A Tight Double-Exponentially Lower Bound for High-Multiplicity Bin Packing

Consider a high-multiplicity Bin Packing instance $I$ with $d$ distinct item types. In 2014, Goemans and Rothvoss gave an algorithm with runtime ${{|I|}^2}^{O(d)}$ for this problem~[SODA'14], where $|I|$ denotes the encoding length of the instance $I$. Although Jansen and Klein~[SODA'17] later developed an algorithm that improves upon this runtime in a special case, it has remained a major open problem by Goemans and Rothvoss~[J.ACM'20] whether the doubly exponential dependency on $d$ is necessary. We solve this open problem by showing that unless the ETH fails, there is no algorithm solving the high-multiplicity Bin Packing problem in time ${{|I|}^2}^{o(d)}$. To prove this, we introduce a novel reduction from 3-SAT. The core of our construction is efficiently encoding all information from a 3-SAT instance with $n$ variables into an ILP with $O(\log(n))$ variables and constraints. This result confirms that the Goemans and Rothvoss algorithm is essentially best-possible for Bin Packing parameterized by the number $d$ of item sizes in the context of XP time algorithms.

💡 Research Summary

The paper addresses the long‑standing open question of whether the doubly‑exponential dependence on the number $d$ of distinct item sizes in the high‑multiplicity Bin Packing problem is inherent. In the high‑multiplicity setting we are given $d$ item types with sizes $s_i$, multiplicities $a_i$, and a bin capacity $B$, and we must pack all items using the minimum number of bins. Goemans and Rothvoss (SODA 2014) showed that for constant $d$ the problem can be solved in time $(|I|^2)^{O(d)}$, which is exponential in $d$ and, when expressed as $|I|^{2^{O(d)}}$, is doubly‑exponential in $d$. Later work by Jansen and Klein gave a modest improvement for a special case, but it remained unknown whether the double‑exponential factor could be eliminated.

The authors settle this by proving, under the Exponential Time Hypothesis (ETH), that no algorithm can solve high‑multiplicity Bin Packing in time $|I|^{2^{o(d)}}$. Their reduction proceeds in two main stages: (1) a compact encoding of a 3‑SAT instance into an integer linear program (ILP) with only $O(\log n)$ variables and constraints, and (2) a transformation of that ILP into a Bin Packing instance with $d = O(\log n)$ item types.

The first stage begins with a standard sparsification of the 3‑SAT formula so that the number of clauses $m$ is $O(n)$. Using an extension of the well‑known “variable‑occurrence” reduction, they transform the formula so that each variable appears exactly twice positively and once negatively (Lemma 5). They then pack the entire formula into a single large integer $Z$; the binary representation of $Z$ is designed so that, with only $O(\log n)$ binary selector variables $\chi_j$ and $O(\log n)$ integer variables $x_j$, one can recover the clause indices and the polarity of each occurrence. Lemma 6 shows how to replace the nonlinear products $x_j\cdot\chi_j$ by $O(\log n)$ linear equations without introducing additional variables. The resulting ILP admits exactly $2\log n$ feasible solutions, each of which corresponds to a distinct truth assignment of the original formula (Lemma 8).

In the second stage the authors apply the “aggregation” technique of Jansen, Pirotton, and Tutas (Lemma 7). They convert each inequality of the ILP into an equality by adding slack variables and then combine all constraints into a single weighted sum using a large base $M$ that prevents carries between digits. This yields a single equality of the form \