Approximation Schemes for Geometric Knapsack for Packing Spheres and Fat Objects

TL;DR

This work delivers polynomial-time $(1+ ext{eps})$-approximation schemes for geometric knapsack in constant dimension for several object families: $d$-dimensional hyperspheres, a broad class of fat convex polygons, and arbitrarily small fat convex objects, all without rotation and without resource augmentation in the base results. The core approach combines large-object guessing with precise algebraic verification, a multilevel grid-based dynamic program, and a hierarchical decomposition that places small objects into a small number of augmented subknapsacks. A key innovation is the discretization technique and the use of corner-empty regions to manage uncertainty about large-object coordinates, alongside a DP that handles configurations efficiently. Extensions to higher dimensions and to polygons rely on exact placements for large objects (via LP/extreme-point properties for polygons) and a common grid-based packing framework for small objects, yielding broad PTAS results and advancing the theory of geometric packing with fat objects. The results hold practical significance for multi-dimensional packing problems where object diversity and nontrivial geometries previously prevented PTAS-style guarantees, and they leave open questions about rational-coordinate feasibility and general convex-object PTAS extensions.

Abstract

We study the geometric knapsack problem in which we are given a set of $d$-dimensional objects (each with associated profits) and the goal is to find the maximum profit subset that can be packed non-overlappingly into a given $d$-dimensional (unit hypercube) knapsack. Even if $d=2$ and all input objects are disks, this problem is known to be \textsf{NP}-hard [Demaine, Fekete, Lang, 2010]. In this paper, we give polynomial time $(1+\varepsilon)$-approximation algorithms for the following types of input objects in any constant dimension $d$: - disks and hyperspheres, - a class of fat convex polygons that generalizes regular $k$-gons for $k\ge 5$ (formally, polygons with a constant number of edges, whose lengths are in a bounded range, and in which each angle is strictly larger than $π/2$), - arbitrary fat convex objects that are sufficiently small compared to the knapsack. We remark that in our \textsf{PTAS} for disks and hyperspheres, we output the computed set of objects, but for a $O_\varepsilon(1)$ of them, we determine their coordinates only up to an exponentially small error. However, it is unclear whether there always exists a $(1+\varepsilon)$-approximate solution that uses only rational coordinates for the disks' centers. We leave this as an open problem that is related to well-studied geometric questions in the realm of circle packing.

Figures (13)

• Figure 1: Left: The squares are stacked compactly inside the knapsack. Middle: The pentagons cannot be stacked as tightly inside the knapsack as the squares. Right: The space in the corner (striped area) cannot be covered by any large circle.
• Figure 2: Packing of fat convex polygons in a knapsack
• Figure 3: Line $\Psi$ separates the two $f$-fat and convex objects $P$ and $P'$. We construct line $\Upsilon$ such that it intersects no common grid cells with line $\Psi$. We proceed to shrink the two objects $P, P'$ by a factor of $1+\eps$ such that they cannot intersect the space between lines $\Upsilon$ and $\Psi$.
• Figure 4: $P,P'$ are shrunk so that they cannot intersect the space between the lines $\Upsilon$ and $\Psi$.
• Figure 5: An illustration of a hierarchical grid. The second level gridcells in the example define a 'discretized' packing of the circles, i.e., no two circles share a common gridcell.

Theorems & Definitions (60)

• theorem 1
• lemma 1
• proof
• lemma 2
• proof
• lemma 3
• proof
• corollary 1
• lemma 4
• proof