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Exploring Vertex Cover, Hamiltonian Cycle, Max Independent Set, Max Clique, & TSP in Graph Theory

This lecture covers Vertex Cover, Hamiltonian Cycle, Max Independent Set, Max Clique problems in graph theory, discussing decision versions, NP-completeness, 2-approximation techniques, performance ratios, complexity, and approximations. It also touches on the Hamiltonian Cycle problem, the Traveling Salesman Problem (TSP), and relevant theorems. Prepare to deepen your understanding of these fundamental graph theory concepts and their applications.

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Exploring Vertex Cover, Hamiltonian Cycle, Max Independent Set, Max Clique, & TSP in Graph Theory

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  1. Lecture 24 Vertex Cover and Hamiltonian Cycle

  2. Vertex Cover • Given a graph G=(V,E), find a minimum subset C of vertices such that every edge is incident to a vertex in C.

  3. Decision Version • Given a graph G=(V,E) and positive integer k < |V|, is there a vertex cover C of size at most k?.

  4. Vertex-Cover is NP-complete Proof.

  5. 2-approximation • The vertex set of a maximal matching gives 2-approximation, i.e., approx / opt < 2

  6. Performance Ratio

  7. ρ-Approximation • ρ-approximation is apolynomial-timeapproximation satisfying: 1 < approx(input)/opt(input) <ρ for MIN or 1 < opt(input)/approx(input) <ρ for MAX

  8. Max Independent Set • An independent set is a vertex subset such that no edge exists between any two in the subset. • A vertex subset is a vertex cover if and only if its complement is an independent set. • Given a graph, find a maximum independent set.

  9. Complexity and Approximation • Max Independent Set is NP-hard. • For any constant c>1, there does not exist a polynomial-time c-approximation unless NP=P.

  10. Max Clique • A clique is a vertex subset which induces a complete subgraph. • A vertex subset is a clique of graph G if and only if it is an independent set of the complement of G. • :Max Clique: Given a graph, find a maximum clique.

  11. Complexity and Approximation • Max Clique is NP-hard. • For any constant c>1, there does not exist a polynomial-time c-approximation unless NP=P.

  12. Hamiltonian Cycle • Given a graph G, does G contain a Hamiltonian cycle? • Hamiltonian cycle is a cycle passing every vertex exactly once.

  13. HC is NP-complete

  14. Traveling Salesman • Given n cities with a distance table, find a minimum total-distance tour to visit each city exactly once.

  15. HC <m TSP p • From a given graph G, we need to construct (n cities, a distance table, k).

  16. Quiz Sample • Is Traveling Salesman Problem NP-complete? • Answer: No • Because NP contains only decision problems, TSP does not belong to NP.

  17. Quiz Sample • TSP is NP-hard. Does this mean that for any A in NP, A<m TSP? • Answer: No! • It is in wide sense that if TSP can be solved in p.-t., then every problem in NP can be solved in p.-t.. p

  18. Special Case Theorem • Traveling around a minimum spanning tree is a 2-approximation.

  19. Theorem • Minimum spanning tree + minimum-length perfect matching on odd vertices is 1.5-approximation

  20. Minimum perfect matching on odd vertices has weight at most 0.5 opt.

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