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Unraveling the Truth Behind Game Theory and Algorithmic Myths in Internet Computing

This article explores and dispels long-held misconceptions about algorithms in game theory and Internet computing. Focused on pivotal works from renowned scholars such as Kuhn, Edmonds, and Shapley, it delves into the primal-dual paradigm and the complexities of matching algorithms like the Hopcroft-Karp method. By analyzing the interplay between theoretical foundations and practical applications in economics and computing, the text highlights the unique characteristics of alternating paths in bipartite versus non-bipartite graphs, offering insights into optimization and market efficiency.

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Unraveling the Truth Behind Game Theory and Algorithmic Myths in 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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