Dynamic Packing: Concepts & Applications

• Dynamic packing is a collection of techniques that manage the arrival and departure of items to maximize packing efficiency under specific constraints.
• It leverages algorithms like dynamic bin and vector bin packing to optimize resource allocation while minimizing costly repacking and migration.
• Stochastic, LP, and physical models in dynamic packing enable precise analysis of spatial correlations, steady-state densities, and phase transitions.

Dynamic packing refers to a broad set of processes and algorithms in which objects, particles, jobs, or resource requests arrive and depart over time, and must be accommodated to maximize (or approximate) packing efficiency according to prescribed constraints. Across diverse domains—stochastic processes, computational geometry, combinatorial optimization, condensed matter physics, and cloud computing—dynamic packing encompasses both the mathematical analysis of evolving packings and the development of efficient, adaptive algorithms. Prominent subfields include dynamic bin (and vector bin) packing, dynamic packing LPs, stochastic spatial packings, and dynamical physical models.

1. Stochastic and Markovian Models of Dynamic Packing

Stochastic packing models study the evolution of spatial packings with random arrivals and departures, typically using Markov processes on infinite or large state spaces. A classical paradigm is the dynamic storage allocation problem: items of variable sizes arrive to occupy portions of a storage medium and depart after random lifetimes. For the linear storage model on the half-axis, where items of size 1 or 2 arrive as Poisson processes and depart after exponential holding times, first-fit discipline (placing each new item in the leftmost feasible gap) is proved to be asymptotically optimal: as the load increases, the empty-to-occupied space ratio over any macroscopic bulk interval vanishes, and the empirical spatial profile converges to the optimal block-packing (all small items contiguous, then all large, zero gaps) (Ernst et al., 4 Apr 2024). The proofs employ Lyapunov-drift arguments for finite intervals, leveraging the generator of the underlying Markov chain to establish concentration of measure and scaling limits.

Dynamic space packing (DSP) processes form another solvable Markovian family with additive and subtractive mechanisms. On a $d$-dimensional lattice, each incoming particle at a random site pushes out its neighbors, enforcing an exclusion constraint; the process admits explicit solutions for the steady-state density, correlation functions, and so-called desorption statistics, and yields steady-state densities exactly matching $1/(2d+1)$ on hypercubic lattices or $1/(q+1)$ on $q$-regular graphs. In continuous space, DSP achieves the Minkowski sphere-packing lower bound asymptotically, i.e., packing fraction $2^{-d}$ in $\mathbb{R}^d$ (Dandekar et al., 2023). These stochastic models enable exact analysis of density, spatial correlations, and cluster-removal events as functions of system dimension and geometric constraints.

2. Algorithmic Frameworks in Dynamic Bin and Vector Bin Packing

Dynamic bin packing (DBP) addresses the online problem of allocating items of variable size, which arrive and depart adversarially or stochastically, into bins of fixed capacity while minimizing the number of active bins and the amount of repacking or migration over time. A central distinction is made between fully dynamic and relaxed (insertion-only or deletion-only) models.

Fully Dynamic Bin Packing (FDBP)

In FDBP, an optimal tradeoff exists between competitive ratio $\alpha$ (relative to offline optimum) and recourse (number or size of items moved per update). It is proved that to achieve asymptotic competitive ratio $1+\epsilon$, algorithms must incur $\Omega(1/\epsilon)$ migration factor per operation (Berndt et al., 2014, Gupta et al., 2017). Tight, non-amortized upper bounds on migration—$O(1/\epsilon^4 \log(1/\epsilon))$—are achievable using dynamic rounding schemes and queue-based handling of small items. These approaches maintain robustness and low computational complexity even with both arrivals and departures and do not require bundling of small items (Feldkord et al., 2017).

For unit-cost (and arbitrary-cost) migration models, the tradeoffs are fully characterized:

• For unit costs, competitive ratio $\alpha^* \approx 1.3871$ is attainable with $O(\epsilon^{-2})$ amortized recourse, and worst-case recourse $O(\epsilon^{-4})$ (Gupta et al., 2017).
• For migration proportional to item size (size-cost), fully dynamic $(1+\epsilon)$-approximation is possible with $O(1/\epsilon)$ amortized migration.

Delicate algorithmic techniques include: grouping items into size classes, LP-guided curve-fitting of small items, overlaying greedy large-item placements, buffer-group/bucket invariants, and dynamic potential functions to minimize repacking actions.

Dynamic Vector Bin Packing (DVBP) and Multi-Resource Models

DVBP generalizes classical bin packing to $d$-dimensional resource vectors, with each active request (job) spanning a time interval. The complex online setting arises in cloud infrastructure, where jobs arrive over time with multidimensional resource needs and varying durations. Any-Fit-style heuristics—First Fit, Next Fit, and Move-To-Front—are analyzed under adversarial and random inputs, with tight competitive ratios scaling as functions of maximum-to-minimum job duration ratio $\mu$: $$\text{MTF}: (2\mu+1)d+1, \quad \text{FF}: (\mu+2)d+1, \quad \text{NF}: 2\mu d+1,$$ and matching lower bounds are established. Move-To-Front empirically minimizes average cost by aligning "similar-timed" jobs and maintaining active intervals effectively (Murhekar et al., 2023).

Data reduction for DVBP through lossy kernelization is possible: by greedily deleting $\epsilon$-fraction of requests, instance sizes drop by an order of magnitude for small $\epsilon$ while retaining $(1+\epsilon)$-approximate solutions, crucial for scaling to real-world traces (Bevern et al., 2022).

