This chapter of the book "A Course in Combinatorial Optimization" by Alexander Schrijver covers the fundamentals of shortest paths and trees. It begins with an introduction to shortest paths in directed graphs, including the concepts of walks, paths, and distances. The chapter discusses algorithms for finding shortest paths, such as Dijkstra's algorithm and its variants, and provides a detailed analysis of their time complexities. It also explores the relationship between shortest paths and cuts in graphs, and presents applications of shortest path problems in various fields, such as network planning and dynamic programming. The chapter then delves into the concept of polytopes and polyhedra, defining convex sets and hyperplanes, and proving the separation theorem. It introduces polytopes and polyhedra as special types of convex sets, and discusses the characterization of vertices in polyhedra. The chapter also covers the relationship between polytopes and polyhedra, showing that each bounded polyhedron is a polytope and vice versa. Finally, the chapter touches on the application of these concepts in solving optimization problems, such as finding minimum spanning trees and maximum reliability in network design. It concludes with exercises that reinforce the theoretical concepts and practical applications discussed in the chapter.This chapter of the book "A Course in Combinatorial Optimization" by Alexander Schrijver covers the fundamentals of shortest paths and trees. It begins with an introduction to shortest paths in directed graphs, including the concepts of walks, paths, and distances. The chapter discusses algorithms for finding shortest paths, such as Dijkstra's algorithm and its variants, and provides a detailed analysis of their time complexities. It also explores the relationship between shortest paths and cuts in graphs, and presents applications of shortest path problems in various fields, such as network planning and dynamic programming. The chapter then delves into the concept of polytopes and polyhedra, defining convex sets and hyperplanes, and proving the separation theorem. It introduces polytopes and polyhedra as special types of convex sets, and discusses the characterization of vertices in polyhedra. The chapter also covers the relationship between polytopes and polyhedra, showing that each bounded polyhedron is a polytope and vice versa. Finally, the chapter touches on the application of these concepts in solving optimization problems, such as finding minimum spanning trees and maximum reliability in network design. It concludes with exercises that reinforce the theoretical concepts and practical applications discussed in the chapter.