Polynomial Upper Bounds on the Number of Differing Columns of Δ-Modular Integer Programs
Bibliographic record
Abstract
We study integer-valued matrices with bounded determinants. Such matrices appear in the theory of integer programs (IPs) with bounded determinants. For example, an IP can be solved in strongly polynomial time if the constraint matrix is bimodular: that is, the determinants are bounded in absolute value by two. Determinants are also used to bound the [Formula: see text] distance between IP solutions and solutions of its linear relaxation. One of the first to quantify the complexity of IPs with bounded determinants was Heller, who identified the maximum number of differing columns in a totally unimodular matrix. Each extension of Heller’s bound to general determinants has been superpolynomial in the determinants or the number of equations. We provide the first column bound that is polynomial in both values. For integer programs with box constraints, our result gives the first [Formula: see text] distance bound that is polynomial in the determinants and the number of equations. Our result can also be used to derive a bound on the height of Graver basis elements that is polynomial in the determinants and the number of equations. Furthermore, we show a tight bound on the number of differing columns in a bimodular matrix; this is the first tight bound since Heller. Our analysis reveals combinatorial properties of bimodular IPs that may be of independent interest. Funding: J. Lee was supported in part by the Office of Naval Research [Grant N00014-21-1-2135] and the Air Force Office of Scientific Research [Grant FA9550-19-1-0175]. J. Paat was supported by a Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant [Grant RGPIN-2021-02475].
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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 | 0.003 | 0.023 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.004 | 0.008 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.001 | 0.006 |
| Insufficient payload (model declined to judge) | 0.011 | 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".