An assignment problem with a wrinkle

In [1], the following problem is posed:

Consider two arrays \(\color{darkblue}a_i\) (length \(\color{darkblue}m\)) and \(\color{darkblue}b_j\) (length \(\color{darkblue}n\)) with \(\color{darkblue}m \lt \color{darkblue}n\). Assign all values \(\color{darkblue}a_i\) to a \(\color{darkblue}b_j\) such that:

• Each \(\color{darkblue}b_j\) can have 0 or 1 \(\color{darkblue}a_i\) assigned to it.
• The assignments need to maintain the original order of \(\color{darkblue}a_i\). I.e. if \(\color{darkblue}a_i \rightarrow \color{darkblue}b_j\) then \(\color{darkblue}a_{i+1}\) must be assigned to a slot in \(\color{darkblue}b\) that is beyond slot \(j\). In the picture below that means that arrows cannot cross.
• Do this while minimizing the sum of the products.

Mixed-integer programming model

This can be viewed as an assignment problem with a side constraint:

MIP Model
\[\begin{align}\min& \sum_{i,j}\color{darkred}x_{i,j}\cdot\color{darkblue}a_i\cdot\color{darkblue}b_j \\ &\sum_j \color{darkred}x_{i,j}=1 &&\forall i\\ & \sum_i \color{darkred}x_{i,j}\le 1 &&\forall j\\ & \color{darkred}v_i = \sum_j j \cdot \color{darkred} x_{i,j}\\ & \color{darkred}v_i \ge \color{darkred}v_{i-1}+1\\ & \color{darkred}x_{i,j} \in \{0,1\} \\ & \color{darkred}v_i \ge 1\end{align}\]

Here \(\color{darkred}x_{i,j}\) indicates the assignment \(\color{darkblue}a_i \rightarrow \color{darkblue}b_j\). The variable \(\color{darkred}v_i\) represents the position in \(\color{darkblue}b\) to which \(\color{darkblue}a_i\) is assigned.

The output of this model can look like: 

It is interesting to see what happens when we feed this model an even larger data set. I generated some random data: \[\begin{align} & \color{darkblue}m=100\\ &\color{darkblue}n=1000 \\ & \color{darkblue}a_i \sim U(0,100) \\ & \color{darkblue}b_j \sim U(0,100)\end{align}\] This is not a small problem: we will have 100,000 binary variables. This model has no longer a network structure, so compared to a pure assignment problem we see much poorer performance:

	Assignment problem as LP	Assignment problem as MIP	Full MIP model
Rows/Columns	1,100/100,000	1,100/100,000	1,300/100,100
Objective	11,524.178	11,524.178	14,371.455
Time (seconds)	1	7	1,853
Nodes	-	0	3,419
Iterations	2,672	2,727	30,7429

The assignment problem was formed by dropping the constraints involving \(\color{darkred}v_i\). Further, the problem was solved as an LP. It is noted that Cplex behaves differently when we use continuous variables \(\color{darkred}x_{i,j}\) or declare them as binary variables, When they are binary variables, the presolver works differently, and as a result, the solution of the relaxed problem is no longer optimal: it has to do some work to find the optimal integer solution. In this case this was done in MIP preprocessing: no branching was needed. When solved as a pure LP, the optimal solution is integer-valued automatically. There is a  difference in solution time: the LP solves in 1 second, while the MIP version takes 7 seconds.