<![CDATA[Integer programming (IP) constitutes a fundamental framework in combinatorial optimization, extending linear programming by requiring decision variables to take integer values. This article presents a comprehensive narrative review of integer programming theory from the perspective of polyhedral analysis and cutting-plane methods. We examine the geometric relationship between the linear programming relaxation and the integer hull, the role of total unimodularity in guaranteeing integral extreme points, and the structural properties of valid inequalities that tighten this relaxation. The discussion covers the derivation of Chvátal-Gomory cuts, the geometry of branch-and-bound as a recursive partitioning of the feasible region, and Lagrangian relaxation as a dual bounding technique for integer programs, including the resulting integrality gap that distinguishes integer from linear duality. We further discuss recent developments connecting polyhedral theory with machine-learning-based cutting-plane selection. The findings show that polyhedral and geometric perspectives provide a coherent theoretical foundation for understanding why integer programs are computationally harder than their linear relaxations and how modern solvers exploit geometric structure to close the integrality gap. This review contributes to a deeper theoretical understanding of integer programming as a cornerstone of discrete optimization.]]>
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