ADVANCED APPROXIMATION AND PARAMETERIZED ALGORITHMS FOR SOLVING NP-HARD COMBINATORIAL OPTIMIZATION PROBLEMS IN LARGE-SCALE GRAPH SYSTEMS

Abstract

Combinatorial optimization problems on large-scale graphs are predominantly NP-hard, rendering exact solutions computationally intractable for real-world instances involving billions of vertices and edges. This paper presents a comprehensive review of recent advances in approximation algorithms and parameterized complexity techniques designed to address these challenges. We explore the theoretical foundations of intractability, the role of approximation guarantees, and breakthroughs in fixed-parameter tractable (FPT) algorithms that exploit structural parameters to achieve practical solvability. Particular emphasis is placed on the integration of machine learning approaches, especially Graph Neural Networks (GNNs), including automated neural architecture search (AutoGNP) and physics-inspired Hamiltonian relaxation methods that scale to millions of variables. Recent approximation improvements for fundamental problems such as metric k-median, correlation clustering, and variants of the Traveling Salesman Problem (TSP) are discussed, alongside modern decomposition techniques like tree decompositions and twin-width. The paper further examines scalability solutions in streaming and Massively Parallel Computation (MPC) models, hybrid neuro-symbolic solvers, and real-world applications in logistics, bioinformatics, and blockchain analysis. By synthesizing classical algorithmic breakthroughs with emerging learning-based methods, this work highlights promising directions for overcoming computational barriers in massive graph systems.