FOLLOW AN IDEA
Combinatorial Optimization
Explore discrete structures; network algorithms; complexity and approximations. Concepts and connections are available independently of video coverage.
Discrete structures
Shortest path models
Dijkstra solves non-negative networks; Bellman-Ford tolerates negative edges and underpins route pricing.
Video explanations to be addedMinimum spanning tree
Connect all nodes at least total cost with Kruskal or Prim; the base of routing and pipeline design.
Video explanations to be addedGraph matchings
A matching is a set of edges with no shared endpoints.
Video explanations to be addedMatroids
A matroid abstracts independence through hereditary and exchange axioms.
Video explanations to be added
Network algorithms
Network flow models
Maximise flow through capacities; the min-cut theorem turns a bottleneck argument into a computable bound.
Video explanations to be addedMax-flow min-cut theorem
For a finite capacitated network, the maximum flow value equals minimum separating cut capacity.
Video explanations to be addedBipartite matching
Matching in a two-part graph pairs vertices across parts without shared endpoints.
Video explanations to be addedAssignment problem
A costed one-to-one pairing can be optimized using matching or linear programming structure.
Video explanations to be added
Complexity and approximations
Traveling salesman routing
Visit each site once at least tour length; heuristics plus bounds are the practical modelling answer.
Video explanations to be addedNP-completeness
An NP-complete decision problem lies in NP and admits polynomial reductions from every NP problem.
Video explanations to be addedApproximation algorithms
An approximation algorithm provides a provable ratio to the optimum under stated assumptions.
Video explanations to be addedInteger relaxations
Removing integrality yields easier problems whose optima can bound the discrete optimum.
Video explanations to be added