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Aggregation of Partial Rankings p-Ratings and Top-m Lists

Aggregation of Partial Rankings p-Ratings and Top-m Lists. Nir Ailon Institute for Advanced Study. Rankings. linear order on element set V (candidates) e.g. over V={v 1 ,v 2 ,v 3 }:  =v 2 ,v 3 ,v 1 voting : expressing preferences IR : sorting search results by relevance

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Aggregation of Partial Rankings p-Ratings and Top-m Lists

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  1. Aggregation of Partial Rankingsp-Ratings and Top-m Lists Nir Ailon Institute for Advanced Study

  2. Rankings linear order on element set V (candidates) e.g. over V={v1,v2,v3}: =v2,v3,v1 voting: expressing preferences IR: sorting search results by relevance learning: concept class for recommendation systems

  3. Rank Aggregation “fuse” list of rankings (votes) into one 1 2 . . k 

  4. Partial Ranking linear order on equivalence classes of elt’s e.g. =[v1v4], [v2v7v5], [v3v6] equivalent elements: ties v1< v7, v3 v6

  5. v300, v250, v1845, […] V250 < V1845 v300 < v5 v5 v6 ranking of 2000 elementsV={v1,…, v2000} top-m lists (m=3)

  6. v1  v2   v3  v4  v5  v6   v7 v8  v9   v10   i ranking of 10 elements V={v1,…, v10} p-ratings (p=3) v10 < v1 v5 v4

  7. Partial Rank Aggregation “fuse” a list of partial rankings into one (partial) 1 (partial) 2 . . (partial) k  (full)

  8. Objective Function • [FKMSV’04]: metrics on partial rankings • equivalence (up to const) of all metrics • 3 approx for metric “Fprof” • [ACN’05]: Approx algorithms for rank agg’

  9. Objective Function (Generalized Kemeny) d(i, ) = #{u,v | u< v , v< u} (not a metric on partial rankings) min d(i, )

  10. Geometric Interpretation of d Pi-1(i) i’ (full) d(i, )  i=Pi(i’) (partial)

  11. Results Overview • 2-rating in P but top-2 NP-Hard • 2-approximation: RepeatChoice • Generalizes pick-a-perm [ACN05, DKNS01] • 3/2-approximation: LPKwikSorth • Generalizes LPKwikSort [ACN05] • Solve LP • “Tweak” optimal solution using function h

  12. Complexity • 2-rating aggregation in P • Input: 8 i i = [Vi1], [Vi2] Vi1[ Vi2=V • 8 v2V nv = #{i: v 2 Vi1} • sort V by decreasing nv • top-2 aggregation NP-Hard • Input: 8 i i= ui, vi, [V\{u_i, v_i}] • Proof: Reduction from Min FAS in tournaments [ACN05, Alon06]

  13. v1  v2   v3    v4  2 v1  v2   v3  v4  v1 v2 v3  v4  3 1 ranking of 4 candidates V={v1, v2, v3, v4} [v2 v3], [v1 v4] v2, v3, [v1 v4] v2, v3, v4, v1 RepeatChoice: 2 approximation

  14. Rounding LP • w(u,v) = #{i: u <i v}/k • Satisfies : w(u,v) · w(u,w) + w(w,v) • Minimum FAS IP • minimize  x(u,v) w(v,u) : x(u,v) · x(u,w) + x(w,v)T: x(u,v) + x(v,u) = 1I: x(u,v) 2 {0,1} LP [0,1]

  15. LPKwikSorth: 3/2 approximation u 1 h v with probability h(x(v,u)) with probability h(x(u,v)) 0 1 0

  16. Other Metrics • d(i, ) does not penalize u,v if u i v • d’(i, ) = #{u,v| u i v} • d+d’/2 metric [FKMSV 05] • d’ depends on input, not output ) 2 and 3/2 approx algorithms for d+d’/2 • Constant approx for all metrics in FKMSV 04

  17. open questions • PTAS • Is Top-m agg’ NP-Hard for k=o(n) voters? • Best of LP-KwikSort, RepeatChoice: 4/3 approx? [ACN 05]

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