Hardware-Agnostic Benders' Decomposition

• The paper introduces a hardware-agnostic Benders’ decomposition algorithm that decouples master and subproblems to consistently perform across diverse compute architectures.
• It leverages QUBO reformulations and precomputed embedding strategies (FX/TT) to reduce embedding time by up to 90% and accelerate subproblem runtimes dramatically.
• The algorithm’s hardware-independence ensures efficient resource utilization—from quantum to classical systems—enabling scalable MILP solutions for high-dimensional problems.

A hardware-agnostic Benders' decomposition algorithm refers to a class of decomposition-based hybrid algorithms for large-scale mixed-integer linear programming (MILP) problems that are designed to perform efficiently across a wide spectrum of compute architectures, including quantum, classical, and accelerated hardware. These frameworks decouple hardware-specific operations through abstraction layers, enabling high-performance MILP solutions that leverage the unique computational strengths of diverse backend platforms while enforcing portability, memory efficiency, and scalability.

1. Problem Structure and Hardware-Agnostic Design Principles

Hardware-agnostic Benders’ decomposition aims to tackle MILPs that rapidly become intractable for standard classical solvers as problem size grows. The core principle exploits classical Benders’ decomposition (BD): partition the problem into a master problem (MP) involving integer (or binary) variables and a subproblem (SP) over continuous variables. The hardware-agnostic philosophy manifests in two distinct computational roles:

• Offload the MP (often a pure-0–1 or integer program) to specialized or accelerated hardware—such as quantum annealers or classical accelerators—but in a representation (e.g., QUBO, bit-packed MIP master, reduced variable footprint) that abstracts over hardware-specific graph, memory, or precision constraints.
• Solve the SP with efficient classical solvers, with minimal cross-hardware data transfer and maximal utilization of vector, parallel, or quantum sampling capabilities.

This approach enables BD frameworks to adapt across classical multicore CPUs, GPUs, distributed computing, FPGAs, and quantum devices—maintaining algorithmic consistency and performance enhancements regardless of backend (López-Baños et al., 20 Jan 2026, Lehmann et al., 2023).

2. Mathematical Formulation and Algorithmic Enhancements

The standard MILP to which hardware-agnostic BD is applied is:

$$(OP)\quad \min\; c^\top x + c'^\top y \quad \text{s.t.}\quad Gx + G'y \leq h,\; Ax + A'y = b,\; x \in \mathbb{Z}^n,\; y \in \mathbb{R}^m$$

Decomposition:

• MP: optimization over $x$ (binary/integer variables), possibly reformulated for the target hardware.
• SP: given $x^i$, solve for $y$ (continuous variables), generally a linear or easily solvable subproblem.

Algorithmic Enhancements:

Hardware-agnostic BD algorithms introduce the following key enhancements:

• QUBO Layer (Quantum-Aided): In quantum-classical hybrids, reformulate the MP as a QUBO instance (using binary encoding for variables and slacks) to suit quantum annealer backends. Penalize QUBO constraint violations to parity with the objective scale for hardware consistency (López-Baños et al., 20 Jan 2026).
• Embedding Strategies: Precompute logical-to-physical variable embeddings (such as FX: fixed-size, and TT: tight-fit) to decouple the minor-embedding step (bottleneck on quantum hardware) from the BD control loop. Empirically, FX/TT strategies reduce total embedding and preprocessing time by an order of magnitude without loss in solution quality (López-Baños et al., 20 Jan 2026).
• Analytical Master Reduction: For problems with extreme variable sizes, analytically eliminate variables (e.g., x, y) to leave only the smallest master (e.g., z) in the outer BD loop, tractable for even memory-limited classical or quantum resources (Lehmann et al., 2023).
• Precision Control and Cut Management: Conservative rounding, dynamic bounds, and integer-precision enforcement for constraint-side variables—the master is always feasible and QUBO size remains hardware-compliant (López-Baños et al., 20 Jan 2026).
• Data-Parallel and Bit-Packed Subproblem Processing: Bit-packed representations, ctz-based (count trailing zeros) sparse scans, and SIMD/OpenMP/MPI parallelism harness hardware-agnostic primitives for SP efficiency and scalability (Lehmann et al., 2023).

3. Embedding and Hardware Mapping Strategies

Embedding refers to mapping a logical BD master (e.g., QUBO variables or MIP bitmaps) to the physical variable (qubit) and connectivity graph of the target hardware.

Embedding Strategy	Description	Impact on Performance
Minor-Miner (MM)	Default heuristic embedding at each iteration	High runtime overhead
Fixed-Size (FX)	Precompute maximal complete-graph embedding once and reuse	~10× faster embedding; no loss in quality
Tight-Fit (TT)	Precompute minimal embeddings for all subproblem sizes	Similar speedup; supports more QUBO sizes

On D-Wave's Pegasus hardware, FX and TT reduce embedding and preprocessing costs significantly relative to iterative or heuristic methods, and enable quantum acceleration for large master problems up to the hardware’s qubit capacity limit. For classical bit-packed masters, these strategies translate to index remapping and require no heavy data movement, preserving the hardware-agnostic guarantee (López-Baños et al., 20 Jan 2026, Lehmann et al., 2023).

