Approximating the Geometric Knapsack Problem in Near-Linear Time and Dynamically

TL;DR

The paper delivers near-linear-time PTAS-like results for the d-dimensional geometric knapsack with constant dimension, achieving a $(1+ε)$-approximation for hypercubes and fast $(2+ε)$ or $(17/9+ε)$-approximations for axis-aligned rectangles, both in static and dynamic settings. Central to the approach are easily guessable packings and an indirect guessing framework that estimates remaining parameters via a sequence of small, solvable IPs/LPs, supported by range-counting data structures and box-based decompositions (N^*, S-boxes, etc.). The results substantially improve the dependence on $n$ in running times (nearly linear in $n$) while preserving guarantees that are close to optimal under current techniques, and they extend to dynamic environments with polylogarithmic updates. These advances have practical implications for fast, scalable packing in multidimensional geometries and offer a framework that could guide further improvements for broader classes of shapes and dimensions.

Abstract

An important goal in algorithm design is determining the best running time for solving a problem (approximately). For some problems, we know the optimal running time, assuming certain conditional lower bounds. In this work, we study the $d$-dimensional geometric knapsack problem where we are far from this level of understanding. We are given a set of weighted d-dimensional geometric items like squares, rectangles, or hypercubes and a knapsack which is a square or a (hyper-)cube. We want to select a subset of items that fit non-overlappingly inside the knapsack, maximizing the total profit of the packed items. We make a significant step towards determining the best running time for solving these problems approximately by presenting approximation algorithms with near-linear running times for any constant dimension d and any constant parameter $ε$. For (hyper)-cubes, we present a $(1+ε)$-approximation algorithm whose running time drastically improves upon the known $(1+ε)$-approximation algorithm which has a running time where the exponent of n depends exponentially on $1/ε$ and $d$. Moreover, we present a $(2+ε)$-approximation algorithm for rectangles in the setting without rotations and a $(17/9+ε)$-approximation algorithm if we allow rotations by 90 degrees. The best known polynomial time algorithms for these settings have approximation ratios of $17/9+ε$ and $1.5+ε$, respectively, and running times in which the exponent of n depends exponentially on $1/ε$. We also give dynamic algorithms with polylogarithmic query and update times and the same approximation guarantees as the algorithms above. Key to our results is a new family of structured packings which we call easily guessable packings. They are flexible enough to guarantee profitable solutions and structured enough so that we can compute these solutions quickly.

Figures (8)

• Figure 1: Packing of items into single boxes
• Figure 2: Packing of boxes into knapsack
• Figure 4: Sorted packing of an $\mathcal{N}^{*}$-box: each item is packed into a single grid cell.
• Figure 5: NFDH packing of an $\mathcal{S}$-box: items are packed greedily into shelves.
• Figure 7: Visualization of packing of $\mathcal{L}$-, $\mathcal{S}$-,$\mathcal{H}$- and $\mathcal{V}$-box, respectively.

Theorems & Definitions (45)

• Lemma 1
• Definition 2: $\mathcal{V}$-boxes and $\mathcal{N}$-boxes jansen2022ptas
• Definition 3: $\mathcal{N}^{*}$-box
• Definition 4: $\mathcal{S}$-boxes
• Lemma 5: implied by Lemma 4 in harren2009approximation
• Lemma 6: Near-optimal packing of hypercubes
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10