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

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].

Figures (7)

• Figure 1: Summary of parameter-preserving reductions. Equivalent problems are highlighted in the same colour.
• Figure 2: Construction of the Multiset ${k}$-way Partition with targets instance in \ref{['lem:ETH_to_BPtargets']}. We represent the bit blocks (highest bits are on the left) of items $z(i, \alpha)$, $d(i, j, x)$, $d'(i, j, x)$ and target $t_\ell$, where $\ell(i) = \ell$, $\ell(j) = \ell'$ and $\ell" > \ell' > \ell$. The content of the communication channels is described in \ref{['fig:ETH_to_kBP_CC']}.
• Figure 3: The communication channel assigned to an external (left) and an internal (right) communication tuple $(C_i, C_j, x)$ in \ref{['lem:ETH_to_BPtargets']} for items $z(i, \alpha)$, $z(j, \beta)$, $d(i, j, x)$, $d'(i, j, x)$ and the targets $t_{\ell(i)}$ and $t_{\ell(j)}$.
• Figure 4: Construction of the Multiset Weak Grouped ${k}$-way Partition with targets instance in \ref{['lem:SETH_to_GroupedkBP']}. We show the bit blocks of items and targets, where the highest bits are on the left, for $i \in [N/a]$, $j \in [M]$, $\alpha \in \{0, 1\}^{N/a}$, $b \in [N/a +M] = [q]$.
• Figure 5: An instance of $P_{k} |r_j| C_{\max}$ with a valid schedule $S$ (left) and the corresponding instance of $P_{k} || \Sigma U_j$ with target objective $\sum_j U_j=0$ with the valid schedule $S'$ (right) in the proof of \ref{['lem:GroupedkPart_to_PrjCmax']}.

Theorems & Definitions (81)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Definition 3.1: Parameter-preserving reduction
• Lemma 3.2