A (1+epsilon)-Approximation for Makespan Scheduling with Precedence Constraints using LP Hierarchies

In a classical problem in scheduling, one has $n$ unit size jobs with a precedence order and the goal is to find a schedule of those jobs on $m$ identical machines as to minimize the makespan. It is one of the remaining four open problems from the book of Garey & Johnson whether or not this problem is $\mathbf{NP}$-hard for $m=3$. We prove that for any fixed $\varepsilon$ and $m$, an LP-hierarchy lift of the time-indexed LP with a slightly super poly-logarithmic number of $r = (\log(n))^{\Theta(\log \log n)}$ rounds provides a $(1 + \varepsilon)$-approximation. For example Sherali-Adams suffices as hierarchy. This implies an algorithm that yields a $(1+\varepsilon)$-approximation in time $n^{O(r)}$. The previously best approximation algorithms guarantee a $2 - \frac{7}{3m+1}$-approximation in polynomial time for $m \geq 4$ and $\frac{4}{3}$ for $m=3$. Our algorithm is based on a recursive scheduling approach where in each step we reduce the correlation in form of long chains. Our method adds to the rather short list of examples where hierarchies are actually useful to obtain better approximation algorithms.

💡 Research Summary

The paper addresses the classic scheduling problem P | prec, p_j = 1 | C_max: given n unit‑length jobs together with a partial order (precedence constraints), schedule them non‑preemptively on m identical machines so as to minimize the makespan (the completion time of the last job). This problem has been studied since Graham’s seminal 1966 paper, which gave a simple list‑scheduling algorithm with a 2 − 1/m approximation. Despite decades of work, the best known polynomial‑time guarantees are a 2 − 7/(3m + 1) approximation for m ≥ 4 and a 4/3 approximation for m = 3. Moreover, it remains an open question (one of the four listed by Garey & Johnson) whether the case m = 3 is NP‑hard at all, and whether a PTAS exists for any fixed m.

The authors show that, for any fixed ε > 0 and any fixed number of machines m, a modest number of rounds of a linear‑programming hierarchy suffices to obtain a (1 + ε)‑approximation. Specifically, they consider the natural time‑indexed LP with variables x_{j,t} indicating whether job j is scheduled at time slot t. This LP has an integrality gap of roughly 2 − 2/(m + 1). By applying the Sherali‑Adams (SA) hierarchy for r = (log n)^{Θ(log log n)} rounds—i.e., a “slightly super‑poly‑logarithmic” number of rounds—the gap collapses to 1 + ε. The same result holds for stronger hierarchies such as Lasserre, but the authors present an explicit SA formulation that is more accessible.

The SA system SA(K(T), r) is defined over partial assignments σ that fix the time slots of at most r jobs. For each σ a vector x_σ of dimension |J|·T is introduced, and three families of constraints are imposed: (I) consistency under extension of σ, (II) that each x_σ lies in the convex cone generated by feasible fractional solutions that respect the fixed assignments, and (III) that the “empty” assignment x_∅ belongs to the original LP polytope K(T). Lemma 2 establishes three crucial properties: (a) if the total mass λ_σ = ∑t x{σ,j,t} is positive then the normalized vector is a 0/1 solution respecting σ; (b) when r = n the whole collection corresponds to a convex combination of integral schedules; (c) any positive variable x_{j*,t*} can be fixed to 1, yielding a feasible SA solution with one fewer level.

The algorithm proceeds recursively on a binary partition of the time horizon. The interval