arXiv subjects

Magnus Rahbek Dalgaard Hansen

Publications and source records attributed to Magnus Rahbek Dalgaard Hansen.

3 recordsLinked to original sources

arXiv2026

Tight Lower Bounds for Algebraic Communication and Applications

Communication complexity studies how much information must be exchanged to solve a problem whose input is split among several parties. The classical setting deals with Boolean inputs split between two parties. We study an algebraic variant, where the inputs are vectors over a field $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$. Alice and Bob have inputs $X\in \mathbb{F}^n$ and $Y\in \mathbb{F}^n$, respectively. We consider two kinds of tasks: the polynomial evaluation problem (compute the value of a polynomial $g\in \mathbb{F}[X,Y]$), and the set-recognition problem (decide whether (X,Y) is in $S$, for $S\subseteq \mathbb{F}^{n} \times \mathbb{F}^n$). In both settings, Alice and Bob send evaluations of polynomials depending only on their own inputs. In the set-recognition problem, a referee receives the messages and may apply polynomial tests to the messages received so far; the outcomes of these tests determine acceptance or rejection. The protocols may be deterministic or probabilistic. We study: - Upper bounds and reductions: We give non-trivial upper bounds for a range of natural polynomial evaluation and set-recognition problems and prove reductions between different problems, which help organize the landscape of the model. - A lower bound framework and tight lower bounds: Our main technical contribution is a general framework for proving lower bounds for algebraic set-recognition problems. We prove several probabilistic lower bounds for natural problems, giving tight or near-tight characterizations of their algebraic communication. - Applications of the framework: Finally, we give two applications of our framework: proving lower bounds for a class of left-to-right algebraic algorithms (algebraic scanners) and a more general algebraic computational setting inspired by the BSS model.

cs.CC↗

arXiv2026

Separation Results for Constant-Depth and Multilinear Ideal Proof Systems

In this work, we establish separation theorems for several subsystems of the Ideal Proof System (IPS), an algebraic proof system introduced by Grochow and Pitassi (J. ACM, 2018). Separation theorems are well-studied in the context of classical complexity theory, Boolean circuit complexity, and algebraic complexity. In an important work of Forbes, Shpilka, Tzameret, and Wigderson (ToC, 2021), two proof techniques were introduced to prove lower bounds for subsystems of the IPS, namely the functional method and the multiples method. We use these techniques and obtain the following results. Hierarchy theorem for constant-depth IPS: Recently, Limaye, Srinivasan, and Tavenas (J. ACM 2025) proved a hierarchy theorem for constant-depth algebraic circuits. We adapt the result and prove a hierarchy theorem for constant-depth $\mathsf{IPS}$. We show that there is an unsatisfiable multilinear instance refutable by a depth-$Δ$ $\mathsf{IPS}$ such that any depth-($Δ/10)$ $\mathsf{IPS}$ refutation for it must have superpolynomial size. This result is proved by building on the multiples method. Separation theorems for multilinear IPS: In an influential work, Raz (ToC, 2006) unconditionally separated two algebraic complexity classes, namely multilinear $\mathsf{NC}^{1}$ from multilinear $\mathsf{NC}^{2}$. In this work, we prove a similar result for a well-studied fragment of multilinear-$\mathsf{IPS}$. Specifically, we present an unsatisfiable instance such that its functional refutation, i.e., the unique multilinear polynomial agreeing with the inverse of the polynomial over the Boolean cube, has a small multilinear-$\mathsf{NC}^{2}$ circuit. However, any multilinear-$\mathsf{NC}^{1}$ $\mathsf{IPS}$ refutation ($\mathsf{IPS}_{\mathsf{LIN}}$) for it must have superpolynomial size. This result is proved by building on the functional method.

cs.CC↗

arXiv2025

On Closure Properties of Read-Once Oblivious Algebraic Branching Programs

We investigate the closure properties of read-once oblivious Algebraic Branching Programs (roABPs) under various natural algebraic operations and prove the following. - Non-closure under factoring: There is a sequence of explicit polynomials $(f_n(x_1,\ldots, x_n))_n$ that have $\mathsf{poly}(n)$-sized roABPs such that some irreducible factor of $f_n$ does not have roABPs of superpolynomial size in any order. - Non-closure under powering: There is a sequence of polynomials $(f_n(x_1,\ldots, x_n))_n$ with $\mathsf{poly}(n)$-sized roABPs such that any super-constant power of $f_n$ does not have roABPs of polynomial size in any order (and $f_n^n$ requires exponential size in any order). - Non-closure under symmetric compositions: There are symmetric polynomials $(f_n(e_1,\ldots, e_n))_n$ that have roABPs of polynomial size such that $f_n(x_1,\ldots, x_n)$ do not have roABPs of subexponential size. (Here, $e_1,\ldots, e_n$ denote the elementary symmetric polynomials in $n$ variables.) These results should be viewed in light of known results on models such as algebraic circuits, (general) algebraic branching programs, formulas and constant-depth circuits, all of which are known to be closed under these operations. To prove non-closure under factoring, we construct hard polynomials based on expander graphs using gadgets that lift their hardness from sparse polynomials to roABPs. For symmetric compositions, we show that the circulant polynomial requires roABPs of exponential size in every variable order.

cs.CC↗