Combinatorial optimization problems such as Maximum Independent Set (MIS) and Maximum Cut (MaxCut) arise across a wide range of applications, yet solving large-scale instances remains computationally challenging. While discrete local-search procedures can produce high-quality solutions, their inherently sequential and iterative nature can become increasingly expensive as problem size and density grow. In this paper, we present a differentiable perspective on combinatorial optimization that leverages continuous relaxation with parallelizable gradient-based updates to improve scalability while reducing dependence on discrete local search. More specifically, for MIS, we show that solutions that remain repairable through a discrete (1,2)-swap can nevertheless be fixed points of projected gradient ascent, revealing a fundamental limitation of purely local gradient dynamics. For MaxCut, we develop a new quadratic formulation whose landscape provably allows projected gradient ascent to escape 1-flip-repairable solutions while ensuring binary local maximizers. Building on these insights, and inspired by discrete perturb-and-improve methods, we introduce a framework that combines projected gradient ascent with differentiable mutations (partial resets) to efficiently explore new regions of the solution space}. Experimental results on MIS and MaxCut demonstrate the potential of this approach for scalable combinatorial optimization, particularly in regimes where conventional discrete local search becomes computationally prohibitive.
Combinatorial optimization problems such as Maximum Independent Set (MIS) and Maximum Cut (MaxCut) arise across a wide range of applications, yet solving large-scale instances remains computationally challenging. While discrete local-search procedures can produce high-quality solutions, their inherently sequential and iterative nature can become increasingly expensive as problem size and density grow. In this paper, we present a differentiable perspective on combinatorial optimization that leverages continuous relaxation with parallelizable gradient-based updates to improve scalability while reducing dependence on discrete local search. More specifically, for MIS, we show that solutions that remain repairable through a discrete (1,2)-swap can nevertheless be fixed points of projected gradient ascent, revealing a fundamental limitation of purely local gradient dynamics. For MaxCut, we develop a new quadratic formulation whose landscape provably allows projected gradient ascent to escape 1-flip-repairable solutions while ensuring binary local maximizers. Building on these insights, and inspired by discrete perturb-and-improve methods, we introduce a framework that combines projected gradient ascent with differentiable mutations (partial resets) to efficiently explore new regions of the solution space}. Experimental results on MIS and MaxCut demonstrate the potential of this approach for scalable combinatorial optimization, particularly in regimes where conventional discrete local search becomes computationally prohibitive.
Binary optimization is a powerful tool for modeling combinatorial problems, yet scalable and theoretically sound solution methods remain elusive. Conventional solvers often rely on heuristic strategies with weak guarantees or struggle with large-scale instances. In this work, we introduce a novel primal-dual framework that reformulates unconstrained binary optimization as a continuous minimax problem, satisfying a strong max-min property. This reformulation effectively smooths the discrete problem, enabling the application of efficient gradient-based methods. We propose a simultaneous gradient descent-ascent algorithm that is highly parallelizable on GPUs and provably converges to a near-optimal solution in linear time. Extensive experiments on large-scale problems--including Max-Cut, MaxSAT, and Maximum Independent Set with up to 50,000 variables--demonstrate that our method identifies high-quality solutions within seconds, significantly outperforming state-of-the-art alternatives.
The Ising machine, as a quantum-inspired computing system, can be used to efficiently solve combinatorial optimization problems. Ongoing studies have positioned it to potentially surpass the performance limitations of traditional computers. However, such Ising machines also suffer from scalability as the solution quality becomes sub-optimal when the problem size increases. In this work, we propose an Ising-based framework that uses a distributed Ising algorithm to find the near-optimal solution for large-scale Max-Cut problems through parallel processing via graph segmentation and reassembly. We selected the coherent Ising machine as the hardware solver and successfully found the solution for a 2000-node instance utilizing 20 computational bits on the Gset standard dataset. Additionally, we investigated the key factors influencing the system's performance and discovered that setting the number of reorganizations to one results in higher accuracy. Building on this configuration, we utilized a 100-bit CIM hardware platform (CPQC-1) to solve instances with varying numbers of nodes. The experimental results agree well with the theoretical simulations, confirming the practical feasibility of our method. Moreover, compared to existing Ising-based solvers, our algorithm significantly reduces computational resource requirements while achieving superior performance on large-scale combinatorial optimization problems, highlighting its effectiveness on quantum-inspired hardware.
