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Electronic Colloquium on Computational Complexity

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REPORTS > 2023:
All reports in year 2023:
TR23-082 | 1st June 2023
Ryan Williams

Self-Improvement for Circuit-Analysis Problems

Many results in fine-grained complexity reveal intriguing consequences from solving various SAT problems even slightly faster than exhaustive search. We prove a ``self-improving'' (or ``bootstrapping'') theorem for Circuit-SAT, $\#$Circuit-SAT, and its fully-quantified version: solving one of these problems faster for ``large'' circuit sizes implies a significant speed-up for ``smaller'' circuit ... more >>>

TR23-081 | 1st June 2023
Noga Amit, Guy Rothblum

Constant-Round Arguments from One-Way Functions

We study the following question: what cryptographic assumptions are needed for obtaining constant-round computationally-sound argument systems? We focus on argument systems with almost-linear verification time for subclasses of $\mathbf{P}$, such as depth-bounded computations.
Kilian's celebrated work [STOC 1992] provides such 4-message arguments for $\mathbf{P}$ (actually, for $\mathbf{NP}$) using collision-resistant hash ... more >>>

TR23-080 | 1st June 2023
Halley Goldberg, Valentine Kabanets

Improved Learning from Kolmogorov Complexity

Carmosino, Impagliazzo, Kabanets, and Kolokolova (CCC, 2016) showed that the existence of natural properties in the sense of Razborov and Rudich (JCSS, 1997) implies PAC learning algorithms in the sense of Valiant (Comm. ACM, 1984), for boolean functions in $\P/\poly$, under the uniform distribution and with membership queries. It is ... more >>>

TR23-079 | 31st May 2023
Russell Impagliazzo, Valentine Kabanets, Ilya Volkovich

Mutual Empowerment between Circuit Obfuscation and Circuit Minimization

We study close connections between Indistinguishability Obfuscation ($IO$) and the Minimum Circuit Size Problem ($MCSP$), and argue that algorithms for one of $MCSP$ or $IO$ would empower the other one. Some of our main results are:

\item If there exists a perfect (imperfect) $IO$ that is computationally secure ... more >>>

TR23-078 | 30th May 2023
Or Meir

Toward Better Depth Lower Bounds: A KRW-like theorem for Strong Composition

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq \mathbf{NC}^{1}$). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4), 1995) suggested to approach this problem by proving that depth complexity of a composition of functions $f \diamond g$ is roughly ... more >>>

TR23-077 | 25th May 2023
Nir Bitansky, Chethan Kamath, Omer Paneth, Ron Rothblum, Prashant Nalini Vasudevan

Batch Proofs are Statistically Hiding

Batch proofs are proof systems that convince a verifier that $x_1,\dots, x_t \in L$, for some $NP$ language $L$, with communication that is much shorter than sending the $t$ witnesses. In the case of statistical soundness (where the cheating prover is unbounded but honest prover is efficient), interactive batch proofs ... more >>>

TR23-076 | 24th May 2023
Lijie Chen, Zhenjian Lu, Igor Carboni Oliveira, Hanlin Ren, Rahul Santhanam

Polynomial-Time Pseudodeterministic Construction of Primes

A randomized algorithm for a search problem is *pseudodeterministic* if it produces a fixed canonical solution to the search problem with high probability. In their seminal work on the topic, Gat and Goldwasser posed as their main open problem whether prime numbers can be pseudodeterministically constructed in polynomial time.

... more >>>

TR23-075 | 17th May 2023
Abhranil Chatterjee, Sumanta Ghosh, Rohit Gurjar, Roshan Raj

Border Complexity of Symbolic Determinant under Rank One Restriction

VBP is the class of polynomial families that can be computed by the determinant of a symbolic matrix of the form $A_0 + \sum_{i=1}^n A_ix_i$ where the size of each $A_i$ is polynomial in the number of variables (equivalently, computable by polynomial-sized algebraic branching programs (ABP)). A major open problem ... more >>>

TR23-074 | 14th May 2023
Abhibhav Garg, Rafael Mendes de Oliveira, Shir Peleg, Akash Sengupta

Radical Sylvester-Gallai Theorem for Tuples of Quadratics

We prove a higher codimensional radical Sylvester-Gallai type theorem for quadratic polynomials, simultaneously generalizing [Han65, Shp20]. Hansen's theorem is a high-dimensional version of the classical Sylvester-Gallai theorem in which the incidence condition is given by high-dimensional flats instead of lines. We generalize Hansen's theorem to the setting of quadratic forms ... more >>>

TR23-073 | 15th May 2023
Xi Chen, Yuhao Li, Mihalis Yannakakis

Reducing Tarski to Unique Tarski (in the Black-box Model)

