This document discusses the probabilistic method for proving deterministic combinatorial results. It covers four problems: how many minimum cuts exist in a graph; how many acute triangles exist in a set of points; proving the existence of a large sum-free subset; and proving a graph contains a large cut. The probabilistic method uses probability theory and randomized algorithms to analyze these types of combinatorial problems.

1. Randomized Algorithms
CS648
Lecture 20
Probabilistic Method
(part 1)
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2. PROBABILISTIC METHOD
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3. Probabilistic methods
Methods that use
• Probability theory
• Randomized algorithm
to prove deterministic combinatorial results
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4. PROBLEM 1
HOW MANY MIN CUTS ?
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5. Min-Cut
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6. Algorithm for min-cut
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7. Analysis of Algorithm for min-cut
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8. Number of min-cuts
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9. PROBLEM 2
HOW MANY ACUTE TRIANGLES ?
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10. How many acute triangles
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14. Two stage sampling
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15. Number of acute triangles
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16. PROBLEM 3
SUM FREE SUBSET OF LARGE SIZE
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17. Large subset that is sum-free
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18. Large subset that is sum-free
To prove it, use
• the fact that mapping is 1-1
and uniform.
• and Linearity of expectation.
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19. Large subset that is sum-free
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22. PROBLEM 4
LARGE CUT IN A GRAPH
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23. Large cut in a graph
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24. Large cut in a graph
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25. Large cut in a graph
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26. Large cut in a graph
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