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A very high-level overview of the modern toolkit involved in solving NP-hard problems.

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- 1. Cheat Sheets for Hard Problems
- 2. Some problems tend to be harder than others.
- 3. NP
- 4. P NP
- 5. P NP NP-complete
- 6. P NP
- 7. X
- 8. TSPX
- 9. TSPX solveTSP{ blah blah blah blah blah }
- 10. Independent Set
- 11. Independent Set Clique
- 12. Independent Set Clique Independent SetClique
- 13. Independent SetClique
- 14. Independent SetClique
- 15. Independent SetClique SolveIndSet { Return Clique( ); }G
- 16. TSPX solveTSP{ blah blah blah blah blah }
- 17. Did you say NP-complete?
- 18. Did you say NP-complete?
- 19. Travelling Salesman Satisfiability Integer Linear Programming
- 20. Minimum Multi-way cut ....
- 21. Heuristics
- 22. Heuristics
- 23. Formal analysis?
- 24. You have Polynomial Time.
- 25. You have Polynomial Time. WORK BACKWARDS!
- 26. Approximation & Randomized Algorithms
- 27. A no-compromise situation?
- 28. A no-compromise situation?
- 29. A no-compromise situation? Exploit additional structure in the input.
- 30. Parameterized & Exact Analysis
- 31. Parameterized & Exact Analysis Chromatic Number is easy on Graphs.Interval
- 32. Parameterized & Exact Analysis Chromatic Number is easy on Graphs.Planar*
- 33. Parameterized & Exact Analysis Chromatic Number is easy on Graphs.Bipartite
- 34. Good solutions tend to involve a combination of several techniques.
- 35. Vertex Cover
- 36. Vertex Cover Every edge has at least one end point in a vertex cover.
- 37. Vertex Cover Every edge has at least one end point in a vertex cover.
- 38. Is there a Vertex Cover with at most k vertices?
- 39. A vertex with more than k neighbors.
- 40. Throw away all vertices with degree (k+1) or more. (And decrease the budget appropriately.)
- 41. Throw away all vertices with degree (k+1) or more. After all the high-degree vertices are gone... (And decrease the budget appropriately.)
- 42. ...any vertex can cover at most k edges.
- 43. ...any vertex can cover at most k edges. Suppose the current budget is (k-x).
- 44. ...any vertex can cover at most k edges. Suppose the current budget is (k-x). If the number of edges in the graph exceeds k(k-x)...?
- 45. Lots of edges - no small vertex cover possible. Few edges - brute force becomes feasible.
- 46. Lots of edges - no small vertex cover possible. Few edges - brute force becomes feasible. win/win situation
- 47. Common Sense
- 48. Common Sense Approximate
- 49. Common Sense Approximate Randomize
- 50. Common Sense Approximate Randomize Exploit Input Structure
- 51. Common Sense Approximate Randomize Exploit Input Structure
- 52. http://neeldhara.com/summer2013 Slides and Other Resources

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