Cheat Sheets for Hard Problems
5 likes2,239 views
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.
Cheat Sheets for Hard Problems
- 2. Cheat Sheets for Hard Problems
- 3. Some problems tend to be harder than others.
- 4. NP
- 5. P NP
- 7. P NP
- 8. X
- 9. TSPX
- 12. Independent Set
- 17. Independent SetClique SolveIndSet { Return Clique( ); }G
- 19. Did you say NP-complete?
- 20. Did you say NP-complete?
- 21. Travelling Salesman Satisfiability Integer Linear Programming
- 24. Heuristics
- 25. Heuristics
- 26. Formal analysis?
- 27. You have Polynomial Time.
- 28. You have Polynomial Time. WORK BACKWARDS!
- 29. Approximation & Randomized Algorithms
- 32. A no-compromise situation? Exploit additional structure in the input.
- 33. Parameterized & Exact Analysis
- 34. Parameterized & Exact Analysis Chromatic Number is easy on Graphs.Interval
- 35. Parameterized & Exact Analysis Chromatic Number is easy on Graphs.Planar*
- 36. Parameterized & Exact Analysis Chromatic Number is easy on Graphs.Bipartite
- 37. Good solutions tend to involve a combination of several techniques.
- 38. Vertex Cover
- 43. Vertex Cover Every edge has at least one end point in a vertex cover.
- 44. Vertex Cover Every edge has at least one end point in a vertex cover.
- 54. Is there a Vertex Cover with at most k vertices?
- 55. A vertex with more than k neighbors.
- 56. Throw away all vertices with degree (k+1) or more. (And decrease the budget appropriately.)
- 57. Throw away all vertices with degree (k+1) or more. After all the high-degree vertices are gone... (And decrease the budget appropriately.)
- 58. ...any vertex can cover at most k edges.
- 59. ...any vertex can cover at most k edges. Suppose the current budget is (k-x).
- 60. ...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)...?
- 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
- 65. Common Sense
- 68. Common Sense Approximate Randomize Exploit Input Structure
- 69. Common Sense Approximate Randomize Exploit Input Structure