Parallelizable Gradient-Based Optimization For Multi-Objective MaxCut

Abstract

Multi-objective combinatorial optimization arises in a wide range of problems and applications, including the canonical multi-objective MaxCut problem. Differentiable single-instance quadratic methods have recently achieved remarkable performance in single-objective combinatorial optimization. In this paper, we develop a differentiable framework for multi-objective MaxCut by combining an adjacency-based quadratic formulation with linear scalarization, thereby reducing the problem to a preference-conditioned single-objective signed-weight MaxCut problem. Theoretically, we characterize projected gradient ascent (PGA) fixed points and their local dynamics under the signed-weight adjacency formulation. We further characterize how these fixed points depend on preferences and establish their connection to Pareto optimality. Computationally, unlike conventional heuristics and branch-and-bound methods, our approach is GPU-parallelizable and can therefore benefit from substantial performance speedups. We term our algorithm Multi-objective QUadratic Combinatorial Optimization (MO-QUCO) and its parallelized variant pMO-QUCO. Empirically, we evaluate our methods on multi-layered graphs with different sizes and edge-weight distributions. Both our CPU-only and GPU-based algorithms outperform state-of-the-art exact and heuristic methods in terms of wall-clock runtime and objective quality. Despite operating under different computational settings, MO-QUCO also outperforms the SOTA quantum method.