1. Optimal Online Contention Resolution Schemes VIA Ex-Ante Prophet Inequalities
(22nd August, 2018)
Sahil Singla
(Carnegie Mellon University ➞ Princeton University)
Joint Work with euiwoong Lee

2. Online Rounding Algorithms USING Prophet Inequalities
(22nd August, 2018)
Sahil Singla
(Carnegie Mellon University ➞ Princeton University)
Joint Work with euiwoong Lee

3. Diamond-Selling Problem
Sell One Diamond: Multiple potential buyers
Buyers Arrive and Make a Take-it-or-Leave-it Bid
Decide Immediately and Irrevocably
Goal:
Maximize value of the accepted bid
Cannot go back to a declined bid
Similar Examples
Selling ad-slots
Finding a secretary/marriage partner

4. Prophet Inequality Model
𝑿 𝒊 = 𝒗 𝒊 w.p. 𝒑 𝒊 𝟎 otherwise
Given Indep Distributions on Values 𝑋 1 , 𝑋 2 , .. , 𝑋 𝑛
Irrevocably decide if picking: Known adversarial order
Objective:
Maximize value of picked buyer 𝐄[𝐀𝐥𝐠]
Compare to 𝐄[𝐌𝐚𝐱], i.e., 𝐄[𝐀𝐥𝐠] 𝐄[𝐌𝐚𝐱]
Cannot Beat ½-Approx
Let 𝑋 1 =1 w.p. 1
Let 𝑋 2 = 1 𝜖 w.p. 𝜖 0 otherwise
Picked
X1=0.3
X1~Unif(0,1)
X2=0.6
X2~Exp(2)
X3=0.2
X3∼0.2
X4~Unif(0,1)
X4=0.9
𝐄[𝐌𝐚𝐱] =𝜖⋅ 1 𝜖 + 1−𝜖 ⋅1≈2
𝐄[𝐀𝐥𝐠 ] ≤1
Krengel-Sucheston’77 gave approx

5. 1 2 -OCRS (Magician’s problem Alaei’11)
Prophet Ineq to OCRS
𝑿 𝒊 = 𝐯 𝒊 w.p. 𝒑 𝒊 𝟎 otherwise
Lemma: 𝐄[𝐌𝐚𝐱] is at most ex-ante relaxation
max: 𝑖 𝑦 𝑖 ⋅ v 𝑖
s.t. 𝑖 𝑦 𝑖 ≤1
𝑦 𝑖 ∈[0, 𝑝 𝑖 ]
At most 1 item
Probability i is max
1 2 -OCRS (Magician’s problem Alaei’11)
Suffices: Any algorithm that picks every i w.p. 𝑦 𝑖 2 while 𝑋 𝑖 = v 𝑖
because, implies 𝐄 𝐀𝐥𝐠 ≥ 1 2 ⋅ 𝑖 𝑦 𝑖 ⋅ v 𝑖 ⟹ prophet inequality
WLOG 𝑦 𝑖 = 𝑝 𝑖
by reducing 𝑝 𝑖
Thm ( 1 4 -OCRS): Consider i w.p on reaching, and pick if 𝑋 𝑖 = v 𝑖 .
PROOF: Pr i considered = Pr Reach i ⋅Pr i considered| reach i ≥ Pr Reach n ⋅ 1 2 ≥ 1 4
≥(1− 1 2 ∑ 𝑦 𝑗 ) ≥ 1 2

6. Matroid Prophet Inequality
Constraints = Matroid
Function = Additive
Given Indep Distributions on Values 𝑋 1 , 𝑋 2 , .. , 𝑋 𝑛
Irrevocably decide if picking: Known adversarial order
Goal: Maximize sum of values of Indep Set
Compare to 𝐄[𝐌𝐚𝐱 𝐈𝐧𝐝𝐞𝐩 𝐒𝐞𝐭] i.e., 𝐄[𝐀𝐥𝐠] 𝐄[𝐌𝐚𝐱 𝐈𝐧𝐝𝐞𝐩 𝐒𝐞𝐭]
𝑿 𝒊 = 𝐯 𝒊 w.p. 𝒑 𝒊 𝟎 otherwise
Lemma: 𝐄[𝐌𝐚𝐱 𝐈𝐧𝐝𝐞𝐩 𝐒𝐞𝐭] is at most ex-ante relaxation
max: 𝑖 𝑦 𝑖 ⋅ v 𝑖
s.t. 𝐲∈Matroid Polytope
𝑦 𝑖 ∈[0, 𝑝 𝑖 ]
𝛼-OCRS ⟹ 𝛼-Approx Ex- Ante Prophet Inequality
Probability i in offline max
Compares to relaxation
instead of 𝐄[𝐌𝐚𝐱 𝐈𝐧𝐝𝐞𝐩 𝐒𝐞𝐭]

