PROPHET INEQUALITIES Items n values drawn independently from known - - PowerPoint PPT Presentation

PROPHET INEQUALITIES

• n values drawn independently from known distributions:

Xi ∼ Fi

• τ-threshold mechanism: Take first item with Xi ≥ τ

THEOREM: The following give 2-approximation

• Median: Pr max Xi ≥ τ = 1/2

Samuel-Cahn’84

• Mean:

τ = E max Xi /2

Kleinberg-Weinberg’12

• WHAT IF VALUES CORRELATED?
• Ω n -approx Hill-Kertz’92
• What about “mild” correlations?

1 Xi ∼ Fi

𝛃-ap approx means: E reward ≥ 1/α ⋅ E max Xi

Items

arriving online Irrevocably select one

“Prophet Inequalities with Linear Correlations and Augmentations” by Nicole Immorlica, Sahil Singla, and Bo Waggoner

LINEAR CORRELATIONS MODEL

• VALUES

𝐘 ≔ A ⋅ 𝐙

Known matrix A ∈ 0,1 n×m and Yi ∼ Fi for known distributions Fi.

• E.g.,

A = identity matrix gives classical prophet inequality

2

m features n items Aij = degree to which item i exhibits feature j (0 ≤ Aij ≤ 1) value of feature Y1 Ym drawn independently value of item X1 Xn arriving online Bateni-Dehghani-Hajighayi-Seddighin’15 Chawla-Malec-Sivan’15

• 𝐭𝐬𝐩𝐱: row sparsity of A
• 𝐭𝐝𝐩𝐦: column sparsity of A

THM 1 (Single Item): 𝚰(𝐧𝐣𝐨{𝐭𝐬𝐩𝐱, 𝐭𝐝𝐩𝐦}) approx THM 2 (Multiple Items): Selecting r items

• FOR 𝐬 ≫ 𝐭𝐝𝐩𝐦: (𝟐 + 𝐩 𝟐 ) approximation
• FOR 𝐬 ≫ 𝐭𝐬𝐩𝐱: 𝚰(𝐭𝐬𝐩𝐱) approximation

MAIN SUBPROBLEM

Augmentation Problem 1. Think of Xi’s = independent part + dependent part: Xi = Zi + Wi 2. Can we recover E max Zi given only Zi distributions? Illustrative Example

• X1 drawn uniformly from [0,1]
• X2 is 104 w.p. 1/100; zero otherwise

Median threshold: τ ≈ 1/2, picks X1 half the time. Mean threshold: τ ≈ 50 never picks X1.

3

+𝟑 all the time AFTER ADDING SOME POSITIVE NOISE

Note: Prophet inequality for Wi = 0

Independent Correlated with past

positive “noise”

AUGMENTATION LEMMA: Threshold τ = E max Zi /2 guarantees 𝐅 𝐁𝐌𝐇𝛖 ≥ 𝐅 𝐧𝐛𝐲 𝐚𝐣 /𝟑

COLUMN SPARSITY

Inclusion-Threshold Mechanism: Run τ-threshold on a ``random subset’’

• Ignore each Xi independently w.p. (1 − 1/scol)
• Assign Yj to first surviving Xi that contains it
• Define Zi = σj→i AijYj and use Augmentation Lemma

Proof Idea: Show E max Zi ≈

E max Xi e⋅scol

1. Max Xi survives with 1/scol probability 2. Pr[Yj in Max Xi assigned to Zi] ≥ 1 − 1/scol scol−1 ≈ 1/e

4

THM (Single Item): 𝐏(𝐭𝐝𝐩𝐦) approximation

Y1 Ym τ-threshold mechanisms have Ω(n) approximation