Combinatorics

Deciphering Non-Constructivity in Graph Minor Theory What are Graph Minors? A graph minor of a graph G is a graph that can be obtained from G by contracting edges, deleting edges and deleting isolated vertices. More formally, a graph H is called a minor of a graph G if H can be formed from G…

Defining Combinatorial and Continuous Complexity Computational geometry algorithms can be categorized into two main types: combinatorial algorithms that operate on discrete, finite sets of geometric objects, and continuous algorithms that work with real-valued coordinates and quantities. This dichotomy leads to different challenges in analyzing computational complexity and resource requirements. Combinatorial computational geometry problems such as…

The Research-Implementation Gap A persistent bifurcation exists between theoretical computer science research and the practical implementation of systems. On the one side, theorists work in rarefied abstraction, developing conceptual models and proving possibilities. On the other, engineers build real-world applications, constrained by the hard limits of current infrastructure. This division between inquiry and invention stymies…

The P vs. NP Problem The most fundamental question in theoretical computer science is whether the complexity classes P and NP are equal. P represents the set of problems that can be solved in polynomial time by a deterministic Turing machine. NP represents problems where solutions can be verified in polynomial time by a non-deterministic…