On the cost of essentially fair clusterings Ioana Bercea, Martin - PowerPoint PPT Presentation

On the cost of essentially fair clusterings Ioana Bercea, Martin Groß, Samir Khuller, Aounon Kumar, Clemens Rösner, Daniel Schmidt, Melanie Schmidt Aussois, January 2019 1

Assigning students to schools Example problem Goals Minimize maximum distance to school Preserve color ratios 2

Assigning students to schools Example problem Goals Minimize maximum distance to school Preserve color ratios 2

Assigning students to schools Example problem Goals Minimize maximum distance to school Preserve color ratios 2

What is a fair assignment? Given point set P col p color of each point p l h , u h (rational) lower/upper bound on ratio of color h c 1 , . . . , c k ⊆ P k locations (“centers”) Assignment φ : P → { 1 , . . . , k } is fair if l h ≤ |{ p ∈ P | col p = h ∧ φ ( p ) = ℓ }| ≤ u h |{ p ∈ P | φ ( p ) = ℓ }| for all colors h and all centers ℓ . [Chierichetti et al., NIPS 2017] 3

Fair clustering Find k centers and a fair assignment φ minimizing max p ∈ P dist( p , φ ( p )) ( k -center) � dist( p , φ ( p )) ( k -median) p ∈ P � dist 2 ( p , φ ( p )) ( k -means) p ∈ P Also: Facility location, supplier variants 4

State-of-the-Art I Classical (colorblind) clustering Approximation Hardness k -center 2 [G85,HS85] 2 [HN79] facility location 1.488 [L11] 1.463 [GK99] k -median 2.675+ ε [BPRST15] ≈ 1.74 [HN79] k -means 6.357 [ANSW17] 1.0013 [ACKS15,LSW15] [ACKS15] Awasthi, Charikar, Krishnaswamy, Sinop. SoCG 2015. [ANSW17] Ahmadian, Norouzi-Fard, Svensson, Ward. FOCS 2017. [BPRST15] Behsaz, Friggstad, Salavatipour, Sivakumar. ICALP 2015. [G85] Gonzalez. Theoretical Computer Science 1985. [GK99] Guha, Khuller. J. Algorithms 1999. [HN79] Hsu, Nemhauser. Discrete Applied Mathematics 1979. [HS85] Hochbaum, Shmoys. Mathematics of Operations Research 1985. [L11] Li. ICALP 2011. [LSW15] Lee, M. Schmidt, Wright. IPL 2017. 5

State-of-the-Art II Fair clustering Approximation exact ratio fair essentially fair 4 ∗ [CKLV17] k -center 14 [RS18] facility location Θ( t ) ∗ [CKLV17] k -median k -means ∗ at most two colors [CKLV17] Chierichetti, Kumar, Lattanzi, Vassilvitskii. NIPS 2017. [RS18] Rösner, M. Schmidt. ICALP 2018. 6

State-of-the-Art II Fair clustering Approximation exact ratio fair essentially fair 4 ∗ [CKLV17] k -center 14 5 3 facility location 3.488 Θ( t ) ∗ [CKLV17] k -median 4.675 k -means 62.856 ∗ at most two colors [CKLV17] Chierichetti, Kumar, Lattanzi, Vassilvitskii. NIPS 2017. [RS18] Rösner, M. Schmidt. ICALP 2018. 6

State-of-the-Art II Fair clustering Approximation exact ratio fair essentially fair 4 ∗ [CKLV17] k -center 14 5 3 ← today facility location 3.488 Θ( t ) ∗ [CKLV17] k -median 4.675 k -means 62.856 ∗ at most two colors [CKLV17] Chierichetti, Kumar, Lattanzi, Vassilvitskii. NIPS 2017. [RS18] Rösner, M. Schmidt. ICALP 2018. 6

Main result Black-box approach: Any clustering C An essentially fair weakly supervised rounding clustering C ′ with cost( C ′ ) ∈ O (cost( C ) + FAIR-OPT ) A fair LP solution Turns any colorblind clustering into an essentially fair clustering on the same set of centers with slightly increased cost. 7

Main result Black-box approach: Any clustering C An essentially fair weakly supervised rounding clustering C ′ with cost( C ′ ) ∈ O (cost( C ) + FAIR-OPT ) A fair LP solution Turns any colorblind clustering into an essentially fair clustering on the same set of centers with slightly increased cost. For instance: Use a 2-approximate k -center solution. 7

Weakly supervised LP-rounding “Guess” optimum radius τ from O ( | P | 2 ) candidates The classical k -center LP with fairness constraints � x ij = 1 for all j ∈ P i ∈ P x ij ≤ y i for all i ∈ P � y i ≤ k i ∈ P x ij = 0 for all i , j with dist( i , j ) > τ � � � x ij ≤ x ij ≤ u h for all i ∈ P and all colors h ℓ h x ij j ∈ P col( p j )=col h j ∈ P 8

