PackingStar: Algorithms & Physical Packings

• PackingStar is a multidisciplinary framework that unifies advanced reinforcement learning, RSA techniques, and graph-theoretic algorithms to tackle geometric, combinatorial, and physical star packing challenges.
• Its reinforcement learning component uses a cooperative two-agent MCTS and actor–critic setup to explore high-dimensional kissing number configurations, yielding record-breaking sphere packings.
• Graph and experimental studies within PackingStar reveal practical insights on density, stability, and approximation performance in both 3D particle aggregates and complex network structures.

PackingStar encompasses multiple advanced concepts and algorithms at the intersection of geometric, combinatorial, and algorithmic packing problems, involving both physical star-shaped particles and abstract star-structures in graphs and high-dimensional geometry. The term refers to (1) reinforcement learning-based algorithms for high-dimensional sphere (kissing) number problems, (2) analytical and numerical studies of 2D and 3D star-shaped object packings, and (3) graph-theoretic star-packing with rigorous approximation algorithms. The following sections provide an encyclopedic synthesis, organized along the principal research threads and methodologies.

1. Game-Theoretic Reinforcement Learning for High-Dimensional Sphere Packing

PackingStar designates a two-agent reinforcement-learning system that efficiently explores the huge configuration space of the Kissing Number Problem in $\mathbb{R}^n$—the maximal number $K(n)$ of non-overlapping unit spheres tangent to a central unit sphere. The setting is formalized via Gram (cosine) matrices $M_{ij}=\langle x_i,x_j \rangle$ with $M_{ii}=1$, $M\succeq0$, and $M_{ij}\le1/2$ for $i\neq j$; the problem is to maximize the order $m$ of such a matrix subject to these constraints.

PackingStar models the matrix construction as a cooperative two-player Markov game in which:

• Player 1 (Filler) expands the Gram matrix by proposing new rows/columns consistent with known feasible cosine set structures and rank/PSD constraints, guided by Monte Carlo Tree Search (MCTS).
• Player 2 (Corrector) prunes suboptimal rows/columns via a learned neural policy to maintain and possibly increase the maximal feasible matrix order. Both agents share the end-of-episode team reward: the final size $m$ of the Gram matrix, corresponding to the kissing number found.

The neural architecture for the Corrector consists of three fully connected ReLU layers ([256, 256, 128]), with separate policy and value heads; the input is either the flattened upper triangle or the dominant eigenvectors of the Gram matrix. Training uses actor–critic methods (PPO), with Player 1's MCTS using 800 playouts per move ($c_{\rm UCB}=1.5$), and standard RL hyperparameters. Player 2 is updated by policy-gradient on the final matrix size.

This hybrid MCTS + RL setup allows rapid exploration of combinatorially vast spaces, and PackingStar has produced new best-found kissing numbers in all studied dimensions $n=25$ to $31$ (e.g., $K(25)$ improved from 197048 to 197056, and $K(31)$ increased to 238078), as well as the first rigorous breakthrough for rational configurations in $n=13$ ($K_r(13)=1146$), and thousands of distinct new configurations in $n=14$ and other midrange cases (Ma et al., 17 Nov 2025).

PackingStar's success comes from direct optimization in the Gram-matrix/cosine-inner-product space, not via explicit $\mathbb{R}^n$ coordinates, and from its two-agent cooperative dynamics, which reduce the effective search space dimensionality. The approach generalizes to other extremal spherical code problems, combinatorial Gram-matrix SDP optimizations, and error-correcting code construction.

2. Random Sequential Adsorption of Star-Polygons in Two Dimensions

PackingStar also refers to the RSA (Random Sequential Adsorption) of $n$-sided convex polygons and regular star-polygons denoted $\{n/k\}$ (vertices evenly spaced, each connected by $k$-step chords). The RSA process sequentially places non-overlapping shapes at random positions and orientations on a planar collector, halting when no further addition is possible. The area of a star-polygon $\{n/k\}$ (unit edge length) is

$$S_p = \frac{n}{4} \left[\cot\left(\frac{\pi}{n}\right) - \tan\left(\frac{(k-1)\pi}{n}\right)\right].$$

Key findings for RSA kinetics and coverage:

• Saturated coverage $\theta_s$ is determined by Feder's law: $\theta(t) = \theta_s - A t^{-1/d}$, where $d$ is the effective degree-of-freedom ($d \approx 3$ for rod-like, anisotropic stars at small $n$ and $d \rightarrow 2$ in the disk limit $n \to \infty$).
• For fixed $n$, increasing star concavity ($k>1$) reduces $\theta_s$ (packing ratio), as star “teeth” block extra area while not increasing self-area (Table below).
• As $n \to \infty$, both convex and star-concave polygons approach the disk limit: $\theta_s \to 0.547$, $d \to 2$.

(n, k)	$\theta_s$ (approx.)	Geometry
(5,1)	0.528	regular pentagon
(5,2)	0.424	pentagram
(10,1)	0.538	decagon
(10,2)	0.507	10-pointed star
(50,k)	≃0.546	nearly disk-like

Analysis of the density autocorrelation function $G(r)$, available-surface function (ASF), and kinetic exponents indicate a smooth crossover from anisotropic/concave behavior to classic disk RSA with increasing $n$. Concave shapes always pack less efficiently than convex counterparts due to enhanced exclusion zones (Cieśla et al., 2014).

