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Algorithmic Game Theory and Internet Computing

Dispelling an Old Myth about an Ancient Algorithm. Algorithmic Game Theory and Internet Computing. Vijay V. Vazirani Georgia Tech. Central to the Theory of Algorithms. Kuhn, 1955: Primal-dual paradigm Edmonds, 1965: Definition of P Valiant, 1979: Definition of #P

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Algorithmic Game Theory and Internet Computing

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  1. Dispelling an Old Myth about an Ancient Algorithm Algorithmic Game Theoryand Internet Computing Vijay V. Vazirani Georgia Tech

  2. Central to the Theory of Algorithms • Kuhn, 1955: Primal-dual paradigm • Edmonds, 1965: Definition of P • Valiant, 1979: Definition of #P • Jerrum, Valiant & Vazirani, 1986: Equivalance of random generation and approximate counting

  3. Matching in game theory and economics Gale & Shapley, 1962: Stable marriage Shapley & Shubik, 1971: Matching markets

  4. Matching in game theory and economics Gale & Shapley, 1962: Stable marriage Shapley & Shubik, 1971: Matching markets Mehta, Saberi, Vazirani & Vazirani, 2005: Adwords market

  5. V U

  6. V U

  7. V U

  8. V U

  9. V U

  10. V U

  11. V U

  12. Maximum matching algorithm

  13. Maximum matching algorithm O(n) iterations, O(m)/iteration, so O(mn) algorithm.

  14. Shortest augmenting paths

  15. M Disjoint. Therefore aug paths w.r.t. M. Maximal set.

  16. Shortest augmenting paths phase

  17. Hopcroft & Karp, 1973:

  18. Micali & Vazirani, 1980:

  19. Micali & Vazirani, 1980:

  20. In G =(U, V, E) w.r.t. current matching M define level(v): length of shortest alternating path from an unmatched U vertex to v. • Find, via an alternating BFS, starting from all unmatched vertices in U. At search level i: level i vertices find level i+1 vertices.

  21. V U

  22. U 0

  23. 1 V U 0

  24. U 2 1 V U 0

  25. Set of predecessors V 3 U 2 1 V U 0

  26. U 4 V 3 U 2 1 V U 0

  27. U 4 V 3 U 2 1 V U 0

  28. U 4 V 3 U 2 1 V U 0

  29. V 5 U 4 V 3 U 2 1 V U 0

  30. V 5 U 4 V 3 U 2 1 V U 0

  31. V 5 U 4 V 3 U 2 1 V U 0

  32. V 5 U 4 V 3 U 2 1 V U 0

  33. V 5 U 4 V 3 U 2 1 V U 0

  34. V 5 U 4 V 3 U 2 1 V U 0

  35. V 5 U 4 V 3 U 2 1 V U 0

  36. General Graphs

  37. G =(V, E). W.r.t. matching M define: • evenlevel(v): length of min even alternating path from an unmatched vertex to v. • oddlevel(v): length of min odd alternating path from an unmatched vertex to v. • minlevel(v): smaller of e(v) and o(v) maxlevel(v): larger of e(v) and o(v)

  38. Qualitative difference BFS honesty: Minimum length alternating paths in bipartite graphs are BFS honest.

  39. Qualitative difference BFS honesty: Minimum length alternating paths in bipartite graphs are BFS honest. Not so in non-bipartite graphs!

  40. Qualitative difference BFS honesty: Not so in non-bipartite graphs! Because shortest f to u path has v on it at opposite parity.

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