Tight (S)ETH-based Lower Bounds for Pseudopolynomial Algorithms for Bin Packing and Multi-Machine Scheduling

2026-03-13

AI Summary

This work establishes tight conditional lower bounds for the bin packing and multiprocessor scheduling problems in the pseudo-polynomial time regime, addressing a long-standing gap in the literature. Under the Exponential Time Hypothesis (ETH) and the Strong Exponential Time Hypothesis (SETH), we present fine-grained reductions and parameterized complexity analyses that yield, for the first time, an ETH-based lower bound of $2^{o(n)} T^{o(k)}$ for bin packing. Furthermore, we derive SETH-based lower bounds for several classical multiprocessor scheduling problems that precisely match the running times of dynamic programming algorithms developed in the 1960s–70s. These results resolve open questions posed by Jansen et al. and by Fischer and Wennmann, effectively ruling out the existence of substantially faster pseudo-polynomial-time algorithms for these fundamental problems.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Bin Packing with $k$ bins is a fundamental optimisation problem in which we are given a set of $n$ integers and a capacity $T$ and the goal is to partition the set into $k$ subsets, each of total sum at most $T$. Bin Packing is NP-hard already for $k=2$ and a textbook dynamic programming algorithm solves it in pseudopolynomial time $\mathcal O(n T^{k-1})$. Jansen, Kratsch, Marx, and Schlotter [JCSS'13] proved that this time cannot be improved to $(nT)^{o(k / \log k)}$ assuming the Exponential Time Hypothesis (ETH). Their result has become an important building block, explaining the hardness of many problems in parameterised complexity. Note that their result is one log-factor short of being tight. In this paper, we prove a tight ETH-based lower bound for Bin Packing, ruling out time $2^{o(n)} T^{o(k)}$. This answers an open problem of Jansen et al. and yields improved lower bounds for many applications in parameterised complexity. Since Bin Packing is an example of multi-machine scheduling, it is natural to next study other scheduling problems. We prove tight lower bounds based on the Strong Exponential Time Hypothesis (SETH) for several classic $k$-machine scheduling problems, including makespan minimisation with release dates ($P_k|r_j|C_{\max}$), minimizing the number of tardy jobs ($P_k||ΣU_j$), and minimizing the weighted sum of completion times ($P_k || Σw_j C_j$). For all these problems, we rule out time $2^{o(n)} T^{k-1-\varepsilon}$ for any $\varepsilon > 0$ assuming SETH, where $T$ is the total processing time; this matches classic $n^{\mathcal O(1)} T^{k-1}$-time algorithms from the 60s and 70s. Moreover, we rule out time $2^{o(n)} T^{k-\varepsilon}$ for minimizing the total processing time of tardy jobs ($P_k||Σp_jU_j$), which matches a classic $\mathcal O(n T^{k})$-time algorithm and answers an open problem of Fischer and Wennmann [TheoretiCS'25].

Problem

• Bin Packing
• Multi-Machine Scheduling
• Pseudopolynomial Algorithms
• Exponential Time Hypothesis
• Lower Bounds

Innovation

• ETH
• SETH
• Bin Packing
• Multi-machine Scheduling
• Tight Lower Bounds