Step 1: Rounding y 1 3 1 3 1 3 Use centers ( ) from given C as integer ˆ y for instance: Use 2-approximation for classical k -center (Fractionally) assign each point ( ) to its center’s ( ) center Obtain integer ˆ y , potentially fractional fair ˆ x Maximum radius bounded by FAIR-OPT + cost( C ) 9

Step 1: Rounding y 1 3 1 3 1 3 1 3 Use centers ( ) from given C as integer ˆ y for instance: Use 2-approximation for classical k -center (Fractionally) assign each point ( ) to its center’s ( ) center Obtain integer ˆ y , potentially fractional fair ˆ x Maximum radius bounded by FAIR-OPT + cost( C ) 9

Step 1: Rounding y 1 3 1 3 1 3 1 3 Use centers ( ) from given C as integer ˆ y for instance: Use 2-approximation for classical k -center (Fractionally) assign each point ( ) to its center’s ( ) center Obtain integer ˆ y , potentially fractional fair ˆ x Maximum radius bounded by FAIR-OPT + cost( C ) 9

Step 1: Rounding y 1 1 3 3 1 3 1 3 1 3 Use centers ( ) from given C as integer ˆ y for instance: Use 2-approximation for classical k -center (Fractionally) assign each point ( ) to its center’s ( ) center Obtain integer ˆ y , potentially fractional fair ˆ x Maximum radius bounded by FAIR-OPT + cost( C ) 9

Step 1: Rounding y 1 1 3 3 1 3 1 3 1 3 Use centers ( ) from given C as integer ˆ y for instance: Use 2-approximation for classical k -center (Fractionally) assign each point ( ) to its center’s ( ) center Obtain integer ˆ y , potentially fractional fair ˆ x Maximum radius bounded by FAIR-OPT + cost( C ) 9

Step 1: Rounding y 1 1 3 3 1 3 1 3 2 1 3 3 Use centers ( ) from given C as integer ˆ y for instance: Use 2-approximation for classical k -center (Fractionally) assign each point ( ) to its center’s ( ) center Obtain integer ˆ y , potentially fractional fair ˆ x Maximum radius bounded by FAIR-OPT + cost( C ) 9

Step 1: Rounding y 1 1 3 3 1 3 1 3 2 3 Use centers ( ) from given C as integer ˆ y for instance: Use 2-approximation for classical k -center (Fractionally) assign each point ( ) to its center’s ( ) center Obtain integer ˆ y , potentially fractional fair ˆ x Maximum radius bounded by FAIR-OPT + cost( C ) 9

Step 1: Rounding y 1 1 3 3 1 3 ≤ τ 1 3 2 3 Use centers ( ) from given C as integer ˆ y for instance: Use 2-approximation for classical k -center (Fractionally) assign each point ( ) to its center’s ( ) center Obtain integer ˆ y , potentially fractional fair ˆ x Maximum radius bounded by FAIR-OPT + cost( C ) 9

Step 1: Rounding y 1 1 3 3 1 3 ≤ τ ≤ cost( C ) 1 3 2 3 Use centers ( ) from given C as integer ˆ y for instance: Use 2-approximation for classical k -center (Fractionally) assign each point ( ) to its center’s ( ) center Obtain integer ˆ y , potentially fractional fair ˆ x Maximum radius bounded by FAIR-OPT + cost( C ) 9

Step 1: Rounding y 1 1 3 3 ≤ FAIR-OPT + cost( C ) 1 3 ≤ τ ≤ cost( C ) 1 3 2 3 Use centers ( ) from given C as integer ˆ y for instance: Use 2-approximation for classical k -center (Fractionally) assign each point ( ) to its center’s ( ) center Obtain integer ˆ y , potentially fractional fair ˆ x Maximum radius bounded by FAIR-OPT + cost( C ) 9

Step 2: Rounding x Define: mass h ( x , C ) := � j ∈ C , col( j )= h x ij Claim: Can find integral ˆ x with ⌊ mass h ( x , C ) ⌋ ≤ mass h (ˆ x , C ) ≤ ⌈ mass h ( x , C ) ⌉ and ⌊ mass( x , C ) ⌋ ≤ mass(ˆ x , C ) ≤ ⌈ mass( x , C ) ⌉ for all colors h and all clusters C without increasing cost. Consequence: Error of at most one point per cluster. = mass h (ˆ x , C ) ratio of color h in cluster C ∧ | C | 10

The flow network for one color 1 −⌊ mass h ( x , C ) ⌋ 1 “remainder” . . . . . . 1 all open centers all points with color h arc ( i , j ) if x ij > 0 Want: ⌊ mass h ( x , C ) ⌋ ≤ mass h (ˆ x , C ) ≤ ⌈ mass h ( x , C ) ⌉ 11