We study the problem of finding a Tarski fixed point over the $k$-dimensional grid $[n]^k$. We give a black-box reduction from the Tarski problem to the same problem with an additional promise that the input function has a unique fixed point. It implies that the Tarski problem and the unique ... more >>>

TR23-072 | 18th May 2023
Yeyuan Chen, Yizhi Huang, Jiatu Li, Hanlin Ren

Range Avoidance, Remote Point, and Hard Partial Truth Tables via Satisfying-Pairs Algorithms

The *range avoidance problem*, denoted as $\mathcal{C}$-$\rm Avoid$, asks to find a non-output of a given $\mathcal{C}$-circuit $C:\{0,1\}^n\to\{0,1\}^\ell$ with stretch $\ell>n$. This problem has recently received much attention in complexity theory for its connections with circuit lower bounds and other explicit construction problems. Inspired by the Algorithmic Method for circuit ... more >>>

TR23-071 | 8th May 2023
Yuval Filmus, Itai Leigh, Artur Riazanov, Dmitry Sokolov

Sampling and Certifying Symmetric Functions

A circuit $\mathcal{C}$ samples a distribution $\mathbf{X}$ with an error $\epsilon$ if the statistical distance between the output of $\mathcal{C}$ on the uniform input and $\mathbf{X}$ is $\epsilon$. We study the hardness of sampling a uniform distribution over the set of $n$-bit strings of Hamming weight $k$ denoted by $\mathbf{U}^n_k$ ... more >>>

TR23-070 | 9th May 2023
Shuichi Hirahara, Zhenjian Lu, Hanlin Ren

Bounded Relativization

Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called *bounded relativization*. For a complexity class $C$, we say that a statement is *$C$-relativizing* if the statement holds relative ... more >>>

TR23-069 | 11th May 2023
Bruno Pasqualotto Cavalar, Igor Carboni Oliveira

Constant-depth circuits vs. monotone circuits

We establish new separations between the power of monotone and general (non-monotone) Boolean circuits:

- For every $k \geq 1$, there is a monotone function in ${\rm AC^0}$ (constant-depth poly-size circuits) that requires monotone circuits of depth $\Omega(\log^k n)$. This significantly extends a classical result of Okol'nishnikova (1982) and Ajtai ... more >>>

TR23-068 | 10th May 2023
Ben Davis, Robert Robere

Colourful TFNP and Propositional Proofs

Recent work has shown that many of the standard TFNP classes — such as PLS, PPADS, PPAD, SOPL, and EOPL — have corresponding proof systems in propositional proof complexity, in the sense that a total search problem is in the class if and only if the totality of the problem ... more >>>

TR23-067 | 7th May 2023
Guy Goldberg

Linear Relaxed Locally Decodable and Correctable Codes Do Not Need Adaptivity and Two-Sided Error

Relaxed locally decodable codes (RLDCs) are error-correcting codes in which individual bits of the message can be recovered by querying only a few bits from a noisy codeword.
Unlike standard (non-relaxed) decoders, a relaxed one is allowed to output a ``rejection'' symbol, indicating that the decoding failed.
To prevent the ... more >>>

TR23-066 | 4th May 2023
Klim Efremenko, Gillat Kol, Dmitry Paramonov, Raghuvansh Saxena

Protecting Single-Hop Radio Networks from Message Drops

Single-hop radio networks (SHRN) are a well studied abstraction of communication over a wireless channel. In this model, in every round, each of the $n$ participating parties may decide to broadcast a message to all the others, potentially causing collisions. We consider the SHRN model in the presence of stochastic ... more >>>

TR23-065 | 4th May 2023
Louis Golowich

From Grassmannian to Simplicial High-Dimensional Expanders

In this paper, we present a new construction of simplicial complexes of subpolynomial degree with arbitrarily good local spectral expansion. Previously, the only known high-dimensional expanders (HDXs) with arbitrarily good expansion and less than polynomial degree were based on one of two constructions, namely Ramanujan complexes and coset complexes. ... more >>>

TR23-064 | 3rd May 2023
Oded Goldreich

On the Lower Bound on the Length of Relaxed Locally Decodable Codes

We revisit the known proof of the lower bound on the length of relaxed locally decodable codes, providing an arguably simpler exposition that yields a slightly better lower bound for the non-adaptive case and a weaker bound in the general case.