4. Memory and Computational Efficiency

Hardware-agnostic BD algorithms are designed to remain efficient across architectures through the following mechanisms:

• Bit-Packed Data Structures: Packing large binary matrices (e.g., fitting matrix $F_{pb}$) into bit-arrays enables O(1) access and O(n/64) memory scaling, allowing representation of multi-gigavariable instances in a few GB.
• Sparsity-Aware Scans: Algorithms utilize ctz preprocessing, skipping zeros, and vectorized min-scans for assignment and dual computations, eliminating unnecessary work and memory traffic (Lehmann et al., 2023).
• OpenMP/GPU/Cluster Parallelism: Subproblem scans and KD-tree constructions (in packing problems) exploit parallelization over units/boxes either via OpenMP on CPUs, thread blocks on GPUs, or distributed over clusters—cut generation and feasibility checks are naturally parallel.
• Elimination of Solver Overhead in SP: Analytical closed-form or O(P·avg_fit) scans eliminate the need for embedded LP solvers, allowing SP to be processed using fixed-point or float ops on any backend (Lehmann et al., 2023).
• Minimal Communication: Only compact summary vectors (e.g., BD cut coefficients $w^i$, constants $s^i$) are communicated between BD levels or across nodes, facilitating hardware-independence.

For example, in a variable-height packaging problem, the hardware-agnostic BD algorithm reduced the subproblem memory footprint from 60 GB to <1 GB and the runtime of subproblem evaluation by three orders of magnitude via these mechanisms (Lehmann et al., 2023).

5. Benchmarking and Performance in Quantum-Classical Hybrids

Hybrid quantum-classical hardware-agnostic BD algorithms have been benchmarked on Transmission Network Expansion Planning (TNEP) and large-scale packaging optimization.

Key empirical findings (López-Baños et al., 20 Jan 2026, Lehmann et al., 2023):

• The quantum-annealer-driven MP (QUBO reformulation) paired with classical SP matching simulated annealing performance for MILPs with up to 7–8 binary variables per BD iteration when embedding overhead was minimized via FX/TT strategies.
• Precomputed, hardware-agnostic embeddings reduced total embedding and preprocessing time by approximately 90% compared to iterative embedding, without degrading the energy or solution success rate.
• Conservative rounding and slack management kept master QUBO (or bit-packed master) sizes tractable for hardware capacity constraints, and ensured no true solution was ever excluded.
• Analytical reduction of the master (e.g., retaining only z-variables in master) cut overall Benders’ runtime by over 40× compared to formulations that retained superfluous variables.
• Speedups from hardware-agnostic SP acceleration (bit-packed with parallel ctz scans) achieved up to 1000× improvement for the subproblem and over 25× acceleration for core bin-packing logic, with immediate portability to CPUs, GPUs, and clusters.

Configuration	Success Rate (≤7 buses)	Embedded Time Speedup	Notes
Classical Gurobi	100%	–	Exact
Quantum (QA-GRB-MM)	>95%	1×	No FX/TT
Quantum (QA-GRB- FX/TT)	≈90% at 8 buses	~10×	Hardware-agnostic

These benchmarks confirmed that hardware-agnostic design enables consistent use of the best available platform—CPU, GPU, quantum annealer—by minimizing memory, leveraging device-native data structures, and confining hardware-specific functionality to pluggable, reusable components.

6. Applicability, Extensions, and Hardware-Independence

The hardware-agnostic BD paradigm is applicable to any MILP (and related combinatorial) structure amenable to BD, with further extensibility to NISQ-era quantum hardware and emerging classical accelerators.

• Portability Features: Bit arrays, integer/float ops, and avoidance of floating-point pivoting or hardware-specific solver libraries facilitate rapid deployment on new or heterogeneous architectures.
• Parallel and Distributed Execution: Methods are naturally parallel—each compute node operates on its own data partition, with only cut vectors communicated, ensuring scalability.
• Analytical Formulation: For problems where closed-form duals or master elimination is possible, the approach avoids reliance on external LP/MIP solvers entirely.
• Future Extensions: The framework admits enhancements such as dynamic penalty tuning in QUBO encodings, multi-cut BD strategies, and integration with gate-model quantum backends for further hardware abstraction (López-Baños et al., 20 Jan 2026).

This suggests that hardware-agnostic BD can provide a foundational blueprint for quantum-classical hybrid optimization methods and high-performance MILP solvers by systematically abstracting hardware variability while preserving or improving solution tractability, efficiency, and scalability.