Discrete Structures and Combinatorics for AI: Graphs, Symmetry, Relaxation, the Limits of Learned Algorithms

Modern AI systems are built from continuous optimization, but many of the objects they manipulate are fundamentally discrete: graphs, token sequences, matchings, cuts, codes, constraints, subsets, plans, and combinatorial search spaces. Understanding these structures is essential for knowing not only what AI systems can compute, but also where their guarantees stop. Discrete Structures and Combinatorics for AI develops the discrete mathematics that underlies contemporary machine learning and connects it directly to the design and analysis of AI systems. The book moves from counting, extremal combinatorics, and the probabilistic method to graph structure, spectral graph theory, random graphs, Boolean functions, and coding. It then develops symmetry and the Weisfeiler-Leman hierarchy, polyhedral relaxations, matroids and submodularity, factor graphs and message passing, before turning to graph neural networks, learned combinatorial solvers, algorithm discovery, multi-agent computation, tokenisation, and discrete diffusion. Throughout, classical results are separated carefully from modern, model-dependent, and open claims. The text repeatedly asks five questions of every transfer from mathematics to AI: What is the object? What metric is controlled? Under which quantifier? In which regime? For which model class? This discipline clarifies distinctions that are often blurred in the literature-between representability and learnability, expressivity and generalisation, heuristic performance and certified approximation, or a discovered construction and a proof of optimality. Written for graduate students and researchers in mathematics, statistics, computer science, and machine learning, the book combines rigorous proofs with operational readings tied to real AI pipelines, including data deduplication, mixture-of-experts routing, retrieval and context selection, graph representation learning, constrained inference, learned optimisation, agent and tool orchestration, tokeniser design, and discrete generation. The result is both a graduate-level introduction to discrete mathematics for AI and a framework for reasoning precisely about the capabilities, limitations, and failure modes of learned algorithms.