. Here, $mVF, mVP$, and $mVNP$ are monotone variants of $VF, VP$, and $VNP$, respectively. For an algebraic complexity class $C$, $\overline{C}$ denotes the closure of $C$. ... more >>>

Let $X$ be a $d$-dimensional simplicial complex. A function $F\colon X(k)\to \{0,1\}^k$ is said to be a direct product function if there exists a function $f\colon X(1)\to \{0,1\}$ such that $F(\sigma) = (f(\sigma_1), \ldots, f(\sigma_k))$ for each $k$-face $\sigma$. In an effort to simplify components of the PCP theorem, Goldreich ... more >>>

We solve the derandomized direct product testing question in the low acceptance regime, by constructing new high dimensional expanders that have no small connected covers. We show that our complexes have swap cocycle expansion, which allows us to deduce the agreement theorem by relying on previous work.

Derandomized direct product ... more >>>

We study the problem of finding multicollisions, that is, the total search problem in which the input is a function $\mathcal{C} : [A] \to [B]$ (represented as a circuit) and the goal is to find $L \leq \lceil A/B \rceil$ distinct elements $x_1,\ldots, x_L \in A$ such that $\mathcal{C}(x_1) = ... more >>>

The generalized pigeonhole principle says that if tN + 1 pigeons are put into N holes then there must be a hole containing at least t + 1 pigeons. Let t-PPP denote the class of all total NP-search problems reducible to finding such a t-collision of pigeons. We introduce a ... more >>>

Let R_eps denote randomized query complexity for error probability eps, and R:=R_{1/3}. In this work we investigate whether a perfect composition theorem R(f o g^n)=Omega(R(f).R(g)) holds for a relation f in {0,1}^n * S and a total inner function g:{0,1}^m \to {0, 1}.

Let D^(prod) denote the maximum distributional query ... more >>>

An elementary proof of quadratic lower bound for determinantal complexity of the permanent in positive characteristic is stated. This is achieved by constructing a sequence of matrices with zero permanent, but the rank of Hessian is bounded below by a degree two polynomial.

We study the complexity of memory checkers with computational security and prove the first general tight lower bound.

Memory checkers, first introduced over 30 years ago by Blum, Evans, Gemmel, Kannan, and Naor (FOCS '91, Algorithmica '94), allow a user to store and maintain a large memory on a remote ... more >>>

We consider the notion of a local-characterization of an infinite family of unlabeled bounded-degree graphs.
Such a local-characterization is defined in terms of a finite set of (marked) graphs yielding a generalized notion of subgraph-freeness, which extends the standard notions of induced and non-induced subgraph freeness.

We survey the work ... more >>>

Several theorems and conjectures in communication complexity state or speculate that the complexity of a matrix in a given communication model is controlled by a related analytic or algebraic matrix parameter, e.g., rank, sign-rank, discrepancy, etc. The forward direction is typically easy as the structural implications of small complexity often ... more >>>

We consider the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Our main results show that for a range of linear algebra problems---including matrix-vector product, matrix inversion, matrix multiplication and powering---existing ... more >>>

The complexity class PPP contains all total search problems many-one reducible to the PIGEON problem, where we are given a succinct encoding of a function mapping n+1 pigeons to n holes, and must output two pigeons that collide in a hole. PPP is one of the “original five” syntactically-defined subclasses ... more >>>

We give a lower bound of ?(?n) on the unambiguous randomised parity-query complexity of the approximate majority problem – that is, on the lowest randomised parity-query complexity of any function over {0,1}? whose value is "0" if the Hamming weight of the input is at most n/3, is "1" if ... more >>>

Given a non-negative real matrix $M$ of non-negative rank at least $r$, can we witness this fact by a small submatrix of $M$? While Moitra (SIAM J. Comput. 2013) proved that this cannot be achieved exactly, we show that such a witnessing is possible approximately: an $m\times n$ matrix always ... more >>>

The field of combinatorial reconfiguration studies search problems with a focus on transforming one feasible solution into another.

Recently, Ohsaka [STACS'23] put forth the Reconfiguration Inapproximability Hypothesis (RIH), which roughly asserts that there is some $\varepsilon>0$ such that given as input a $k$-CSP instance (for some constant $k$) over ... more >>>

We introduce the entangled quantum polynomial hierarchy $QEPH$ as the class of problems that are efficiently verifiable given alternating quantum proofs that may be entangled with each other. We prove $QEPH$ collapses to its second level. In fact, we show that a polynomial number of alternations collapses to just two. ... more >>>

This paper presents the first Distributed Oblivious RAM (DORAM) protocol that achieves sub-logarithmic communication overhead without computational assumptions.
That is, given $n$ $d$-bit memory locations, we present an information-theoretically secure protocol which requires $o(d \cdot \log(n))$ bits of communication per access (when $d = \Omega(\log^2(n)$).

This comes as a surprise, ... more >>>

An $n$-variate polynomial $g$ of degree $d$ is a $(n,d,t)$ design polynomial if the degree of the gcd of every pair of monomials of $g$ is at most $t-1$. The power symmetric polynomial $\mathrm{PSym}_{n,d} := \sum_{i=1}^{n} x^d_i$ and the sum-product polynomial $\mathrm{SP}_{s,d} := \sum_{i=1}^{s}\prod_{j=1}^{d} x_{i,j}$ are instances of design polynomials ... more >>>

A long-standing open problem dating back to the 1960s is whether there exists a search-to-decision reduction for the time-bounded Kolmogorov complexity problem - that is, the problem of determining whether the length of the shortest time-$t$ program generating a given string $x$ is at most $s$.

In this work, we ... more >>>

Recently, Pasarkar, Papadimitriou, and Yannakakis (ITCS 2023) have introduced the new TFNP subclass called PLC that contains the class PPP; they also have proven that several search problems related to extremal combinatorial principles (e.g., Ramsey’s theorem and the Sunflower lemma) belong to PLC. This short paper shows that the class ... more >>>

We describe CNFs in n variables which, over a range of parameters, have small resolution refutations but are such that any small refutation must have height larger than n (even exponential in n), where the height of a refutation is the length of the longest path in it. This is ... more >>>