Recall that a locally decodable code is an error ... more >>>

TR23-063 | 2nd May 2023
Jacobo Toran, Florian Wörz

Cutting Planes Width and the Complexity of Graph Isomorphism Refutations

The width complexity measure plays a central role in Resolution and other propositional proof systems like Polynomial Calculus (under the name of degree). The study of width lower bounds is the most extended method for proving size lower bounds, and it is known that for these systems, proofs with small ... more >>>

TR23-062 | 2nd May 2023
Benny Applebaum, Eliran Kachlon

Conflict Checkable and Decodable Codes and Their Applications

Let $C$ be an error-correcting code over a large alphabet $q$ of block length $n$, and assume that, a possibly corrupted, codeword $c$ is distributively stored among $n$ servers where the $i$th entry is being held by the $i$th server. Suppose that every pair of servers publicly announce whether the ... more >>>

TR23-061 | 2nd May 2023
Abhimanyu Choudhury, Meena Mahajan

Dependency schemes in CDCL-based QBF solving: a proof-theoretic study

We formalize the notion of proof systems obtained by adding normal dependency schemes into the QCDCL proof system underlying algorithms for solving Quantified Boolean Formulas, by exploring the addition of the dependency schemes via two approaches: one as a preprocessing tool, and second in propagation and learnings in the QCDCL ... more >>>

TR23-060 | 17th April 2023
Sagnik Saha, Nikolaj Schwartzbach, Prashant Nalini Vasudevan

The Planted $k$-SUM Problem: Algorithms, Lower Bounds, Hardness Amplification, and Cryptography

In the average-case $k$-SUM problem, given $r$ integers chosen uniformly at random from $\{0,\ldots,M-1\}$, the objective is to find a set of $k$ numbers that sum to $0$ modulo $M$ (this set is called a ``solution''). In the related $k$-XOR problem, given $k$ uniformly random Boolean vectors of length $\log{M}$, ... more >>>

TR23-058 | 23rd April 2023
Xin Li, Yan Zhong

Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs

Affine extractors give some of the best-known lower bounds for various computational models, such as AC$^0$ circuits, parity decision trees, and general Boolean circuits. However, they are not known to give strong lower bounds for read-once branching programs (ROBPs). In a recent work, Gryaznov, Pudl\'{a}k, and Talebanfard (CCC' 22) introduced ... more >>>

TR23-057 | 27th April 2023
Iddo Tzameret, Luming Zhang

Stretching Demi-Bits and Nondeterministic-Secure Pseudorandomness

We develop the theory of cryptographic nondeterministic-secure pseudorandomness beyond the point reached by Rudich's original work (Rudich 1997), and apply it to draw new consequences in average-case complexity and proof complexity. Specifically, we show the following:

?*Demi-bit stretch*: Super-bits and demi-bits are variants of cryptographic pseudorandom generators which are ... more >>>

TR23-056 | 26th April 2023
Geoffrey Mon, Dana Moshkovitz, Justin Oh

Approximate Locally Decodable Codes with Constant Query Complexity and Nearly Optimal Rate

We present simple constructions of good approximate locally decodable codes (ALDCs) in the presence of a $\delta$-fraction of errors for $\delta<1/2$. In a standard locally decodable code $C \colon \Sigma_1^k \to \Sigma_2^n$, there is a decoder $M$ that on input $i \in [k]$ correctly outputs the $i$-th symbol of a ... more >>>

TR23-055 | 20th April 2023
Amey Bhangale, Subhash Khot, Dor Minzer

On Approximability of Satisfiable $k$-CSPs: II

Let $\Sigma$ be an alphabet and $\mu$ be a distribution on $\Sigma^k$ for some $k \geq 2$. Let $\alpha > 0$ be the minimum probability of a tuple in the support of $\mu$ (denoted by $supp(\mu)$). Here, the support of $\mu$ is the set of all tuples in $\Sigma^k$ that ... more >>>

TR23-054 | 20th April 2023
Amey Bhangale, Subhash Khot, Dor Minzer

On Approximability of Satisfiable $k$-CSPs: III

In this paper we study functions on the Boolean hypercube that have the property that after applying certain random restrictions, the restricted function is correlated to a linear function with non-negligible probability. If the given function is correlated with a linear function then this property clearly holds. Furthermore, the property ... more >>>

TR23-053 | 19th April 2023
Leroy Chew

Proof Simulation via Round-based Strategy Extraction for QBF

The proof complexity of Quantified Boolean Formulas (QBF) relates to both QBF solving and OBF certification. One method to p-simulate a QBF proof system is by formalising the soundness of its strategy extraction in propositional logic. In this work we illustrate how to use extended QBF Frege to simulate LD-Q(Drrs)-Res, ... more >>>

TR23-052 | 19th April 2023
Noah Fleming, Vijay Ganesh, Antonina Kolokolova, Chunxiao Li, Marc Vinyals

Limits of CDCL Learning via Merge Resolution

In their seminal work, Atserias et al. and independently Pipatsrisawat and Darwiche in 2009 showed that CDCL solvers can simulate resolution proofs with polynomial overhead. However, previous work does not address the tightness of the simulation, i.e., the question of how large this overhead needs to be. In this paper, ... more >>>

