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Mar 15, 2023 •105 likes •1.41k views

Ensuring your favorite player wins: Tournament Rigging and Bribery Sushmita Gupta, Sanjukta Roy, Saket Saurabh, Meirav Zehavi (Knockout)Tournaments How can we ensure our favorite player/team wins the tournament? How to ensure that favorite

  1. T OURNAMENT F IXING I n terms of an SBA I NPUT : Win-lose graph (O LD ) Q UESTION : Does there exist a seeding ensuring that favorite wins the tournament? (N EW ) Q UESTION : Does the win-lose graph have a subgraph that is an SBA with favorite as the root ?

  2. T OURNAMENT F IXING I n terms of an SBA I NPUT : Win-lose graph (O LD ) Q UESTION : Does there exist a seeding ensuring that favorite wins the tournament? (N EW ) Q UESTION : Does the win-lose graph have a subgraph that is an SBA with favorite as the root ?

  3. Sketch of the Algorithm

  4. Sketch of the Algorithm I. GUESS:

  5. Sketch of the Algorithm I. GUESS: II. VERIFY:

  6. Sketch of the Algorithm I. GUESS: II. VERIFY: III. GREEDY:

  7. Sketch of the Algorithm (i) A template — a partial structure of some SBA where certain paths and subtrees are compressed. I. GUESS: (ii) We know the position of the affected vertices & position of their Least common ancestor. (iii) Length of those paths and sizes of subtrees II. VERIFY: III. GREEDY:

  8. Sketch of the Algorithm (i) A template — a partial structure of some SBA where certain paths and subtrees are compressed. I. GUESS: (ii) We know the position of the affected vertices & position of their Least common ancestor. (iii) Length of those paths and sizes of subtrees II. VERIFY: (i) Our guess is ``realizable” in terms of an SBA III. GREEDY:

  9. Sketch of the Algorithm (i) A template — a partial structure of some SBA where certain paths and subtrees are compressed. I. GUESS: (ii) We know the position of the affected vertices & position of their Least common ancestor. (iii) Length of those paths and sizes of subtrees II. VERIFY: (i) Our guess is ``realizable” in terms of an SBA (i) Fill up the paths & subtrees of the template III. GREEDY: (ii) If final outcome is an SBA, then done. (iii) Or else, guess again

  10. What if no favorable seeding exists for favorite ?

  11. Can favorite win with bribery?

  12. Can favorite win with bribery? I NPUT : Win-lose graph

  13. Can favorite win with bribery? I NPUT : Win-lose graph Q UESTION : Is it possible to fix some (say l) matches so that there is a seeding that enables favorite to win ?

  14. Can favorite win with bribery? I NPUT : Win-lose graph Q UESTION : Is it possible to fix some (say l) matches so that there is a seeding that enables favorite to win ? In terms of an SBA ….

  15. Can favorite win with bribery? I NPUT : Win-lose graph Q UESTION : Is it possible to fix some (say l) matches so that there is a seeding that enables favorite to win ? In terms of an SBA …. Can we reverse l arcs in the win-lose graph so that there will be an SBA with favorite as the root ?

  16. Can favorite win with bribery?

  17. Can favorite win with bribery? Answered in 2 n poly(n) time & poly(n) space, n := number of players.

  18. Can favorite win with bribery? Answered in 2 n poly(n) time & poly(n) space, n := number of players. Answered in 2 O(k2log k) poly(n) time & poly(n) space, k := FAS of win-lose graph

  19. Can favorite win with bribery? Answered in 2 n poly(n) time & poly(n) space, n := number of players. Answered in 2 O(k2log k) poly(n) time & poly(n) space, k := FAS of win-lose graph Uses our algorithm for T OURNAMENT F IXING

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