Generate all solutions that sum up to one

In a post, the following question was posed:

We can select unique values  \(\displaystyle\frac{1}{i}\) for \(i=1,\dots,n\). Find all combinations that add up to 1.

A complete enumeration scheme was slow even for \(n=10\). Can we use a MIP model for this or something related?

A single solution is easily found using the model:

Mathematical Model
\[ \begin{align} & \sum_{i=1}^n \frac{1}{i} \cdot \color{darkred}x_i = 1 \\ & \color{darkred}x_i \in \{0,1\} \end{align}\]

Finding common patterns

In [1], the following problem is stated:

Given a boolean matrix, with \(m\) rows and \(n\) columns, find the largest pattern of ones that is found in at least \(\color{darkblue}K\) rows. We can ignore cells where the pattern has a zero value: they don't count.

A small example [1] is given: 

row 1:[0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1]
row 2:[0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1]
row 3:[0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1]
row 4:[1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1]
row 5:[1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1]
row 6:[1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1]

 With \(K=3\), we can form a pattern with 10 nonzero elements:

 

row 1:  [0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1]
row 2:  [0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1]
row 3:  [0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1]
row 4:  [1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1]
row 5:  [1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1]
row 6:  [1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1] 
pattern:[1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1]

The pattern is shared by rows 4, 5, and 6.

Can we formulate a MIP model for this? My first attempt is as follows.

Solving as network with lowerbounds

In [1], we looked at the following problem:

Mathematical Model
\[ \begin{align} \min& \sum_{i,j} \color{darkblue}a_{i,j} \cdot \color{darkred}x_{i,j} \\ & \sum_j \color{darkblue}a_{i,j}\cdot \color{darkred}x_{i,j} \ge \color{darkblue}r_i && \forall i \\ & \sum_i \color{darkblue}a_{i,j}\cdot \color{darkred}x_{i,j} \ge \color{darkblue}c_j && \forall j \\ & \color{darkred}x_{i,j} \in \{0,1\} \end{align}\]