TR23-051 | 18th April 2023
Benjamin Böhm, Olaf Beyersdorff

QCDCL vs QBF Resolution: Further Insights

We continue the investigation on the relations of QCDCL and QBF resolution systems. In particular, we introduce QCDCL versions that tightly characterise QU-Resolution and (a slight variant of) long-distance Q-Resolution. We show that most QCDCL variants - parameterised by different policies for decisions, unit propagations and reductions -- lead to ... more >>>

TR23-050 | 18th April 2023
Manasseh Ahmed, TsunMing Cheung, Hamed Hatami, Kusha Sareen

Communication complexity of half-plane membership

We study the randomized communication complexity of the following problem. Alice receives the integer coordinates of a point in the plane, and Bob receives the integer parameters of a half-plane, and their goal is to determine whether Alice's point belongs to Bob's half-plane.

This communication task corresponds to determining ... more >>>

TR23-049 | 17th April 2023
Mika Göös, Artur Riazanov, Anastasia Sofronova, Dmitry Sokolov

Top-Down Lower Bounds for Depth-Four Circuits

We present a top-down lower-bound method for depth-$4$ boolean circuits. In particular, we give a new proof of the well-known result that the parity function requires depth-$4$ circuits of size exponential in $n^{1/3}$. Our proof is an application of robust sunflowers and block unpredictability.

more >>>

TR23-048 | 4th April 2023
Hadley Black, Deeparnab Chakrabarty, C. Seshadhri

A $d^{1/2+o(1)}$ Monotonicity Tester for Boolean Functions on $d$-Dimensional Hypergrids

Monotonicity testing of Boolean functions on the hypergrid, $f:[n]^d \to \{0,1\}$, is a classic topic in property testing. Determining the non-adaptive complexity of this problem is an important open question. For arbitrary $n$, [Black-Chakrabarty-Seshadhri, SODA 2020] describe a tester with query complexity $\widetilde{O}(\varepsilon^{-4/3}d^{5/6})$. This complexity is independent of $n$, but ... more >>>

TR23-047 | 2nd April 2023
Hunter Monroe

Ruling Out Short Proofs of Unprovable Sentences is Hard

If no optimal propositional proof system exists, we (and independently Pudlák) prove that ruling out length $t$ proofs of any unprovable sentence is hard. This mapping from unprovable to hard-to-prove sentences powerfully translates facts about noncomputability into complexity theory. For instance, because proving string $x$ is Kolmogorov random ($x{\in}R$) is ... more >>>

TR23-046 | 13th April 2023
Yizhi Huang, Rahul Ilango, Hanlin Ren

NP-Hardness of Approximating Meta-Complexity: A Cryptographic Approach

It is a long-standing open problem whether the Minimum Circuit Size Problem ($\mathrm{MCSP}$) and related meta-complexity problems are NP-complete. Even for the rare cases where the NP-hardness of meta-complexity problems are known, we only know very weak hardness of approximation.

In this work, we prove NP-hardness of approximating meta-complexity with ... more >>>

TR23-045 | 13th April 2023
Vinayak Kumar

Tight Correlation Bounds for Circuits Between AC0 and TC0

Revisions: 1

We initiate the study of generalized $AC^0$ circuits comprised of arbitrary unbounded fan-in gates which only need to be constant over inputs of Hamming weight $\ge k$ (up to negations of the input bits), which we denote $GC^0(k)$. The gate set of this class includes biased LTFs like the $k$-$OR$ ... more >>>

TR23-044 | 28th March 2023
Sourav Chakraborty, Chandrima Kayal, Manaswi Paraashar

Separations between Combinatorial Measures for Transitive Functions

The role of symmetry in Boolean functions $f:\{0,1\}^n \to \{0,1\}$ has been extensively studied in complexity theory.
For example, symmetric functions, that is, functions that are invariant under the action of $S_n$ is an important class of functions in the study of Boolean functions.
A function $f:\{0,1\}^n \to \{0,1\}$ ... more >>>

TR23-043 | 9th April 2023
Yotam Dikstein, Irit Dinur

Coboundary and cosystolic expansion without dependence on dimension or degree

We give new bounds on the cosystolic expansion constants of several families of high dimensional expanders, and the known coboundary expansion constants of order complexes of homogeneous geometric lattices, including the spherical building of $SL_n(F_q)$. The improvement applies to the high dimensional expanders constructed by Lubotzky, Samuels and Vishne, and ... more >>>

TR23-042 | 3rd April 2023
Johan Håstad

On small-depth Frege proofs for PHP

We study Frege proofs for the one-to-one graph Pigeon Hole Principle
defined on the $n\times n$ grid where $n$ is odd.
We are interested in the case where each formula
in the proof is a depth $d$ formula in the basis given by
$\land$, $\lor$, and $\neg$. We prove that ... more >>>

