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Cheat Sheets for Hard Problems
This document provides an overview of techniques for solving hard computational problems. It discusses the complexity classes P, NP, and NP-complete, and provides examples of NP-complete problems like the travelling salesman problem. The document then discusses heuristic approaches like approximation and randomized algorithms. It also discusses exploiting additional structure in problem inputs and parameterized/exact analysis. Finally, it provides an example of using vertex cover techniques like degree bounds to solve the vertex cover problem in polynomial time for certain cases.
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Cheat Sheets for Hard Problems
- 2. Cheat Sheets for Hard Problems
- 3. Some problems tendto be harder than others.
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- 17. Independent SetClique SolveIndSet {Return Clique( ); }G
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- 19. Did you sayNP-complete?
- 20. Did you sayNP-complete?
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- 27. You have PolynomialTime.
- 28. You have PolynomialTime. WORK BACKWARDS!
- 29. Approximation & RandomizedAlgorithms
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- 32. A no-compromise situation? Exploitadditional structure in the input.
- 33. Parameterized & ExactAnalysis
- 34. Parameterized & ExactAnalysis Chromatic Number is easy on Graphs.Interval
- 35. Parameterized & ExactAnalysis Chromatic Number is easy on Graphs.Planar*
- 36. Parameterized & ExactAnalysis Chromatic Number is easy on Graphs.Bipartite
- 37. Good solutions tendto involve a combination of several techniques.
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- 43. Vertex Cover Every edgehas at least one end point in a vertex cover.
- 44. Vertex Cover Every edgehas at least one end point in a vertex cover.
- 54. Is there aVertex Cover with at most k vertices?
- 55. A vertex withmore than k neighbors.
- 56. Throw away allvertices with degree (k+1) or more. (And decrease the budget appropriately.)
- 57. Throw away allvertices with degree (k+1) or more. After all the high-degree vertices are gone... (And decrease the budget appropriately.)
- 58. ...any vertex cancover at most k edges.
- 59. ...any vertex cancover at most k edges. Suppose the current budget is (k-x).
- 60. ...any vertex cancover at most k edges. Suppose the current budget is (k-x). If the number of edges in the graph exceeds k(k-x)...?
- 61. Lots of edges- no small vertex cover possible. Few edges - brute force becomes feasible.
- 62. Lots of edges- no small vertex cover possible. Few edges - brute force becomes feasible. win/win situation
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