Reducing
Cover the Zeros
Covering zeros is the test step. If all zeros can be covered by fewer than n rows and columns, the current zeros do not yet contain a complete assignment. The diagram marks the cover on headers so the lines are visible even when the covered body cells are zeros, making the certificate readable.
Fewest cover lines
The fewest cover uses row W2 and column J2. Why: the covered row and column headers show which lines cover all current zeros. This is a structural test, not a cost calculation: it asks whether the zero pattern is rich enough to contain a complete assignment.
Not enough lines
There are 2 lines, which is less than n=3. Why: fewer than n lines means these zeros do not yet contain a full assignment. If a perfect zero assignment existed, covering its independent zeros would require a full set of assignment positions, so the short cover is a certificate that adjustment is still needed.
Minimum uncovered
The minimum uncovered cells have value 1. Why: this is the amount used to create more zeros. It is the smallest safe push: subtracting more would skip over information, while subtracting this amount creates at least one new opportunity without changing the optimal assignment.
Diagram note
Legend: zero body cells, minimum-uncovered cells, and covering-line headers are marked by role. The cover count is the certificate of whether the current zeros are sufficient; here it says the matrix must be adjusted before selecting an assignment. Pixel positions are rounded for layout; every number shown is exact.