The Complexity of Symmetry Breaking in Massive Graphs
Bibliographic record
Abstract
The goal of this paper is to understand the complexity of symmetry breaking problems, specifically maximal independent set (MIS) and the closely related $β$-ruling set problem, in two computational models suited for large-scale graph processing, namely the $k$-machine model and the graph streaming model. We present a number of results. For MIS in the $k$-machine model, we improve the $\tilde{O}(m/k^2 + Δ/k)$-round upper bound of Klauck et al. (SODA 2015) by presenting an $\tilde{O}(m/k^2)$-round algorithm. We also present an $\tildeΩ(n/k^2)$ round lower bound for MIS, the first lower bound for a symmetry breaking problem in the $k$-machine model. For $β$-ruling sets, we use hierarchical sampling to obtain more efficient algorithms in the $k$-machine model and also in the graph streaming model. More specifically, we obtain a $k$-machine algorithm that runs in $\tilde{O}(βnΔ^{1/β}/k^2)$ rounds and, by using a similar hierarchical sampling technique, we obtain one-pass algorithms for both insertion-only and insertion-deletion streams that use $O(β\cdot n^{1+1/2^{β-1}})$ space. The latter result establishes a clear separation between MIS, which is known to require $Ω(n^2)$ space (Cormode et al., ICALP 2019), and $β$-ruling sets, even for $β= 2$. Finally, we present an even faster 2-ruling set algorithm in the $k$-machine model, one that runs in $\tilde{O}(n/k^{2-ε} + k^{1-ε})$ rounds for any $ε$, $0 \le ε\le 1$.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".