arXiv subjects

Roman Edenhofer

Publications and source records attributed to Roman Edenhofer.

3 recordsLinked to original sources

arXiv2026

A Space-space Trade-off for Directed st-Connectivity

We prove a space-space trade-off for directed $st$-connectivity in the catalytic space model. For any integer $k \leq n$, we give an algorithm that decides directed $st$-connectivity using $O(\log n \cdot \log k+\log n)$ regular workspace and $O\left(\frac{n}{k} \cdot \log^2 n\right)$ bits of catalytic memory. This interpolates between the classical $O(\log^2 n)$-space bound from Savitch's algorithm and a catalytic endpoint with $O(\log n)$ workspace and $O(n\cdot \log^2 n)$ catalytic memory. As a warm-up, we present a catalytic variant of Savitch's algorithm achieving the endpoint above. Up to logarithmic factors, this matches the smallest catalyst size currently known for catalytic logspace algorithms, due to Cook and Pyne (ITCS 2026). Our techniques also extend to counting the number of walks from $s$ to $t$ of a given length $\ell\leq n$.

cs.DS↗

arXiv2025

Dequantization and Hardness of Spectral Sum Estimation

We give new dequantization and hardness results for estimating spectral sums of matrices, such as the log-determinant. Recent quantum algorithms have demonstrated that the logarithm of the determinant of sparse, well-conditioned, positive matrices can be approximated to $\varepsilon$-relative accuracy in time polylogarithmic in the dimension $N$, specifically in time $\poly(\log(N), s, \kappa, 1/\varepsilon)$, where $s$ is the sparsity and $\kappa$ the condition number of the input matrix. We provide a simple dequantization of these techniques that preserves the polylogarithmic dependence on the dimension. Our classical algorithm for the log-determinant runs in time $\polylog(N)\cdot s^{O(\sqrt{\kappa}\log(\kappa/\varepsilon))}$ which constitutes an exponential improvement over previous classical algorithms in certain parameter regimes. We complement our classical upper bounds with complexity-theoretic limitations. We prove that estimating normalized traces of polynomial powers and inverses of log-local Hamiltonians to inverse-polynomial additive accuracy is DQC1-complete, resolving an open problem of Cade and Montanaro (TQC 2018) concerning the complexity of Schatten-$p$ norm estimation. Finally, we prove a general PP-completeness result for unnormalized spectral sums: under mild polynomial-approximability and nondegeneracy assumptions on $f$, estimating $\mathrm{tr}[f(A)]$ to constant additive accuracy is PP-complete.

quant-ph↗

arXiv2024

Directed st-connectivity with few paths is in quantum logspace

We present a $\mathsf{BQSPACE}(O(\log n))$-procedure to count $st$-paths on directed graphs for which we are promised that there are at most polynomially many paths starting in $s$ and polynomially many paths ending in $t$. For comparison, the best known classical upper bound in this case just to decide $st$-connectivity is $\mathsf{DSPACE}(O(\log^2 n/ \log \log n))$. The result establishes a new relationship between~$\mathsf{BQL}$ and unambiguity and fewness subclasses of $\mathsf{NL}$. Further, we also show how to \emph{recognize} directed graphs with at most polynomially many paths between any two nodes in $\mathsf{BQSPACE}(O(\log n))$. This yields the first natural candidate for a language separating $\mathsf{BQL}$ from $\mathsf{L}$ and~$\mathsf{BPL}$. Until now, all candidates potentially separating these classes were inherently promise problems.

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