3. Stability and Structure of 3D Star-Shaped Particle Packings

PackingStar extends to experimental and analytical studies of 3D star-shaped particle aggregates, with particular attention to pile stability, collapse, and load-bearing limits. Each particle comprises six orthogonal tapered arms; frictional and vibrational effects are systematically varied.

The central quantitative metric is the collapse ratio

$$r = \frac{M_{\rm fallen}}{M_{\rm total}}, \qquad 0 \le r \le 1,$$

where $M_{\rm fallen}$ is the mass of particles fallen off at collapse. The critical pile height $h_c$ at which $r = 1/2$ obeys a power law in the cylinder diameter $\Delta$:

$h_c = C (\Delta - s_0)^\alpha,$

with $s_0 \sim$ particle size, $C$ a constant, and $0.6 \lesssim \alpha \lesssim 0.8$ (material/vibration dependent).

Experimental results demonstrate:

• Higher inter-particle friction ($\mu$ from $0.4$ to $1.0$) and vibration during assembly both increase $C$ and thus attainable $h_c$.
• Packing density $\rho$ increases slightly with vibration; mean interlocking distance correspondingly decreases, enhancing pile stability.
• For fixed cylinder $\Delta$, increasing particle size $s$ above $O(\Delta)$ reduces base contacts and destabilizes the pile.
• Collapse and tilting thresholds exhibit sharp, sublinear dependence on system geometry and material parameters (Zhao et al., 2015).

4. Graph-Theoretic Star Packing and Approximation Algorithms

PackingStar encompasses the discrete, algorithmic problem of packing stars in undirected graphs, where a $k$-star is a $K_{1,k}$, a $k^+$-star is any $K_{1,\ell}$ with $\ell\geq k$, and a $k^-/t$-star is any $K_{1,\ell}$ for $1\leq\ell\leq k$ but excluding $\ell=t$.

Two central (NP-hard) problems are

• Maximizing the number of covered vertices by vertex-disjoint $k^+$-stars;
• Maximizing coverage with $k^-$-stars but no $t$-stars ($k>t\ge2$).

Recent results provide local search algorithms with rigorous approximation guarantees:

• For $k^+$-star packing, a $(1 + \frac{k^2}{2k+1})$-approximation for $k\geq 3$ and a $3/2$-approximation for $k=2$.
• For $k^-/t$-star packing, a $(1 + \frac{1}{t+1+1/k})$-approximation is achieved.

These algorithms proceed by alternating local operations: “Collect” (greedily forming large stars), various “Pull” moves (trading stars for higher coverage), and in the $k^-/t$ case, “Revise” moves to eliminate forbidden $t$-stars. All moves are efficiently local and guarantee strict progress, with worst-case running times $O(n^3 m)$ (where $n$ is the number of vertices, $m$ edges), often much faster in practice (Hu et al., 17 Nov 2024). The $k^+$- and $k^-/t$-star packing problems generalize classical matching and path-packing; these local-search bounds are the strongest known for such non-sequential star families.

5. Cross-Domain Themes and Physical Insights

PackingStar unifies several recurring physical and algorithmic themes:

• Concavity and Exclusion: In both geometric RSA and granular 3D star assemblies, increased concavity or anisotropy (i.e., higher $k$ for fixed $n$ in polygons, larger arm count/length in 3D) reduces packing density or stability due to enlarged exclusion zones and inefficient coverage, visible in both coverage metrics and critical stability exponents.
• Connectivity and Interlocking: Inter-particle friction, vibration-induced densification, and minimization of internal voids directly increase pile robustness in experimental 3D systems, paralleling how “collecting” larger stars in $k^+$-star packing algorithms subsumes more vertices.
• Computational Tractability: The algorithmic techniques—MCTS-RL in high-dimensional geometry and local search in graphs—tackle combinatorial explosion by decomposing search into tractable subspaces or sequences of local moves, whether in the Gram-matrix PSD domain or adjacency neighborhoods.
• Scaling Limits: All star-packing phenomena admit continuous scaling limits (disk limit for $n\to\infty$ in 2D RSA, sublinear law for $h_c$ in large 3D piles, stabilization of approximation ratios for increasing $k$ in graphs), indicating robust and universal structure across models.

6. Outlook and Open Problems

PackingStar methodologies have demonstrated state-of-the-art performance in geometric, physical, and combinatorial packing, surpassing previous human-constructed bounds for high-dimensional sphere packings and providing tight algorithmic control in discrete star families.

Key open directions include:

• Formal proofs of optimality for newly discovered high-dimensional sphere configurations;
• Extension of cooperative RL and MCTS techniques to other extremal code or SDP-based packing problems;
• Identification of inapproximability thresholds for non-sequential $k$-star packing problems;
• Further mechanical and architectural studies exploiting the self-stabilizing properties of 3D star aggregates, including beyond sublinear $h_c(\Delta)$ scaling and the role of shape anisotropy.

PackingStar thus serves as a paradigmatic instance of unified advances in high-dimensional geometry, granular matter theory, and combinatorial optimization, driven by both algorithmic innovation and physical insight.