Determine whether the binary optimization problem can be solved through its continuous relaxation

Determine whether, when all objective-function coefficients are distinct, the optimal solution of the continuous relaxation of the matrix-form binary optimization problem (equation (19), denoted \(\ref{eq:MinOptMvMatrix}\)) with positive real variables always coincides with the optimal solution of the original binary optimization problem.

Background

The paper formulates the construction of a combinatorial multivector field as a binary linear optimization problem. Because optimization with binary variables is NP-hard, the authors discuss relaxing the binary variables to positive real variables and solving the resulting continuous linear program before attempting to recover an integer solution.

Although the constraint matrix is not totally unimodular, the authors report that their experiments consistently produced integer solutions after the relaxation step. They therefore explicitly state a conjecture asserting that distinct objective coefficients may guarantee agreement between the relaxed and binary optima. The conjecture remains unresolved in the paper and is presented as a condition under which the computational procedure could become substantially more efficient.