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

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Assigning students to schools

Example problem Goals

Minimize maximum distance to school Preserve color ratios

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Assigning students to schools

Example problem Goals

Minimize maximum distance to school Preserve color ratios

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Assigning students to schools

Example problem Goals

Minimize maximum distance to school Preserve color ratios

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What is a fair assignment?

Given

P point set colp color of each point p lh, uh (rational) lower/upper bound on ratio of color h c1, . . . , ck ⊆ P k locations (“centers”)

Assignment φ : P → {1, . . . , k} is fair if

lh ≤ |{p ∈ P | colp = h ∧ φ(p) = ℓ}| |{p ∈ P | φ(p) = ℓ}| ≤ uh for all colors h and all centers ℓ. [Chierichetti et al., NIPS 2017]

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Fair clustering

Find k centers and a fair assignment φ minimizing

max

p∈P dist(p, φ(p))

(k-center)

• p∈P

dist(p, φ(p)) (k-median)

• p∈P

dist2(p, φ(p)) (k-means) Also: Facility location, supplier variants

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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.

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State-of-the-Art II

Fair clustering

Approximation exact ratio fair essentially fair k-center 14 [RS18] 4∗ [CKLV17] facility location k-median Θ(t)∗ [CKLV17] k-means

∗at most two colors

[CKLV17] Chierichetti, Kumar, Lattanzi, Vassilvitskii. NIPS 2017. [RS18] Rösner, M. Schmidt. ICALP 2018.

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State-of-the-Art II

Fair clustering

Approximation exact ratio fair essentially fair k-center 14 5 4∗ [CKLV17] 3 facility location 3.488 k-median Θ(t)∗ [CKLV17] 4.675 k-means 62.856

∗at most two colors

[CKLV17] Chierichetti, Kumar, Lattanzi, Vassilvitskii. NIPS 2017. [RS18] Rösner, M. Schmidt. ICALP 2018.

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State-of-the-Art II

Fair clustering

Approximation exact ratio fair essentially fair k-center 14 5 4∗ [CKLV17] 3 ← today facility location 3.488 k-median Θ(t)∗ [CKLV17] 4.675 k-means 62.856

∗at most two colors

[CKLV17] Chierichetti, Kumar, Lattanzi, Vassilvitskii. NIPS 2017. [RS18] Rösner, M. Schmidt. ICALP 2018.

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Main result

Black-box approach:

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

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Main result

Black-box approach:

Any clustering C A fair LP solution weakly supervised rounding An essentially fair clustering C′ with cost(C′) ∈ O(cost(C) + FAIR-OPT) 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.

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Weakly supervised LP-rounding

“Guess” optimum radius τ from O(|P|2) candidates

The classical k-center LP with fairness constraints

• i∈P

xij = 1 for all j ∈ P xij ≤ yi for all i ∈ P

• i∈P

yi ≤ k xij = 0 for all i, j with dist(i, j) > τ ℓh

• j∈P

xij ≤

• col(pj)=colh

xij ≤ uh

• j∈P

xij for all i ∈ P and all colors h

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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)

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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)

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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)

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Step 1: Rounding y

1 3 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)

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Step 1: Rounding y

1 3 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)

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Step 1: Rounding y

1 3 1 3 1 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)

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Step 1: Rounding y

1 3 1 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)

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Step 1: Rounding y

1 3 1 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)

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Step 1: Rounding y

1 3 1 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)

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Step 1: Rounding y

1 3 1 3 1 3

≤ τ ≤ cost(C)

1 3

≤ FAIR-OPT + cost(C)

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)

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Step 2: Rounding x

Define: massh(x, C) :=

j∈C,col(j)=h xij

Claim: Can find integral ˆ x with

⌊massh(x, C)⌋ ≤ massh(ˆ x, C) ≤ ⌈massh(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.

ratio of color h in cluster C ∧ = massh(ˆ x, C) |C|

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The flow network for one color

1 1 . . . 1 all points with color h . . . all open centers “remainder” −⌊massh(x, C)⌋ arc (i, j) if xij > 0 Want: ⌊massh(x, C)⌋ ≤ massh(ˆ x, C) ≤ ⌈massh(x, C)⌉

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