A minimum witness for the 3/2 configuration-linear-program gap in two-weight graph balancing, unique at its size
This study addresses the 3/2 integrality gap between the configuration linear programming (Config-LP) relaxation and integer optimal solutions in restricted assignment scheduling, specifically for instances where each job can be processed on at most two machines and only two distinct processing times exist. By employing graph-theoretic modeling, symmetry reduction, exhaustive search verified with both exact rational and floating-point arithmetic, and complexity analysis, the authors establish the existence of a unique minimal witness instance \( I^* \) with six jobs—improving upon the previously known seven-job example. They further prove that no instance with five or fewer jobs can achieve this gap and that at least four machines are necessary. Additionally, they provide a complete classification of all witness instances with seven jobs (13 in total) and eight jobs (154 in total).