For a job assignment problem, total unimodularity would guarantee that the solution from a relaxed problem will be the same as the original integer problem. In my case, the data matrix is given and the orginal problem is stated as follows
$\begin{array}{l} \begin{array}{*{20}{c}} {{\rm{Min}}}&{13{x_{A1}} + 16{x_{A2}} + 15{x_{B1}} + 16{x_{B2}}} \end{array}\\ \begin{array}{*{20}{c}} {s.t}&{\left\{ {\begin{array}{*{20}{c}} {{x_{A1}} + {x_{A2}} = 1}\\ {{x_{B1}} + {x_{B2}} = 1} \end{array}} \right.}\\ {}&{\left\{ {\begin{array}{*{20}{c}} {{x_{A1}} + {x_{B1}} = 1}\\ {{x_{A2}} + {x_{B2}} = 1} \end{array}} \right.}\\ {}&{{x_{A1}},{x_{A2}},{x_{B1}},{x_{B2}} \in \left\{ {0,1} \right\}} \end{array} \end{array}$
The first set of constraints specifies that each assignee is to perform exactly one task. The second set requires each task to be performed by exactly one assignee.
Case 1:
If I change the first set of constraints to each assignee is to perform at most one task . And the second set of constraints to each task to be performed by at least one assignee, will total unimodularity still preserve ?
$\begin{array}{l} \begin{array}{*{20}{c}} {{\rm{Min}}}&{13{x_{A1}} + 16{x_{A2}} + 15{x_{B1}} + 16{x_{B2}}} \end{array}\\ \begin{array}{*{20}{c}} {s.t}&{\left\{ {\begin{array}{*{20}{c}} {{x_{A1}} + {x_{A2}} \le 1}\\ {{x_{B1}} + {x_{B2}} \le 1} \end{array}} \right.}\\ {}&{\left\{ {\begin{array}{*{20}{c}} {{x_{A1}} + {x_{B1}} \ge 1}\\ {{x_{A2}} + {x_{B2}} \ge 1} \end{array}} \right.}\\ {}&{{x_{A1}},{x_{A2}},{x_{B1}},{x_{B2}} \in \left\{ {0,1} \right\}} \end{array} \end{array}$
Case 2:
If I change the first set of constraints to each assignee is to perform at most one task and completely drop the second set of constraint, will total unimodularity still preserve ?
$\begin{array}{l} \begin{array}{*{20}{c}} {{\rm{Min}}}&{13{x_{A1}} + 16{x_{A2}} + 15{x_{B1}} + 16{x_{B2}}} \end{array}\\ \begin{array}{*{20}{c}} {s.t}&{\left\{ {\begin{array}{*{20}{c}} {{x_{A1}} + {x_{A2}} \le 1}\\ {{x_{B1}} + {x_{B2}} \le 1} \end{array}} \right.}\\ {}&{{x_{A1}},{x_{A2}},{x_{B1}},{x_{B2}} \in \left\{ {0,1} \right\}} \end{array} \end{array}$
Would you kindly help me with this ?
Thank you for your enthusiasm !
You can apply the property of total unimodularity if your matrix is totally unimodular and your right hand side is integer.
Changing equalities to inequalities does not change the integral property (By adding slack variables, if your matrix in your initial formulation was $A$, the matrix in standard form will be $[A\mid I]$, which is still totally unimodular).
Removing constraints keeps the integral property since any submatrix of a totally unimodular matrix is still totally unimodular.