Bibliographic record
Abstract
Spectral sparsification for directed Eulerian graphs is a key component in the design of fast algorithms for solving directed Laplacian linear systems. Directed Laplacian linear system solvers are crucial algorithmic primitives to fast computation of fundamental problems on random walks, such as computing stationary distribution, hitting and commute time, and personalized PageRank vectors. While spectral sparsification is well understood for undirected graphs and it is known that for every graph $G,$ $(1+\varepsilon)$-sparsifiers with $O(n\varepsilon^{-2})$ edges exist [Batson-Spielman-Srivastava, STOC '09] (which is optimal), the best known constructions of Eulerian sparsifiers require $Ω(n\varepsilon^{-2}\log^4 n)$ edges and are based on short-cycle decompositions [Chu et al., FOCS '18]. In this paper, we give improved constructions of Eulerian sparsifiers, specifically: 1. We show that for every directed Eulerian graph $\vec{G},$ there exist an Eulerian sparsifier with $O(n\varepsilon^{-2} \log^2 n \log^2\log n + n\varepsilon^{-4/3}\log^{8/3} n)$ edges. This result is based on combining short-cycle decompositions [Chu-Gao-Peng-Sachdeva-Sawlani-Wang, FOCS '18, SICOMP] and [Parter-Yogev, ICALP '19], with recent progress on the matrix Spencer conjecture [Bansal-Meka-Jiang, STOC '23]. 2. We give an improved analysis of the constructions based on short-cycle decompositions, giving an $m^{1+δ}$-time algorithm for any constant $δ> 0$ for constructing Eulerian sparsifiers with $O(n\varepsilon^{-2}\log^3 n)$ edges.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.009 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".