Applied Hilbert's Nullstellensatz for Combinatorial Problems
Bibliographic record
Abstract
Various feasibility problems in Combinatorial Optimization can be stated using systems of polynomial equations. Determining the existence of a \\textit{stable set} of a given size, finding the \\textit{chromatic number} of a graph or more generally, determining the feasibility of an \\textit{Integer Programming problem} are classical examples of this. In this thesis we study a powerful tool from Algebraic Geometry, called \\textit{Hilbert's Nullstellensatz}. It characterizes the \\textit{infeasibility} of a system of polynomial equations by the \\textit{feasibility} of a possibly very large system of \\textit{linear equations}. The solutions to this linear system provide \\textit{certificates} for the infeasibility of the polynomial system, called \\textit{Nullstellensatz Certificates}. \n \nIn this thesis we focus on the study of Nullstellensatz Certificates for the existence of \\textit{proper colorings} of graphs. We use basic ideas from \\textit{duality theory} to determine various properties of the Nullstellensatz Certificates. We give new proofs to several known results in the current literature and present some new results that shed some light on the relationship between the sparsity of a graph and the \\textit{size} of the Nullstellensatz Certificates for \\textit{$k$-colorability}.
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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 | 0.006 | 0.021 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.003 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.010 |
| Scholarly communication | 0.004 | 0.008 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.002 | 0.009 |
| Insufficient payload (model declined to judge) | 0.010 | 0.001 |
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".