7. Online Contention Resolution Scheme
Given 𝒚∈ ℛ 𝑛 inside matroid polytope
Each element 𝑖 active independently w.p. 𝑦 𝑖 : Revealed one-by-one
Decide immediately and irrevocably to select current active 𝑖
Defn (𝜶-OCRS): An online rounding algorithm that always remains feasible and ensures ∀𝒊: 𝐏𝐫 𝒊 𝐢𝐬 𝐬𝐞𝐥𝐞𝐜𝐭𝐞𝐝 & 𝐚𝐜𝐭𝐢𝐯𝐞 ≥𝜶⋅ 𝒚 𝒊
Introduced by Feldman-Svenson-Zenklusen’16:
Gave ¼-OCRS for matroids
Applications:
Stochastic problems with “commitment constraints”
Getting OCRS?

8. Our Main Results We can also design OCRSs from prophet inequalities.
Thm 1: 𝜶-Ex-Ante Proph-Ineq ⟹ 𝜶-OCRS
Thm 2: There Exists a 𝟏 𝟐 -Ex-Ante Prophet Inequality for a Matroid
Corollary: There Exists a 𝟏 𝟐 -OCRS for a Matroid

9. Outline Introduction Why Ex-ante Prophet Inequality implies OCRS
What is a Prophet Inequality and an OCRS
Why OCRS implies Prophet Inequality
Our Main Results
Why Ex-ante Prophet Inequality implies OCRS
CVZ Style LP for OCRS
Using LP Duality
How to Design Ex-ante Prophet Inequalities
Challenges
Proof Ideas
Conclusions
Extension to Random Order
Open Problems

10. Outline Introduction Why Ex-ante Prophet Inequality implies OCRS
What is a Prophet Inequality and an OCRS
Why OCRS implies Prophet Inequality
Our Main Results
Why Ex-ante Prophet Inequality implies OCRS
CVZ Style LP for OCRS
Using LP Duality
How to Design Ex-ante Prophet Inequalities
Challenges
Proof Ideas
Conclusions
Extension to Random Order
Open Problems

11. Valid distribution over Φ
CVZ Style LP for OCRS
Thm 1: 𝜶-Ex-Ante Proph-Ineq ⟹ 𝜶-OCRS
CVZ style exponential sized LP:
Let 𝜙∈Φ denote deterministic online rounding scheme
Let 𝑞 𝑖,𝜙 =Pr[𝑖 selected by 𝜙] and 𝜆 𝜙 =Pr[ϕ is running]
𝐦𝐚 𝐱 𝝀,𝑐 : 𝑐
s.t. ∀𝑖: 𝜙 𝑞 𝑖,𝜙 𝜆 𝜙 ≥ 𝑦 𝑖 ⋅𝑐
𝜙 𝜆 𝜙 =1
∀𝜙: 𝜆 𝜙 ≥0
c-selectability
Valid distribution over Φ
Suffices to show LP value = Dual LP value ≥𝛼
because implies 𝛼-OCRS

12. Using LP Duality Claim: 𝜶-Ex-Ante Proph-Ineq ⟹ Optimal LP value ≥𝛼
𝐦𝐚 𝐱 𝝀,𝑐 : 𝑐
s.t. ∀𝑖: 𝜙 𝑞 𝑖,𝜙 𝜆 𝜙 ≥ 𝑦 𝑖 ⋅𝑐
𝜙 𝜆 𝜙 =1
∀𝜙: 𝜆 𝜙 ≥0
𝐦𝐢𝐧 𝒛,𝜇 : 𝜇
s.t. ∀𝜙: 𝑖 𝑞 𝑖,𝜙 ⋅ 𝑧 𝑖 ≤𝜇
𝑖 𝑦 𝑖 𝑧 𝑖 =1
∀𝑖: 𝑧 𝑖 ≥0
Suffices to show for every feasible 𝒚 and 𝒛, ∃𝜙 s.t. 𝑖 𝑞 𝑖,𝜙 ⋅ 𝑧 𝑖 ≥𝛼
Let 𝑋 𝑖 equals 𝑧 𝑖 w.p. 𝑦 𝑖 : Algo 𝜙 value = 𝑖 𝑧 𝑖 ⋅ 𝑞 𝑖,𝜙 ≥ 𝛼⋅ 𝑖 𝑦 𝑖 𝑧 𝑖 = 𝛼
Ex-ante objective
Thm 1: 𝜶-Ex-Ante Proph-Ineq ⟹ 𝜶-OCRS

