Fast Makespan Minimization via Short ILPs

Short integer linear programs are programs with a relatively small number of constraints. We show how recent improvements on the running-times of solvers for such programs can be used to obtain fast pseudo-polynomial time algorithms for makespan minimization on a fixed number of parallel machines, and other related variants. The running times of our algorithms are all of the form $\widetilde{O}(p^{O(1)}{\max}+n)$ or $\widetilde{O}(p^{O(1)}{\max} \cdot n)$, where $p_{\max}$ is the maximum processing time in the input. These improve upon the time complexity of previously known algorithms for moderate values of $p_{\max}$.

💡 Research Summary

The paper “Fast Makespan Minimization via Short ILPs” investigates how recent advances in solving short integer linear programs (ILPs) can be leveraged to obtain fast pseudo‑polynomial time algorithms for makespan minimization on a fixed number of parallel machines and several related scheduling variants. A short ILP is defined as an ILP whose constraint matrix contains only a constant number of rows, independent of the input size. The authors build on the breakthrough of Jansen and Rohwedder (2019) who showed that an unbounded ILP with M constraints and maximal coefficient (a_{\max}) can be solved in (\widetilde O\big((\sqrt{M},a_{\max})^{2M}+MN\big)) time.

The classic ILP formulation for the problem (R_m|;C_{\max}) (non‑identical machines) uses a binary variable (x_{i,j}) for each job‑machine pair and contains (n+m) constraints, where the (n) constraints enforce that each job is assigned exactly once. Because the number of constraints depends on (n), this formulation is not short and cannot directly benefit from the short‑ILP solver.

The authors focus first on the related‑machines case (Q_m|;C_{\max}), where processing times are of the form (p_{i,j}=p_j/q_i). They replace the (n) assignment constraints by a single global equality (\sum_{i,j} x_{i,j}=n). For each machine (i) they introduce a slack variable (s_i) and write the load constraint as (\sum_j p_j x_{i,j}+s_i = q_i C), where (C) is the candidate makespan. The resulting ILP (called ILP 1) has only (m+1) constraints, and the largest absolute coefficient in the matrix is (p_{\max}) (the maximum processing time among all jobs). The objective function is a linear combination of job indices, designed so that any feasible solution with non‑negative objective corresponds to a schedule respecting the original assignment constraints. Because the objective coefficients do not affect the running time of the Jansen‑Rohwedder solver, ILP 1 can be solved as an unbounded ILP in (\widetilde O\big(p_{\max}^{2(m+1)}+n\big)) time. By binary searching over the possible makespan values in the interval (