TR23-041 | 1st April 2023
Lila Fontes, Sophie Laplante, Mathieu Lauriere, Alexandre Nolin

The communication complexity of functions with large outputs

We study the two-party communication complexity of functions with large outputs, and show that the communication complexity can greatly vary depending on what output model is considered. We study a variety of output models, ranging from the open model, in which an external observer can compute the outcome, to the ... more >>>

TR23-040 | 28th March 2023
Edward Pyne, Ran Raz, Wei Zhan

Certified Hardness vs. Randomness for Log-Space

Let $\mathcal{L}$ be a language that can be decided in linear space and let $\epsilon >0$ be any constant. Let $\mathcal{A}$ be the exponential hardness assumption that for every $n$, membership in $\mathcal{L}$ for inputs of length~$n$ cannot be decided by circuits of size smaller than $2^{\epsilon n}$.
We ... more >>>

TR23-039 | 28th March 2023
Arkadev Chattopadhyay, Yogesh Dahiya, Meena Mahajan

Query Complexity of Search Problems

We relate various complexity measures like sensitivity, block sensitivity, certificate complexity for multi-output functions to the query complexities of such functions. Using these relations, we improve upon the known relationship between pseudo-deterministic query complexity and deterministic query complexity for total search problems: We show that pseudo-deterministic query complexity is at ... more >>>

TR23-038 | 28th March 2023
Rahul Ilango, Jiatu Li, Ryan Williams

Indistinguishability Obfuscation, Range Avoidance, and Bounded Arithmetic

The range avoidance problem (denoted by Avoid) asks to find a string outside of the range of a given circuit $C:\{0,1\}^n\to\{0,1\}^m$, where $m>n$. Although at least half of the strings of length $m$ are correct answers, it is not clear how to deterministically find one. Recent results of Korten (FOCS'21) ... more >>>

TR23-037 | 28th March 2023
Shuichi Hirahara

Capturing One-Way Functions via NP-Hardness of Meta-Complexity

A one-way function is a function that is easy to compute but hard to invert *on average*. We establish the first characterization of a one-way function by *worst-case* hardness assumptions, by introducing a natural meta-computational problem whose NP-hardness (and the worst-case hardness of NP) characterizes the existence of a one-way ... more >>>

TR23-036 | 27th March 2023
Dean Doron, Roei Tell

Derandomization with Minimal Memory Footprint

Existing proofs that deduce BPL=L from circuit lower bounds convert randomized algorithms into deterministic algorithms with large constant overhead in space. We study space-bounded derandomization with minimal footprint, and ask what is the minimal possible space overhead for derandomization.
We show that $BPSPACE[S] \subseteq DSPACE[c \cdot S]$ for $c \approx ... more >>>

TR23-035 | 22nd March 2023
Shuichi Hirahara, Rahul Ilango, Zhenjian Lu, Mikito Nanashima, Igor Carboni Oliveira

A Duality Between One-Way Functions and Average-Case Symmetry of Information

Symmetry of Information (SoI) is a fundamental property of Kolmogorov complexity that relates the complexity of a pair of strings and their conditional complexities. Understanding if this property holds in the time-bounded setting is a longstanding open problem. In the nineties, Longpré and Mocas (1993) and Longpré and Watanabe (1995) ... more >>>

TR23-034 | 24th March 2023
Oded Goldreich

On teaching the approximation method for circuit lower bounds

Revisions: 1

This text provides a basic presentation of the the approximation method of Razborov (Matematicheskie Zametki, 1987) and its application by Smolensky (19th STOC, 1987) for proving lower bounds on the size of ${\cal AC}^0[p]$-circuits that compute sums mod~$q$ (for primes $q\neq p$).
The textbook presentations of the latter result ... more >>>

TR23-033 | 24th March 2023
Sumanta Ghosh, Prahladh Harsha, Simao Herdade, Mrinal Kumar, Ramprasad Saptharishi

Fast Numerical Multivariate Multipoint Evaluation

Revisions: 1

We design nearly-linear time numerical algorithms for the problem of multivariate multipoint evaluation over the fields of rational, real and complex numbers. We consider both \emph{exact} and \emph{approximate} versions of the algorithm. The input to the algorithms are (1) coefficients of an $m$-variate polynomial $f$ with degree $d$ in each ... more >>>

TR23-032 | 24th March 2023
Vishwas Bhargava, Shubhangi Saraf, Ilya Volkovich

Linear Independence, Alternants and Applications

We develop a new technique for analyzing linear independence of multivariate polynomials. One of our main technical contributions is a \emph{Small Witness for Linear Independence} (SWLI) lemma which states the following.
If the polynomials $f_1,f_2, \ldots, f_k \in \F[X]$ over $X=\{x_1, \ldots, x_n\}$ are $\F$-linearly independent then there exists ... more >>>

