Assuming P ≠ NP, NP-complete (NPC) problems are considered intractable. To manage these problems, various strategies sacrifice at least one aspect of tractability. Monotone Local Search (MLS) is one such strategy, utilizing randomized sampling and FPT algorithms to create exact solutions. This talk is a brief introduction to MLS.

We explore treewidth, its definition, its connections to NP-hard problems, and its application in solving 3-Coloring through a case study. This talk aims to highlight treewidth as a valuable property for understanding a graph's "treelike" nature and its utility in addressing complex algorithmic challenges, demonstrating its significance since the mid-80s.

This talk is a brief introduction to piecewise analysis.