计算复杂性

The Quantum k-SAT problem is the quantum generalization of the k-SAT problem. It is the problem whether a given local Hamiltonian is frustration-free. Frustration-free means that the ground state of the k-local Hamiltonian minimizes the…

In this paper, we provide a uniform framework for investigating small circuit classes and bounds through the lens of ordinary differential equations (ODEs). Following an approach recently introduced to capture the class of polynomial-time…

We show that the GCD of two univariate polynomials can be computed by (piece-wise) algebraic circuits of constant depth and polynomial size over any sufficiently large field, regardless of the characteristic. This extends a recent result of…

We show that algebraic formulas and constant-depth circuits are closed under taking factors. In other words, we show that if a multivariate polynomial over a field of characteristic zero has a small constant-depth circuit or formula, then…

We investigate the computational complexity of the timed prediction problem in two-dimensional sandpile models. This question refines the classical prediction problem, which asks whether a cell q will eventually become unstable after adding…

We introduce a new concept of approximation applicable to decision problems and functions, inspired by Bayesian probability. From the perspective of a Bayesian reasoner with limited computational resources, the answer to a problem that…

We propose the Transcendental Encoding Conjecture for decision problems, which asserts that every language in complexity class P encodes to an algebraic real (possibly rational or algebraic irrational) under its binary characteristic…

We give simple deterministic reductions demonstrating the NP-hardness of approximating the nearest codeword problem and minimum distance problem within arbitrary constant factors (and almost-polynomial factors assuming NP cannot be solved…

Finding sparse vectors is a fundamental problem that arises in several contexts including codes, subspaces, and lattices. In this work, we prove strong inapproximability results for all these variants using a novel approach that even…

To understand and engineer biological and artificial nucleic acid systems, algorithms are employed for prediction of secondary structures at thermodynamic equilibrium. Dynamic programming algorithms are used to compute the most favoured, or…

Polynomial factorization is a fundamental problem in computational algebra. Over the past half century, a variety of algorithmic techniques have been developed to tackle different variants of this problem. In parallel, algebraic complexity…

We show that the shortest $s$-$t$ path problem has the overlap-gap property in (i) sparse $\mathbf{G}(n,p)$ graphs and (ii) complete graphs with i.i.d. Exponential edge weights. Furthermore, we demonstrate that in sparse $\mathbf{G}(n,p)$…

We present a simple, self-contained, linear reduction from 3-SAT to Treewidth. Specifically, it shows that 1.00005-approximating Treewidth is NP-hard, and solving Treewidth exactly requires $2^{\Omega(n)}$ time, unless the Exponential-Time…

The low-rank matrix completion problem asks whether a given real matrix with missing values can be completed so that the resulting matrix has low rank or is close to a low-rank matrix. The completed matrix is often required to satisfy…

For a given graph $G = (V, E)$, a subset of the vertices $D\subseteq V$ is called a semitotal dominating set, if $D$ is a dominating set and every vertex $v \in D$ is within distance two to another witness $v' \in D$. We want to find a…

We introduce a technically and conceptually simple approach to magnification of circuit and formula lower bounds. Central to the method are so-called distinguishers, sparse matrices that retain some of the key properties of error-correcting…

Lower bounds against strong algebraic proof systems and specifically fragments of the Ideal Proof System (IPS), have been obtained in an ongoing line of work. All of these bounds, however, are proved only over large (or characteristic $0$)…

The current paper investigates the bounded distance decoding (BDD) problem for ensembles of lattices whose generator matrices have sub-Gaussian entries. We first prove that, for these ensembles the BDD problem is NP-hard in the worst case.…

In this work, we prove upper and lower bounds over fields of positive characteristics for several fragments of the Ideal Proof System (IPS), an algebraic proof system introduced by Grochow and Pitassi (J. ACM 2018). Our results extend the…

Williams (STOC 2025) recently proved that time-$t$ multitape Turing machines can be simulated using $O(\sqrt{t \log t})$ space using the Cook-Mertz (STOC 2024) tree evaluation procedure. As Williams notes, applying this result to fast…