1 / 27

How to Rank with Fewer Errors A PTAS for Minimum Feedback Arc Set in Tournaments

How to Rank with Fewer Errors A PTAS for Minimum Feedback Arc Set in Tournaments. Claire Mathieu, Warren Schudy Brown University. Thanks to: Nir Ailon, Marek Karpinski, and Eli Upfal (for useful discussions)

aderyn
Download Presentation

How to Rank with Fewer Errors A PTAS for Minimum Feedback Arc Set in Tournaments

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. How to Rank with Fewer ErrorsA PTAS for Minimum Feedback Arc Set in Tournaments Claire Mathieu, Warren Schudy Brown University • Thanks to: • Nir Ailon, Marek Karpinski, and Eli Upfal (for useful discussions) • Cora Borradaile, Aparna Das, Micha Elsner, Dave McClosky, Fabio Vandin, and Matt Wronka (for useful comments on practice talks)

  2. Feedback arc set in tournaments FAS problem: minimize number of upsets NP-hard [Ailon Charikar Newman ’05, Alon ’06, Charbit Thomassè Yeo ‘07] Applications Ranking by pairwise comparisons Kemeny rank aggregation 1 upset C A A B C D D Best … Worst B (Animating…)

  3. Algorithms for FAS-T Slater and Alway 1961 Heuristic algorithm Fernandez de la Vega ’96, Arora Frieze Kaplan ’96 and many others small additive error Ailon Charikar Newman ‘05 Quicksort 3 approx LP + quicksort 2.5 approx Coppersmith Fleischer Rudra ‘06 Sort by Wins 5-approx Our Main Result: a PTAS for FAS-T:

  4. Aside: FAS in general graphs Log n log log n approx [Seymour ‘95, Even, Noar, Rao, Schieber ‘95] At least as hard as vertex cover, which cannot be approximated better than 1.36 [Karp ’72, Dinur ‘02] D C A B

  5. Outline Intro Algorithm Analysis Application (Kemeny Rank Aggregation)

  6. Towards our algorithm (1/6) An easy instance: when there is an ordering with zero upsets This means David beat Charlie 1 win 0 wins C A B D A C B D 3 wins 2 wins Wins: 3 2 1 0 Algorithm 1: Sort by Wins 5-approx [CFR]

  7. Towards our algorithm (2/6) A difficult instance for sort by wins Cost Θ(n²), so additive approx Θ(εn²) works! Algorithm 2: return better of additive approximation and sort by wins E A A B C D E D C B 2 2 2 2 2 2 2 2 2 2 OPT Sort by Wins

  8. Towards our algorithm (3/6) A hard instance for Algorithm 2: Algorithm 3: Sort by wins. Recursively divide in two until Cost=Ω(n²), then run additive approximation E D C B A J I H 7 7 7 7 7 4 3 1 1 1 G A B C D E F F G H I J Sorted by wins: Additive Approx (Animating…)

  9. Towards our algorithm (4/6) Algorithm 4: like Algorithm 3 but divide into two parts of random sizes 8 8 7 6 5 4 3 2 1 1 • A hard instance for algorithm 3: Uh-oh, we just “committed” ourselves to paying 3 but OPT=2 Sorted by wins: A B C D E F G H I J (Animating…)

  10. Towards our algorithm (5/6) Sorted by wins A B C D E F G H I J K L M N O Uh-oh, with constant probability we commit to paying Θ(n) (Animating…)

  11. Towards our algorithm (6/6) Final ingredient: after sorting by wins do single vertex move local optimization A B C D E F G I J H H K L M N O (Animating…)

  12. Our algorithm Algorithm: Sort by Wins Single vertex moves Result ordering: ImproveRec(all) ImproveRec(vertex set S): If (Cost(S, )≤Θ(ε²|S|²)) Choose k at random between 1 and |S| Recurse on the first k vertices of S Recurse on the other vertices else Run additive approx. Θ(ε³|S|²) on S E D C B A 7 7 7 7 6 3 3 3 2 0 A B C D E F G H I J Etc… (Animating…)

  13. Outline Intro Algorithm Analysis Application (Kemeny Rank Aggregation)

  14. Proof plan After first divide step “commit” to putting some vertices before others. For talk, will show there is a good ordering “consistent” with the first divide step Therefore Defined later Lemmas Non-Stopping condition Constant-factor approx

  15. Example 1 For this section of talk, “align” permutations. Ignore this detail. It is cheaper to put A to the left of B,C,D than to the right k A (Animating…)

  16. Example 2 k

  17. Example 3 k

  18. Second order pairs error are the root of all evil Definition: {u,v} are second order pair if u and v are alternating and at least one of them is split AFAFalternating, A split-> {A,F} 2nd order pair Ex. 1 EBBE not alternating-> {B,E} not 2nd order pair Ex. 2 AEAE alternating, A split-> {A,E} 2nd order pair Ex. 3 AEAE alternating, neither split-> {A,E} not 2nd order pair Ex. 3’

  19. Bounding (1/2) (Example 2 again) Bound this

  20. If {u,v} second-order pair, bound by 3 Otherwise, show zero by cases: e.g. {E,D} (Neither split) Proof: by def’ns and e.g. {C,D} (u<u<v<v pattern) Proof: edge {D,C} oriented same way in all orderings so terms cancel e.g. {C,A} (u<v<v<u pattern) Proof: and Bounding (2/2) (Animating…)

  21. Outline Intro Algorithm Analysis Application (Kemeny Rank Aggregation)

  22. Bounding (1/2) Def’n:

  23. Claim: Proof: [Diaconis and Graham ’77]: If the pair are in different orders, at least one of the orders has to pay for the pair(This is where the restriction to tournament graphs helps) Bounding (2/2)

  24. Reminder: overall proof Lemmas Non-Stopping condition Constant-factor approx Questions on proof?

  25. Outline Intro Algorithm Analysis Application (Kemeny Rank Aggregation)

  26. An application to voting Kemeny Rank Aggregation: (1959) Voters submit rankings of candidates Translate rankings into tournaments Add those tournaments together Find feedback arc set of resulting weighted tourney Nice properties, e.g. ranks Condorcet winner first [Young & Levenglick ’78, Young ‘95] Our PTAS generalizes to this! A>C>B C>A>B A>B>C A A C C A C B B B 1 C 2 A 1 0 3 2 B 0 1 1 B C A 2 2 3

  27. Open Question Real rankings often have ties, e.g. restaurant guides with ratings 1-5 Exists 1.5-approx [Ailon 07]; is there a PTAS? A: 5 C: 4B: 5 D: 3 A C D B

More Related