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

2. 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

3. 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]:
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?

4. 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

5. 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

6. 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

7. 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

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

9. 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

10. 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?

11. 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

12. 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]

13. 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

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

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

16. 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

17. {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

18. 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

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

20. 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

21. 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?

22. 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

23. 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?

24. 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