A range of quantum algorithms, especially those leveraging variational parameterization and circuit-based optimization, are being studied as alternatives for solving classically intractable combinatorial optimization problems (COPs). However, their applicability is limited by hardware constraints, including shallow circuit depth, limited qubit counts, and noise. To mitigate these issues, we propose a hybrid classical--quantum framework based on graph shrinking to reduce the number of variables and constraints in QUBO formulations of COPs, while preserving problem structure. Our approach introduces three key ideas: (i) constraint-aware shrinking that prevents merges that will likely violate problem-specific feasibility constraints, (ii) a verification-and-repair pipeline to correct infeasible solutions post-optimization, and (iii) adaptive strategies for recalculating correlations and controlling the graph shrinking process. We apply our approach to three standard benchmark problems: Multidimensional Knapsack (MDKP), Maximum Independent Set (MIS), and the Quadratic Assignment Problem (QAP). Empirical results show that our approach improves solution feasibility, reduces repair complexity, and enhances quantum optimization quality on hardware-limited instances. These findings demonstrate a scalable pathway for applying near-term quantum algorithms to classically challenging constrained optimization problems.
Many combinatorial optimization problems admit quadratic unconstrained binary formulations (QUBO) which can often be relaxed to the box [0,1]n and optimized using scalable gradient-based methods. However, the resulting non-convex landscape can often contain local optima that are spurious or infeasible. In this paper, we establish sufficient structural conditions on quadratic penalties that rule out these failures, guaranteeing that every local minimizer of the relaxed problem is both binary and feasible. For each problem we study, we examine existing QUBO formulations when available, identify why they fail when they do, and propose alternative relaxed QUBOs that satisfy our conditions. We show for several common combinatorial problems, including open-pit mining, 0--1 knapsack, and traveling salesman formulations, that these constructions allow gradient-based methods such as projected gradient descent and Adam to be safely applied to obtain valid binary solutions. Our results clarify when differentiable optimization is a reliable local solver for quadratic combinatorial objectives.
Learning-based methods have gained attention as general-purpose solvers due to their ability to automatically learn problem-specific heuristics, reducing the need for manually crafted heuristics. However, these methods often face scalability challenges. To address these issues, the improved Sampling algorithm for Combinatorial Optimization (iSCO), using discrete Langevin dynamics, has been proposed, demonstrating better performance than several learning-based solvers. This study proposes a different approach that integrates gradient-based update through continuous relaxation, combined with Quasi-Quantum Annealing (QQA). QQA smoothly transitions the objective function, starting from a simple convex function, minimized at half-integral values, to the original objective function, where the relaxed variables are minimized only in the discrete space. Furthermore, we incorporate parallel run communication leveraging GPUs to enhance exploration capabilities and accelerate convergence. Numerical experiments demonstrate that our method is a competitive general-purpose solver, achieving performance comparable to iSCO and learning-based solvers across various benchmark problems. Notably, our method exhibits superior speed-quality trade-offs for large-scale instances compared to iSCO, learning-based solvers, commercial solvers, and specialized algorithms.
We present a method to approximately solve general instances of combinatorial optimization problems using the physical dynamics of three-dimensional (3D) rotors obeying Landau-Lifshitz-Gilbert dynamics. Conventional techniques to solve discrete optimization problems that use simple continuous relaxation of the objective function followed by gradient-descent minimization are inherently unable to avoid local optima, thus producing poor-quality solutions. Our method considers the physical dynamics of macrospins capable of escaping from local minima, thus facilitating the discovery of high-quality, nearly optimal solutions, as supported by extensive numerical simulations on a prototypical quadratic unconstrained binary optimization (QUBO) problem. Our method produces solutions that compare favorably with those obtained using state-of-the-art minimization algorithms (such as simulated annealing) while offering the advantage of being physically realizable by means of arrays of stochastic magnetic tunnel-junction devices.