[Rule] MinimumMatrixCover to ILP

[isPANN]

Motivation

Direct ILP formulation for MinimumMatrixCover. Companion issue for #931.

Source

MinimumMatrixCover

Target

ILP

Reference

Standard QBOP linearization for quadratic binary optimization.

Reduction Algorithm

Input: n×n nonneg matrix A.

1. Binary variables x_i ∈ {0, 1} for i = 1, ..., n (mapped to f(i) = 2x_i - 1).
2. Substitute f(i)f(j) = (2x_i-1)(2x_j-1) = 4x_ix_j - 2x_i - 2x_j + 1.
3. Linearize x_ix_j with standard McCormick: introduce y_{ij} = x_i x_j, add y_{ij} ≤ x_i, y_{ij} ≤ x_j, y_{ij} ≥ x_i + x_j - 1.
4. Objective: minimize the linearized quadratic form.

Size Overhead

Code metric	Formula
num_variables	n + n*(n-1)/2
num_constraints	3n(n-1)/2

Validation Method

Closed-loop test.

Example

Source: 2×2 matrix A = [[0,1],[1,0]].
Optimal: f=(+1,-1) or (-1,+1), value = -2. Min(-2).