Exact and Approximate High-Multiplicity Scheduling on Identical Machines

TL;DR

This work advances exact high-multiplicity scheduling on identical machines by refining the Goemans–Rothvoss PQ-framework through three tools: (i) problem-specific preprocessing to shrink the encoding length, (ii) proximity-based coefficient reduction via an LP relaxation and Frank–Tardos, and (iii) new vertex-bound bounds for the integer hull that enable an alternative, vertex-count–driven algorithm. Together, these yield significantly faster runtimes for core problems like ${P||C_{\max}}$, improving dependencies from $\log(C_{\max})$ to $\log(p_{\max})$ in the exponent and removing dependence on large RHS values. The paper also establishes new parameterized lower bounds and shows that the MB18 question on FPT w.r.t. the number of job types is equivalent to a corresponding question for $Q||C_{\max}$ with job and machine types, linking the identical- and uniform-machines settings. Overall, the combination of preprocessing, proximity, and vertex-bound techniques broadens the applicability of PQ-representations to a wider class of scheduling problems with strong theoretical guarantees and practical potential.

Abstract

Goemans and Rothvoss (SODA'14) gave a framework for solving problems which can be described as finding a point in int$.$cone$(P\cap\mathbb{Z}^N)\cap Q$, where $P,Q\subset\mathbb{R}^N$ are (bounded) polyhedra. The running time for solving such a problem is $enc(P)^{2^{O(N)}}enc(Q)^{O(1)}$. This framework can be used to solve various scheduling problems, but the encoding length $enc(P)$ usually involves large parameters like the makespan. We describe three tools to improve the framework: - Problem-specific preprocessing can be used to greatly reduce $enc(P)$. - By solving a certain LP relaxation and then using the classical result by Frank and Tardos (J. Comb. '87), we get a more compact encoding of $P$ in general. - A result by Jansen and Klein (SODA'17) makes the running time depend on the number of vertices of the integer hull of $P$. We provide a new bound for this number that is similar to the one by Berndt et al. (SOSA'21) but better for our setting. For example, applied to the scheduling problem $P||C_{\max}$, these tools improve the running time from $(\log(C_{\max}))^{2^{O(d)}}enc(I)^{O(1)}$ to the possibly much better $(\log(p_{\max}))^{2^{O(d)}}enc(I)^{O(1)}$. Here, $p_{\max}$ is the largest processing time, $d$ is the number of different processing times, $C_{\max}$ is the makespan and $enc(I)$ is the encoding length of the instance. On the complexity side, we use reductions from the literature to provide new parameterized lower bounds for $P||C_{\max}$. Finally, we show that the big open question asked by Mnich and van Bevern (Comput. Oper. Res. '18) whether $P||C_{\max}$ is FPT w.r.t. the number of job types $d$ has the same answer as the question whether $Q||C_{\max}$ is FPT w.r.t. the number of job and machine types $d+τ$ (all in high-multiplicity encoding). The same holds for objective $C_{\min}$.

Figures (3)

• Figure 1: After computing the solution $x^*$ of the relaxation, $\max\{x^*-D,\mathbf{0}\}$ is taken as a fixed part of the solution $z^*$ and only the bounded rest $y$ has to be computed.
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Theorems & Definitions (112)

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