191014K02 | Day 1 Lecture 2

191014K02: Day 1 Lecture 2

Integer Linear Programs

An integer linear program involves n variables x_1, x_2, \ldots, x_n \in \mathbb{Z} and a linear objective function to be optimized.

In particular, we would like to minimize or maximize a function that looks like: \sum_{i = 1}^n {\color{indianred}c_i} x_i,

subject to m linear inequalities:

\begin{aligned} a_1^1 x_1+a_2^1 x_2+ \cdots + a_i^1 x_i + \cdots+a_n^1 x_n & \leqslant b_1 \\ a_1^2 x_1+a_2^2 x_2+ \cdots + a_i^2 x_i + \cdots+a_n^2 x_n & \leqslant b_2 \\ \vdots & \\ a_1^j x_1+a_2^j x_2+ \cdots + a_i^j x_i + \cdots+a_n^3 x_n & \leqslant b_j\\ \vdots & \\ a_1^m x_1+a_2^m x_2+ \cdots + a_i^m x_i + \cdots+a_n^m x_n & \leqslant b_m. \end{aligned}

Here {\color{indianred}c_1,\cdots,c_n} are some constants in \mathbb{Z} or \mathbb{Q}

So given the a_i^j’s as input (1 \leqslant i \leqslant n; 1 \leqslant j \leqslant m), the goal is to set the x_i’s such that:

all the inequalities are satisfied, and
the objective function is optimized¹.

Footnotes

i.e, maximized or minimized↩︎