# DR-Submodularity on the Integer Lattice

## Related Groups

- dr-submodular
- mixed

## Type

- definition
- relation
- technique

## Statement

DR-submodularity tren integer lattice mo rong y tuong diminishing returns tu set functions sang vector / lattice domains. Trong boi canh nay, tang mot toa do khi vector con nho se co loi hon so voi khi vector da lon.

## Intuition

Day la cach tu duy "submodular but beyond sets". No giu duoc mot phan truc giac greedy / thresholding cua submodular cover, nhung yeu cau ky thuat moi de xu ly mien integer lattice va dependence theo range.

## Equivalent Views

- diminishing returns on `Z_+^E`
- lattice-domain extension of submodularity

## Standard Examples

- generalized submodular cover on integer lattice
- machine-learning models voi multiple levels / budgets / counts
- non-monotone DR-submodular maximization under size constraint

## Related Results

- Soma-Yoshida la paper moc trong inbox hien tai cho huong nay
- giup noi cover literature voi mo rong DR-submodular
- co the noi tiep voi fast non-monotone maximization qua khung coarse-routine + plus-version thresholding

## Where It Is Used

- `2015-soma-yoshida-generalized-submodular-cover-integer-lattice`
- `2026-tran-pham-fast-nonmonotone-dr-submodular-size`