TR23-031 | 23rd March 2023
Benny Applebaum, Eliran Kachlon, Arpita Patra

The Round Complexity of Statistical MPC with Optimal Resiliency

In STOC 1989, Rabin and Ben-Or (RB) established an important milestone in the fields of cryptography and distributed computing by showing that every functionality can be computed with statistical (information-theoretic) security in the presence of an active (aka Byzantine) rushing adversary that controls up to half of the parties. We ... more >>>

TR23-030 | 21st March 2023
Jan Krajicek

A proof complexity conjecture and the Incompleteness theorem

Given a sound first-order p-time theory $T$ capable of formalizing syntax of
first-order logic we define a p-time function $g_T$ that stretches all inputs by one
bit and we use its properties to show that $T$ must be incomplete. We leave it as an
open problem whether ... more >>>

TR23-029 | 18th March 2023
Nicollas Sdroievski, Dieter van Melkebeek

Instance-Wise Hardness versus Randomness Tradeoffs for Arthur-Merlin Protocols

A fundamental question in computational complexity asks whether probabilistic polynomial-time algorithms can be simulated deterministically with a small overhead in time (the BPP vs. P problem). A corresponding question in the realm of interactive proofs asks whether Arthur-Merlin protocols can be simulated nondeterministically with a small overhead in time (the ... more >>>

TR23-028 | 15th March 2023
Rahul Santhanam

An Algorithmic Approach to Uniform Lower Bounds

We propose a new family of circuit-based sampling tasks, such that non-trivial algorithmic solutions to certain tasks from this family imply frontier uniform lower bounds such as ``NP is not in uniform ACC^0" and ``NP does not have uniform polynomial-size depth-two threshold circuits". Indeed, the most general versions of our ... more >>>

TR23-027 | 8th March 2023
Joseph Zalewski

Some Lower Bounds Related to the Missing String Problem

We prove that a $O(k \log k)$-probe local algorithm for $k$-Missing String presented by Vyas and Williams is asymptotically optimal among a certain class of algorithms for this problem. The best lower bound we are aware of for the general case is still $\Omega(k)$.

more >>>

TR23-026 | 15th March 2023
Shuichi Hirahara, Nobutaka Shimizu

Hardness Self-Amplification: Simplified, Optimized, and Unified

Strong (resp. weak) average-case hardness refers to the properties of a computational problem in which a large (resp. small) fraction of instances are hard to solve. We develop a general framework for proving hardness self-amplification, that is, the equivalence between strong and weak average-case hardness. Using this framework, we prove ... more >>>

TR23-025 | 10th March 2023
Vikraman Arvind, Pushkar Joglekar

Multivariate to Bivariate Reduction for Noncommutative Polynomial Factorization

Based on a theorem of Bergman we show that multivariate noncommutative polynomial factorization is deterministic polynomial-time reducible to the factorization of bivariate noncommutative polynomials. More precisely, we show the following:

(1) In the white-box setting, given an n-variate noncommutative polynomial f in F over a field F (either a ... more >>>

TR23-024 | 9th March 2023
Mark Bun, Nadezhda Voronova

Approximate degree lower bounds for oracle identification problems

Revisions: 1

The approximate degree of a Boolean function is the minimum degree of real polynomial that approximates it pointwise. For any Boolean function, its approximate degree serves as a lower bound on its quantum query complexity, and generically lifts to a quantum communication lower bound for a related function.

We ... more >>>

TR23-023 | 13th March 2023
Xin Li

Two Source Extractors for Asymptotically Optimal Entropy, and (Many) More

Revisions: 1

A long line of work in the past two decades or so established close connections between several different pseudorandom objects and applications, including seeded or seedless non-malleable extractors, two source extractors, (bipartite) Ramsey graphs, privacy amplification protocols with an active adversary, non-malleable codes and many more. These connections essentially show ... more >>>

TR23-022 | 11th March 2023
Jiatu Li, Igor Carboni Oliveira

Unprovability of strong complexity lower bounds in bounded arithmetic

While there has been progress in establishing the unprovability of complexity statements in lower fragments of bounded arithmetic, understanding the limits of Jerabek's theory $\textbf{APC}_1$ (2007) and of higher levels of Buss's hierarchy $\textbf{S}^i_2$ (1986) has been a more elusive task. Even in the more restricted setting of Cook's theory ... more >>>

TR23-021 | 9th March 2023
Karthik Gajulapalli, Alexander Golovnev, Satyajeet Nagargoje, Sidhant Saraogi

