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The Support of Bin Packing is Exponential

Klaus Jansen, Lis Pirotton, Malte Tutas

TL;DR

This work proves that the support of high-multiplicity Bin Packing solutions is exponential in the number of item sizes, establishing a lower bound of $2^{ ext{Ω}(d)}$ that matches the known upper bound up to constants. The authors introduce a novel aggregation technique that merges multiple equalities and upper bounds into a single knapsack constraint, enabling a direct knapsack-based reduction from a Cone-and-Polytope Intersection construction. They construct a knapsack polytope with $2^d$ representative points and show that any exact representation of a target requires all these points, which translates into an exponential number of configurations in BP. The main result answers open questions about polynomial-in-$d$ bounds and implies that existing BP algorithms cannot be improved to singly-exponential running time solely via smaller support. The approach suggests broader applicability to other ILPs and d-dimensional knapsack problems through the new aggregation method.

Abstract

Consider the classical Bin Packing problem with $d$ different item sizes $s_i$ and amounts of items $a_i.$ The support of a Bin Packing solution is the number of differently filled bins. In this work, we show that the lower bound on the support of this problem is $2^{Ω(d)}$. Our lower bound matches the upper bound of $2^d$ given by Eisenbrand and Shmonin [Oper.Research Letters '06] up to a constant factor. This result has direct implications for the time complexity of several Bin Packing algorithms, such as Goemans and Rothvoss [SODA '14], Jansen and Klein [SODA '17] and Jansen and Solis-Oba [IPCO '10]. To achieve our main result, we develop a technique to aggregate equality constrained ILPs with many constraints into an equivalent ILP with one constraint. Our technique contrasts existing aggregation techniques as we manage to integrate upper bounds on variables into the resulting constraint. We believe this technique can be useful for solving general ILPs or the $d$-dimensional knapsack problem.

The Support of Bin Packing is Exponential

TL;DR

This work proves that the support of high-multiplicity Bin Packing solutions is exponential in the number of item sizes, establishing a lower bound of that matches the known upper bound up to constants. The authors introduce a novel aggregation technique that merges multiple equalities and upper bounds into a single knapsack constraint, enabling a direct knapsack-based reduction from a Cone-and-Polytope Intersection construction. They construct a knapsack polytope with representative points and show that any exact representation of a target requires all these points, which translates into an exponential number of configurations in BP. The main result answers open questions about polynomial-in- bounds and implies that existing BP algorithms cannot be improved to singly-exponential running time solely via smaller support. The approach suggests broader applicability to other ILPs and d-dimensional knapsack problems through the new aggregation method.

Abstract

Consider the classical Bin Packing problem with different item sizes and amounts of items The support of a Bin Packing solution is the number of differently filled bins. In this work, we show that the lower bound on the support of this problem is . Our lower bound matches the upper bound of given by Eisenbrand and Shmonin [Oper.Research Letters '06] up to a constant factor. This result has direct implications for the time complexity of several Bin Packing algorithms, such as Goemans and Rothvoss [SODA '14], Jansen and Klein [SODA '17] and Jansen and Solis-Oba [IPCO '10]. To achieve our main result, we develop a technique to aggregate equality constrained ILPs with many constraints into an equivalent ILP with one constraint. Our technique contrasts existing aggregation techniques as we manage to integrate upper bounds on variables into the resulting constraint. We believe this technique can be useful for solving general ILPs or the -dimensional knapsack problem.

Paper Structure

This paper contains 10 sections, 6 theorems, 17 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Preliminaries
  4. Aggregation of an ILP
  5. Bounding the Support for Bin Packing
  6. The support of closed polytopes
  7. Construction of the ILP
  8. Showing the necessity of all variables of the knapsack polytope
  9. Aggregation of the Constraints
  10. Main proof

Key Result

Theorem 1

There exists a high-multiplicity Bin Packing instance of dimension $d$ such that the solution-vector $x$ (a vector of configurations) contains at least $2^{\Omega(d)}$ non-zero entries, i.e., $|\text{supp}(x)| \in 2^{\Omega(d)}.$

Figures (3)

  • Figure 1: An illustration of the Bin Packing problem with three item types, four bins and capacity $C$. Colors indicate item types. Each bin has a unique configuration, i.e., combination of items, assigned to it. Only the leftmost bin is completely filled. The sizes of each item type are shown on the right.
  • Figure 2: An illustration of the Cone and Polytope Intersection problem with three points $p_i\in P$ and the target $t\in Q.$ The values of each vector are given. A possible solution is shown.
  • Figure 3: A 2-dimensional example of the knapsack constraint with item types $s_1,s_2.$ The shaded area denotes the knapsack polytope $P$. The points resemble integer points and show all feasible configurations. Integer points in the shaded region are the configurations in the set $X$. Integer points on the red line form the set $X^*$ of configurations. Only configurations in $X^*$ fill the knapsack constraint with equality.

Theorems & Definitions (11)

  • Theorem 1
  • Definition 2: Bin Packing
  • Definition 3: Integer Programming
  • Definition 4: Unbounded Knapsack
  • Lemma 4
  • Lemma 4
  • Example 5
  • Lemma 8
  • Lemma 9
  • Example 10
  • ...and 1 more