13. Outline Introduction Why Ex-Ante Prophet Inequality implies OCRS
What is a Prophet Inequality and an OCRS
Why OCRS implies Prophet Inequality
Our Main Results
Why Ex-Ante Prophet Inequality implies OCRS
CVZ Style LP for OCRS
Using LP Duality
How to Design Ex-Ante Prophet Inequalities
Challenges
Proof Ideas
Conclusions
Extension to Random Order
Open Problems

14. Challenges THM [KW’12]: There Exists 𝟏 𝟐 -Matroid Prophet Inequality
Thm 2: There Exists a 𝟏 𝟐 -Ex-Ante Prophet Inequality for a Matroid
THM [KW’12]: There Exists 𝟏 𝟐 -Matroid Prophet Inequality
Standard proph-ineq compares to 𝐄[𝐌𝐚𝐱 𝐈𝐧𝐝𝐞𝐩 𝐒𝐞𝐭]
Ex-ante proph-ineq compares to 𝑖 𝑦 𝑖 𝑣 𝑖
In general 1− 1 𝑒 ⋅ Ex-Ante Relax ≤ 𝐄[𝐌𝐚𝐱 𝐈𝐧𝐝𝐞𝐩 𝐒𝐞𝐭] ≤ Ex-Ante Relax
Implies a ⋅ 1− 1 𝑒 ex-ante prophet inequality
Correlation Gap

15. Proof Ideas
Thm 2: There Exists a 𝟏 𝟐 -Ex-Ante Prophet Inequality for a Matroid
Find probability distribution 𝓓 over independent sets s.t. 𝒚= 𝑆 is Independent 𝓓 𝑆 ⋅ 𝟏 𝑆
Ex-ante objective is 𝑖 𝑦 𝑖 𝑣 𝑖 = 𝑆 is Independent 𝓓 𝑆 ⋅ 𝑖∈𝑆 𝑣 𝑖
Modify KW’12 to compare to correlated distrib 𝓓 instead of product distrib
Corollary: There Exists a 𝟏 𝟐 -OCRS for a Matroid

16. Outline Introduction Why Ex-ante Prophet Inequality implies OCRS
What is a Prophet Inequality and an OCRS
Why OCRS implies Prophet Inequality
Our Main Results
Why Ex-ante Prophet Inequality implies OCRS
CVZ Style LP for OCRS
Using LP Duality
How to Design Ex-ante Prophet Inequalities
Challenges
Proof Ideas
Conclusions
Extension to Random Order
Open Problems

17. Optimal RCRS QUES: Better than 𝟏 𝟐 -OCRS for uniformly random arrival?
THM 3: There exists a (1− 𝟏 𝒆 )-RCRS for a Matroid
Proof uses Two Ideas
Exponential LP with variables all deterministic online algorithms
Show dual is at least (1− 𝟏 𝒆 ), which is an ex-ante prophet- secretary inequality
THM [EHKS’18]: There exists (1− 𝟏 𝒆 )-Prophet-Secretary for
Constraints = Matroid
Function = Additive

18. Summary and Open Problems
Connections between OCRS and Prophet Inequalities
OCRS ⟹ Prophet Ineq
Ex-ante Prophet Ineq ⟹ OCRS
Designing Ex-Ante Prophet Inequalities
1 2 -approx for adversarial order over a matroid
1− 1 𝑒 -approx for random order over a matroid
Open Questions
Can we avoid assuming adversarial order is known?
Can we get “greedy” OCRSs that are useful for submodular functions?
Questions?

19. References
S. Alaei. `Bayesian combinatorial auctions: Expanding single buyer mechanisms to many buyers’. FOCS’11
C. Chekuri, J. Vondrak, and R. Zenklusen. `Submodular function maximization via the multilinear relaxation and contention resolution schemes’. SICOMP’14.
S. Ehsani, T. Kesselheim, M.T. Hajiaghayi, and S. Singla. `Prophet Secretary for Matroids and Combinatorial Auctions’. SODA’18.
M. Feldman, O. Svensson, and R. Zenklusen. `Online contention resolution schemes’. SODA’16
R.D. Kleinberg and M. Weinberg. `Matroid prophet inequalities’. STOC’12
U. Krengel and L. Sucheston. `Semiamarts and finite values’. Bull. Amer. Math. Soc.’77
E. Lee and S. Singla. `Optimal Online Contention Resolution Schemes via Ex-Ante Prophet Inequalities'. ESA’18.