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Aggregation of Partial Rankings p-Ratings and Top-m Lists Nir Ailon Institute for Advanced Study

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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 learning: concept class for recommendation systems

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

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Partial Ranking linear order on equivalence classes of elt’s e.g. =[v 1 v 4 ], [v 2 v 7 v 5 ], [v 3 v 6 ] equivalent elements: ties v 1 < v 7, v 3 v 6 v 1 < v 7, v 3 v 6

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ranking of 2000 elements V={v 1,…, v 2000 } top-m lists (m=3) v 300, v 250, v 1845, […] V 250 < V 1845 v 300 < v 5 v 5 v 6

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ranking of 10 elements V={v 1,…, v 10 } p-ratings (p=3) v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 10 < v 1 v 5 v 4 ii

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

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

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

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Geometric Interpretation of d i ’ (full) i P i ( i ’) (partial) d( i, ) P i -1 ( i )

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

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Complexity 2-rating aggregation in P 2-rating aggregation in P Input: 8 i i = [V i1 ], [V i2 ] V i1 [ V i2 =V Input: 8 i i = [V i1 ], [V i2 ] V i1 [ V i2 =V 8 v 2 V n v = #{i: v 2 V i1 } 8 v 2 V n v = #{i: v 2 V i1 } sort V by decreasing n v sort V by decreasing n v top-2 aggregation NP-Hard top-2 aggregation NP-Hard Input: 8 i i = u i, v i, [V\{u_i, v_i}] Input: 8 i i = u i, v i, [V\{u_i, v_i}] Proof: Reduction from Min FAS in tournaments [ACN05, Alon06] Proof: Reduction from Min FAS in tournaments [ACN05, Alon06]

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v 1 v 2 v 3 v 4 22 v 1 v 2 v 3 v 4 v 1 v 2 v 3 v 4 11 33 [v 2 v 3 ], [v 1 v 4 ] v 2, v 3, [v 1 v 4 ] v 2, v 3, v 4, v 1 ranking of 4 candidates V={v 1, v 2, v 3, v 4 } RepeatChoice: 2 approximation

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

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u v with probability h(x(u,v)) with probability h(x(v,u)) h LPKwikSort h : 3/2 approximation

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

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

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