Range Avoidance for Constant-Depth Circuits: Hardness and Algorithms

Range Avoidance (AVOID) is a total search problem where, given a Boolean circuit $C\colon\{0,1\}^n\to\{0,1\}^m$, $m>n$, the task is to find a $y\in\{0,1\}^m$ outside the range of $C$. For an integer $k\geq 2$, $NC^0_k$-AVOID is a special case of AVOID where each output bit of $C$ depends on at most $k$ ... more >>>

TR23-020 | 3rd March 2023
Scott Aaronson, Shih-Han Hung

Certified Randomness from Quantum Supremacy

We propose an application for near-term quantum devices: namely, generating cryptographically certified random bits, to use (for example) in proof-of-stake cryptocurrencies. Our protocol repurposes the existing "quantum supremacy" experiments, based on random circuit sampling, that Google and USTC have successfully carried out starting in 2019. We show that, whenever the ... more >>>

TR23-019 | 2nd March 2023
Pooya Hatami, William Hoza

Theory of Unconditional Pseudorandom Generators

Revisions: 2

This is a survey of unconditional *pseudorandom generators* (PRGs). A PRG uses a short, truly random seed to generate a long, "pseudorandom" sequence of bits. To be more specific, for each restricted model of computation (e.g., bounded-depth circuits or read-once branching programs), we would like to design a PRG that ... more >>>

TR23-018 | 1st March 2023
Qipeng Liu, Ran Raz, Wei Zhan

Memory-Sample Lower Bounds for Learning with Classical-Quantum Hybrid Memory

In a work by Raz (J. ACM and FOCS 16), it was proved that any algorithm for parity learning on $n$ bits requires either $\Omega(n^2)$ bits of classical memory or an exponential number (in~$n$) of random samples. A line of recent works continued that research direction and showed that for ... more >>>

TR23-017 | 21st February 2023
Deepanshu Kush, Shubhangi Saraf

Near-Optimal Set-Multilinear Formula Lower Bounds

The seminal work of Raz (J. ACM 2013) as well as the recent breakthrough results by Limaye, Srinivasan, and Tavenas (FOCS 2021, STOC 2022) have demonstrated a potential avenue for obtaining lower bounds for general algebraic formulas, via strong enough lower bounds for set-multilinear formulas.

In this paper, we make ... more >>>

TR23-016 | 22nd February 2023
Yuval Filmus, Edward Hirsch, Artur Riazanov, Alexander Smal, Marc Vinyals

Proving Unsatisfiability with Hitting Formulas

Hitting formulas have been studied in many different contexts at least since [Iwama 1989]. A hitting formula is a set of Boolean clauses such that any two of the clauses cannot be simultaneously falsified. [Peitl and Szeider 2022] conjectured that the family of unsatisfiable hitting formulas should contain the hardest ... more >>>

TR23-015 | 20th February 2023
Scott Aaronson, Harry Buhrman, William Kretschmer

A Qubit, a Coin, and an Advice String Walk Into a Relational Problem

Relational problems (those with many possible valid outputs) are different from decision problems, but it is easy to forget just how different. This paper initiates the study of FBQP/qpoly, the class of relational problems solvable in quantum polynomial-time with the help of polynomial-sized quantum advice, along with its analogues for ... more >>>

TR23-014 | 16th February 2023
Tameem Choudhury, Karteek Sreenivasaiah

Depth-3 Circuit Lower Bounds for k-OV

The 2-Orthogonal Vectors (2-OV) problem is the following: given two tuples $A$ and $B$ of $n$ vectors each of dimension $d$, decide if there exists a vector $u\in A$, and $v\in B$ such that $u$ and $v$ are orthogonal. This problem, and its generalization $k$-OV defined analogously for $k$ tuples, ... more >>>

TR23-013 | 7th February 2023
Noam Mazor

A Lower Bound on the Share Size in Evolving Secret Sharing

Revisions: 1

Secret sharing schemes allow sharing a secret between a set of parties in a way that ensures that only authorized subsets of the parties learn the secret. Evolving secret sharing schemes (Komargodski, Naor, and Yogev [TCC ’16]) allow achieving this end in a scenario where the parties arrive in an ... more >>>

TR23-012 | 16th February 2023
Yogesh Dahiya, Vignesh K, Meena Mahajan, Karteek Sreenivasaiah

Linear threshold functions in decision lists, decision trees, and depth-2 circuits

We show that polynomial-size constant-rank linear decision trees (LDTs) can be converted to polynomial-size depth-2 threshold circuits LTF$\circ$LTF. An intermediate construct is polynomial-size decision lists that query a conjunction of a constant number of linear threshold functions (LTFs); we show that these are equivalent to polynomial-size exact linear decision lists ... more >>>

TR23-011 | 13th February 2023
Mikhail Dektiarev, Nikolay Vereshchagin

Half-duplex communication complexity with adversary? can be less than the classical communication complexity

