Combinatorial Prophet Inequalities (18th Jan, 2017) Sahil Singla (Carnegie Mellon University) Joint work with Aviad Rubinstein

Prophet Inequality Krengel and Sucheston [KS-BAMS’77] Given distributions of X1, X2, .. , Xn Sequentially revealed & irrevocably picked/ dropped Pick one to maximize expected value Compare to E[Max{X1, X2, .. , Xn}] 1/2 competitive tight algorithm known Picked X1=0.3 X1~Unif(0,1) X2=0.6 X2~Exp(2) X3=0.2 X3∼0.2 X4~Bern{0,1} X4=1

Additive Prophets Given Constraints, how to go Beyond Single Item? Add item values of a feasible set in ℱ Cardinality [HKS-EC’07, Alaei-FOCS’11]: 1− 1 𝑘+3 Matroid [KW-STOC’12]: 2 Downward-Closed [Rubinstein-STOC’16]: O( log n⋅log r ) Applications Truthful online auction mechanisms using sequentially posted-prices [HKS-EC’07, CHMS-STOC’10] How to go Beyond Additive Functions?

Combinatorial functions Submodular Functions: ∀ A,B ⊆ V: f(A ∪ B) + f(A ∩ B) ≤ f(A) + f(B) XOS Functions of Width W: Given wi : V→ R+ for 1≤i≤W ∀ S ⊆ V : f(S) = maxi{ wi(S) } {0,1} XOS: Additionally, each 𝐰 𝐢 ∈ 𝟎,𝟏 𝐕 Subadditive Functions: ∀ A,B ⊆ V: f(A ∪ B) ≤ f(A) + f(B) Remark: When monotone, Submodular ⊆ XOS ⊆ Subadditive

Simple Combinatorial prophets Combinatorial function f on n items Each item i is Bernoulli w.p. 𝐩 𝐢 Coin toss tells item i is active (participating) or not Value v v(S) = f(S ∩ A), where A = set of active items V A S

General Combinatorial prophets Each item i has k types Coin toss tells type of an item i Combinatorial function f on [n]×[k] items Note: We assume Submodularity/ Subadditivity on the extended support of [n]×[k] items Assume: Simple Combinatorial Prophets for the rest of the talk

Our results Theorem 1: ∃ O(1) prophet Inequalities for Constraints = Matroid Function = Non-Negative Submodular Moreover, we can find it in polynomial time. Theorem 2: ∃ O(𝐥𝐨𝐠 𝐧⋅𝐥𝐨 𝐠 𝟐 𝐫) prophet Inequalities for Constraints = Downward-Closed Function = Non-Negative Monotone Subadditive

OUTLINE Additive & Combinatorial Prophets Submodular Functions Over Matroids Subadditive Functions Over Downward-Closed Extensions and Open Problems

Proof idea Monotone Submodular Functions f S ≤f(T) if S⊆T Let x denote OPT’s marginals OPT F(x) Alg Multilinear Ext. 𝐅 𝐱 := 𝐄 𝑺~𝒙 [𝐟(𝐒)] Correlation gap = 1-1/e OCRS = 1/4

Correlation Gap Monotone Submodular Functions OPT F(x) Alg Corr Gap OCRS Correlation Gap Monotone Submodular Functions For worst possible marginals x Ratio of expected f(S) over independent distributions F(x) & Max-correlated distribution with marginals x Example: Let f(S) = min{1,|S|} Let x = (1/n , 1/n, … , 1/n) Now, F(x) = 1 − 1 e but Max-corr = 1. Non-Monotone Submodular Functions How do we define Correlation Gap?

Non-monot Corr Gap Submodular Functions If we define similarly OPT F(x) Alg Corr Gap OCRS Non-monot Corr Gap Submodular Functions If we define similarly Example: Let f be directed cut function Let x = (ϵ,1−ϵ) Now, F(x) = ϵ 2 but Max-corr = ϵ Instead, let fmax(S) := maxT⊆S {f(T)} For worst possible marginals x Ratio of expected fmax(S) over independent distributions Fmax (x) & Max-correlated distribution with marginals x 1−ϵ ϵ

Bounding Corr Gap Monotone Submodular Functions OPT F(x) Alg Corr Gap OCRS Bounding Corr Gap Monotone Submodular Functions Define a continuous relaxation [CCPV-IPCO’07] f*(x) := min𝑆⊆𝑉 { f(S) + ∑ i∈V∖S f S i ⋅ x i } Show Max-corr ≤ f*(x) ≤ F(x)/ (1-1/e) Non-Monotone Submodular Functions We define a new continuous relaxation f1/2*(x) := min𝑆⊆𝑉 { ET~S/2[ f(T) + ∑ i∈V∖S f T i ⋅ x i ] } Max-corr ≤ O(1)⋅ f1/2*(x) ≤ O(1)⋅ F(x) Intuition [FMV-FOCS’07]