3. Resource Migration and Power of Limited Repacking

The ability to repack (or migrate) items between bins at runtime is critical for effective dynamic packing, particularly in environments with high churn or temporally heterogeneous item lifetimes. Recent work fully characterizes the "power of migrations" (Mellou et al., 23 Aug 2024):

• Sublinear Migrations: Any algorithm restricted to $o(n)$ migrations (job or total item size) offers no asymptotic improvement over zero-migration strategies; competitive ratio remains $\Theta(\mu)$, directly tied to the spread $\mu$ of item durations.
• Linear Migrations: Allowing $O(\alpha n)$ migrations, algorithms attain competitive ratio $O(1/\alpha)$, completely decoupling efficiency from item-duration ratios. This tradeoff is shown to be best possible up to constants.
• Migration Delay Model: Imposing a blackout penalty of $C$ units per migration, the best possible competitive ratio is $O(\min(\sqrt{C},\mu))$.

Such results have direct implications for dynamic resource allocation in cloud systems and data centers, where VM or task migration unavoidably incurs downtime and logistical cost.

4. Dynamic Packing in Physical and Materials Science Systems

Dynamic packing phenomena are directly observable in physics, especially in granular materials and supercooled liquids. Coupled computational and theoretical approaches elucidate how time-dependent forces, disorder, and defect populations affect packing structure and mechanical properties.

• Granular Media: Under sinusoidal dynamic forcing, granular packings soften as evidenced by a drop in resonant frequency; contact number $Z_c$ at resonance decreases with increasing amplitude of dynamic forcing, revealing disruption of force chains. Spatial localization ("soft spots") proliferates under higher amplitude, promoting dissipation and nonaffine response (Reichhardt et al., 2014).
• Defect-Mediated Dynamical Heterogeneity in Supercooled Liquids: Introduction of size-mismatched probe molecules at low concentrations can dramatically tune both transport coefficients and the magnitude of dynamic heterogeneity. Large probes locally reduce free volume, slow diffusion, and increase cluster-scale correlated motion; small probes have the converse effect, suppressing heterogeneity and accelerating local relaxation. This supports theoretical links between "soft spots," packing defects, and spatially heterogeneous dynamics (Taamalli et al., 2017).
• Particle Packing under Freezing Fronts (Colloid Physics): In non-equilibrium directional freezing, particles accumulate at the ice front, forming a packed layer with a pattern set by competition between Brownian relaxation and front-driven kinetics. The Péclet number $\mathrm{Pe} = d V_p / D$ captures the nonequilibrium character: high $\mathrm{Pe}$ results in amorphous, defect-rich packings; low $\mathrm{Pe}$ enables large ordered domains. Macroscopic migration and layer structures are quantitatively modeled as functions of freezing speed and initial concentration (You et al., 2016).
• DEM/MD Simulations of Fine Particle Packing: In two-dimensional gravity-driven packings, hard/soft and polydisperse particles are studied with explicit dissipative contact and long-range (van der Waals) interactions. Packing density, mean coordination number, and relaxation rates are sensitive to these parameters, with contact elasticity being the dominant factor in achievable close-packing configurations (Ferraz et al., 2016).

5. Dynamic Packing in Combinatorial Structures and LPs

Dynamic packing extends deeply into combinatorial optimization and mathematical programming.

• Dynamic Matroid Base Packing: For matroids under dynamic ground set updates (insertions and deletions), the fractional base-packing number $\Phi$—the maximum number of disjoint bases—can be tracked to $(1\pm \epsilon)$ precision using greedy collections of $O(\Phi \log n/ \epsilon^2)$ bases, with amortized sublinear update time via rank oracles. These results generalize dynamic arboricity and matching problems (Vos et al., 19 Nov 2025, Vos et al., 15 May 2024).
• Dynamic Packing LPs (Generalized Frameworks): Dynamic variants of covering and packing LPs—for which entries of the constraint matrix or right-hand side may be updated—are handled, under partially dynamic restrictions, using multiplicative-weights update schemes. In the restricting-only or relaxing-only regimes, fully general $(1+\epsilon)$-approximate solutions can be maintained in amortized polylogarithmic time per update. SETH-based lower bounds show that in fully general, worst-case dynamic LPs, no such efficiency is possible (Bhattacharya et al., 2022).

6. Connections, Applications, and Limitations

Dynamic packing processes, both in algorithmic and physical manifestations, are characterized by complex tradeoffs between efficiency, adaptivity, and stability. Key themes include:

• The need to balance optimal resource utilization (minimum bins, maximum density) with operational overhead (migration, repacking, relaxation times).
• Structural and algorithmic invariants (queue/buffer schemes, Lyapunov functions, greedy base collections) that permit nearly-optimal dynamic performance.
• Emergence of nonequilibrium phase transitions, defect formation, and kinetic constraints in physical systems, governed by dimensionless numbers (e.g., Péclet number) or control parameters (stiffness, forcing amplitude).
• Limits on compressibility and data reduction: certain packing instances in high-dimensions or with considerable overlap admit no polynomial-size kernelization, except under parameter restriction or approximate settings (Bevern et al., 2022).

Dynamic packing continues to be a focal point in optimization, physical modeling, and high-performance computing, with ongoing advances arising from unifying structural, probabilistic, and algorithmic perspectives.