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

PROPHET INEQUALITIES Items • n values drawn independently from known distributions : X i ∼ F i 𝛃 -ap approx means: E reward ≥ 1/α ⋅ E max X i • τ -threshold mechanism: Take first item with X i ≥ τ Irrevocably select one T HEOREM : The following give 2 -approximation X i ∼ F i • Median: Pr max X i ≥ τ = 1/2 Samuel- Cahn’84 • Kleinberg- Weinberg’12 τ = E max X i /2 Mean: • W HAT IF VALUES C ORRELATED ? arriving online • Ω n -approx Hill- Kertz’92 • What about “mild” correlations? 1 “Prophet Inequalities with Linear Correlations and Augmentations” by Nicole Immorlica, Sahil Singla, and Bo Waggoner

Bateni-Dehghani-Hajighayi- Seddighin’15 LINEAR CORRELATIONS MODEL Chawla-Malec- Sivan’15 value of feature value of item m features • V ALUES X 1 Y 1 𝐘 ≔ A ⋅ 𝐙 Known matrix A ∈ 0,1 n×m and Y i ∼ F i for known distributions F i . n items A ij = degree to which item i exhibits feature j • A = identity matrix gives E.g., classical prophet inequality ( 0 ≤ A ij ≤ 1) • X n 𝐭 𝐬𝐩𝐱 : row sparsity of A Y m • 𝐭 𝐝𝐩𝐦 : column sparsity of A drawn independently arriving online THM 2 (Multiple Items): Selecting r items THM 1 (Single Item): 𝚰(𝐧𝐣𝐨{𝐭 𝐬𝐩𝐱 , 𝐭 𝐝𝐩𝐦 }) approx • F OR 𝐬 ≫ 𝐭 𝐝𝐩𝐦 : (𝟐 + 𝐩 𝟐 ) approximation • F OR 𝐬 ≫ 𝐭 𝐬𝐩𝐱 : 𝚰(𝐭 𝐬𝐩𝐱 ) approximation 2

MAIN SUBPROBLEM positive “noise” Augmentation Problem Think of X i ’s = independent part + dependent part: X i = Z i + W i 1. Independent Can we recover E max Z i given only Z i distributions? Correlated with past 2. Note: Prophet inequality for W i = 0 Illustrative Example • +𝟑 X 1 drawn uniformly from [0,1] A FTER A DDING SOME X 2 is 10 4 w.p. 1/100 ; zero otherwise • P OSITIVE N OISE all the time Median threshold: τ ≈ 1/2 , picks X 1 half the time. τ ≈ 50 never picks X 1 . Mean threshold: A UGMENTATION L EMMA : Threshold τ = E max Z i /2 guarantees 𝐅 𝐁𝐌𝐇 𝛖 ≥ 𝐅 𝐧𝐛𝐲 𝐚 𝐣 /𝟑 3

COLUMN SPARSITY τ -threshold mechanisms have Ω(n) approximation THM (Single Item): 𝐏(𝐭 𝐝𝐩𝐦 ) approximation Inclusion-Threshold Mechanism: Run τ - threshold on a ``random subset’’ • Ignore each X i independently w.p. (1 − 1/s col ) Y 1 • Assign Y j to first surviving X i that contains it • Define Z i = σ j→i A ij Y j and use Augmentation Lemma E max X i Proof Idea: Show E max Z i ≈ e⋅s col Y m Max X i survives with 1/s col probability 1. Pr[ Y j in Max X i assigned to Z i ] ≥ 1 − 1/s col s col −1 ≈ 1/e 2. 4