Online crs Goal: Get ALG close to F(x) Note: OPT F(x) Alg Corr Gap OCRS Online crs Goal: Get ALG close to F(x) Note: x is in the matroid polytope Each element i active w.p. at least x i For submodular functions, F(x/4) ≥ ¼ F(x) On average, can we pick each item i w.p. ≥ 𝒙 𝒊 𝟒 ? Yes, using Online Contention Resolution Schemes [FSZ-SODA’16] Extends to non-monotone functions by losing 1/16

OUTLINE Additive & Combinatorial Prophets Submodular Functions Over Matroids Subadditive Functions Over Downward-Closed Extensions and Open Problems

Proof Idea Subadditive {0,1}-XOS Additive Each 𝐰 𝐢 ∈ 𝟎,𝟏 𝐕

Subadditive to {0,1}-xos Subadditive to XOS XOS to {0,1}-XOS. We use a reduction from [Dobzinski-APPROX’07] Loses O(𝐥𝐨𝐠 𝐧) XOS to {0,1}-XOS. We make buckets of weights in range 2i to 2i+1 Consider the best bucket Loses O(𝐥𝐨𝐠 𝐫) Subadditive {0,1}-XOS Additive

{0,1}-Xos to Additive Let set Ai represent weight wi . Hence, f(S) = maxi{ wi(S) } = maxi{ |Ai∩S|} Intuition: View As a New Downward-Closed Set-System Set S is feasible iff S∈ℱ and ∃ i s.t. S⊆ A i . Works, but needs care for General Prophets Subadditive {0,1}-XOS Additive

Additive over downward-closed We modify Rubinstein-STOC’16 For Bernoulli items O(log r) competitive algorithm Idea Always maintain a target τ Pick if doesn’t decrease Pr[achieving 𝛕] by a “lot” Argue decreasing 𝛕 increases Pr[achieving τ] by a “lot” Proof uses a dynamic potential function

OUTLINE Linear & Combinatorial Prophets Submodular Functions Over Matroids Subadditive Functions Over Downward-Closed Extensions and Open Problems

Extensions to Secretary problem Theorem 3: ∃ O(𝐥𝐨𝐠 𝐧⋅𝐥𝐨 𝐠 𝟐 𝐫) Competitive Algorithm for Secretary Problem Constraints = Downward-Closed Function = Non-Negative Monotone Subadditive Subadditive to {0,1}-XOS {0,1}-XOS to Additive Additive Functions over Downward-Closed

Open problems Question 1: Given a combinatorial function Constraints = Matroid Function = Non-Negative Monotone Submodular Can we get a 2 Prophet Inequality ? Question 2: Given a combinatorial function Constraints = Downward-Closed Function = Non-Negative Monotone Subadditive Can we get a O(𝐥𝐨𝐠 𝐧⋅𝐥𝐨𝐠 𝐫) Prophet Inequality?

Open problems Question 3: Given a combinatorial function Constraints = Matroid Function = Submodular Only in the Support Can we get O(1) Prophets ? Recollect: Our Proofs assume Submodularity/ Subadditivity on the extended support of [n]×[k] items Remark: For subadditive functions one can extend a function that is subadditive only in the support to being subadditive in the extended support

summary Questions? New Framework for Combinatorial Prophets O(1) Prophets: Submodular over Matroids New Correlation Gap OCRS O(log n⋅ log2r) Prophets: Subadditive & Downward-Closed Extends to Secretary Problem Open Problems Can we make our results tight? What if submodular only in the support ? Questions?

references S. Agrawal, Y. Ding, A. Saberi, Y. Ye. `Price of correlations in stochastic optimization’. OR’12 S. Alaei. `Bayesian combinatorial auctions: Expanding single buyer mechanisms to many buyers’. FOCS’11 G. Calinescu, C. Chekuri, M. Pal, J. Vondrak. `Maximizing a submodular set function subject to a matroid constraint’. IPCO’07 S. Chawla, J.D. Hartline, D.L. Malec, B. Sivan. `Multi-parameter mechanism design and sequential posted pricing’. STOC’10 S. Dobzinski. `Two randomized mechanisms for combinatorial auctions’. APPROX’07 U. Feige,V. S. Mirrokni, J. Vondrak. `Maximizing non-monotone submodular functions’.  SICOMP’11 M. Feldman, O. Svensson, R. Zenklusen. `Online contention resolution schemes’. SODA’16 M.T. Hajiaghayi, R.D. Kleinberg, T. Sandholm. `Automated online mechanism design and prophet inequalities’. AAI’07 R.D. Kleinberg, M. Weinberg. `Matroid prophet inequalities’. STOC’12 U. Krengel, L. Sucheston. `Semiamarts and finite values’. Bull. Amer. Math. Soc.’77 A. Rubinstein. `Beyond matroids: secretary problem and prophet inequality with general constraints’. STOC’16