Half-duplex communication complexity with adversary was defined in [Hoover, K., Impagliazzo, R., Mihajlin, I., Smal, A. V. Half-Duplex Communication Complexity, ISAAC 2018.] Half-duplex communication protocols generalize classical protocols defined by Andrew Yao in [Yao, A. C.-C. Some Complexity Questions Related to Distributive Computing (Preliminary Report), STOC 1979]. It has been ... more >>>

TR23-010 | 13th February 2023
Per Austrin, Kilian Risse

Sum-of-Squares Lower Bounds for the Minimum Circuit Size Problem

We prove lower bounds for the Minimum Circuit Size Problem (MCSP) in the Sum-of-Squares (SoS) proof system. Our main result is that for every Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$, SoS requires degree $\Omega(s^{1-\epsilon})$ to prove that $f$ does not have circuits of size $s$ (for any $s > \text{poly}(n)$). ... more >>>

TR23-009 | 14th February 2023
Hervé Fournier, Nutan Limaye, Guillaume Malod, Srikanth Srinivasan, Sébastien Tavenas

Towards Optimal Depth-Reductions for Algebraic Formulas

Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth O(log s) with only a polynomial blow-up in size. In this paper, we consider a fine-grained version of this result ... more >>>

TR23-008 | 2nd February 2023
Ond?ej Ježil

Limits of structures and Total NP Search Problems

For a class of finite graphs, we define a limit object relative to some computationally restricted class of functions. The properties of the limit object then reflect how a computationally restricted viewer "sees" a generic instance from the class. The construction uses Krají?ek's forcing with random variables [7]. We prove ... more >>>

TR23-007 | 3rd February 2023
Jan Krajicek

Extended Nullstellensatz proof systems

For a finite set $\cal F$ of polynomials over a fixed finite prime field of size $p$ containing all polynomials $x^2 - x$, a Nullstellensatz proof of the unsolvability of the system
f = 0\ ,\ \mbox{ all } f \in {\cal F}
is a linear combination ... more >>>

TR23-006 | 20th January 2023
Nader Bshouty

Superpolynomial Lower Bounds for Learning Monotone Classes

Revisions: 1

Koch, Strassle, and Tan [SODA 2023], show that, under the randomized exponential time hypothesis, there is no distribution-free PAC-learning algorithm that runs in time $n^{\tilde O(\log\log s)}$ for the classes of $n$-variable size-$s$ DNF, size-$s$ Decision Tree, and $\log s$-Junta by DNF (that returns a DNF hypothesis). Assuming a natural ... more >>>

TR23-005 | 13th January 2023
Paul Beame, Niels Kornerup

Cumulative Memory Lower Bounds for Randomized and Quantum Computation

Cumulative memory---the sum of space used over the steps of a computation---is a fine-grained measure of time-space complexity that is a more accurate measure of cost for algorithms with infrequent spikes in memory usage in the context of technologies such as cloud computing that allow dynamic allocation and de-allocation of ... more >>>

TR23-004 | 13th January 2023
Yinan Li, Youming Qiao, Avi Wigderson, Yuval Wigderson, Chuanqi Zhang

On linear-algebraic notions of expansion

A fundamental fact about bounded-degree graph expanders is that three notions of expansion---vertex expansion, edge expansion, and spectral expansion---are all equivalent. In this paper, we study to what extent such a statement is true for linear-algebraic notions of expansion.

There are two well-studied notions of linear-algebraic expansion, namely dimension expansion ... more >>>

TR23-003 | 11th January 2023
Shachar Lovett, Jiapeng Zhang

Streaming Lower Bounds and Asymmetric Set-Disjointness

Frequency estimation in data streams is one of the classical problems in streaming algorithms. Following much research, there are now almost matching upper and lower bounds for the trade-off needed between the number of samples and the space complexity of the algorithm, when the data streams are adversarial. However, in ... more >>>

TR23-002 | 5th January 2023
Noga Alon, Olivier Bousquet, Kasper Green Larsen, Shay Moran, Shlomo Moran

Diagonalization Games

Revisions: 2

We study several variants of a combinatorial game which is based on Cantor's diagonal argument. The game is between two players called Kronecker and Cantor. The names of the players are motivated by the known fact that Leopold Kronecker did not appreciate Georg Cantor's arguments about the infinite, and even ... more >>>

TR23-001 | 5th January 2023
Prerona Chatterjee, Pavel Hrubes

New Lower Bounds against Homogeneous Non-Commutative Circuits

We give several new lower bounds on size of homogeneous non-commutative circuits. We present an explicit homogeneous bivariate polynomial of degree $d$ which requires homogeneous non-commutative circuit of size $\Omega(d/\log d)$. For an $n$-variate polynomial with $n>1$, the result can be improved to $\Omega(nd)$, if $d\leq n$, or $\Omega(nd \frac{\log ... more >>>

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