🤖 AI Summary
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).
📝 Abstract
In restricted assignment - makespan minimization where each job has one size and a set of allowed machines - the configuration LP is the tightest studied relaxation, and its integrality gap is open in general. On two-weight graph balancing - each job allowed on at most two machines, sizes from two values - the value is known, both bounds due to Jansen, Land, and Maack (2016): their Table 1 instance attains 3/2, and their Corollary 11 bound of 2 - s/b for sizes s < b meets it at {1,2}. We ask how small such an instance - a witness - can be. We give I*, a six-job witness: the complete graph on four machines, unit jobs on a Hamiltonian cycle, weight-2 jobs on the complementary perfect matching, with integral optimum 3 against relaxation value 2. That is one job fewer than the smallest previously in print, and we prove it minimum and unique at its size. No instance of the class with at most five jobs reaches gap 3/2, on any number of machines; at six jobs, again on any number of machines, I* is the only witness, up to relabeling machines and adding machines no job can use. At seven jobs uniqueness fails: exactly thirteen witnesses, classified - the Jansen-Land-Maack instance among them - and at eight jobs exactly 154. Three machines never suffice, at any size: four are necessary for the gap. Results of this shape are in print for the same relaxation in one-dimensional cutting stock, where the extremal non-round-up instances have been enumerated and classified for small demand; the Discussion sets out the relation. Recognizing witnesses at relaxation value 2 - where all of ours live - is coNP-complete, so no min-max characterization exists unless NP = coNP. Every feasibility decision behind the exhaustive claims was made twice, in floating point and in exact rational arithmetic, with full agreement, and the pipeline must rediscover I* before its negatives are believed.