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
X2~Exp(2)
X1X
=0.3
1~Unif(0,1) X2=0.6
X3=0.2
X3∼0.2
XX44~Bern{0,1}
=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−
Matroid [KW-STOC’12]:
2
1
𝑘+3
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:
• XOS FUNCTIONS
f(A ∪ B) + f(A ∩ B) ≤ f(A) + f(B)
OF
WIDTH W:
Given wi : V→ R+ for 1 ≤ i ≤ W
∀S⊆V:
{0,1} XOS:
f(S) = maxi{ wi(S) }
Additionally, each 𝐰𝐢 ∈ 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. 𝐩𝐢
 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
OUR RESULTS
7
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 ⋅ 𝐥𝐨𝐠 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
Correlation
gap = 1-1/e
F(x)
ALG
OCRS = 1/4
Multilinear Ext.
𝐅 𝐱 : = 𝐄𝑺~𝒙 [𝐟(𝐒)]
OPT
10
Corr Gap
CORRELATION GAP
F(x)
ALG
OCRS
• 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?
OPT
11
Corr Gap
NON-MONOT CORR GAP
ALG
F(x)
OCRS
• SUBMODULAR FUNCTIONS
 If we define similarly
EXAMPLE:
Let f be directed cut function
ϵ
1−ϵ
Let x = (ϵ,1 − ϵ)
Now, F(x) = ϵ2 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
Corr Gap
OPT
12
BOUNDING CORR GAP
F(x)
ALG
OCRS
• MONOTONE SUBMODULAR FUNCTIONS
 Define a continuous relaxation [CCPV-IPCO’07]
f*(x) := min𝑆
⊆ 𝑉
{ f(S) + ∑i∈V∖S fS i ⋅ xi }
 Show
Max-corr
f*(x)
≤
≤
F(x)/ (1-1/e)
• NON-MONOTONE SUBMODULAR FUNCTIONS
Intuition
[FMV-FOCS’07]
fT i ⋅ xi ] }
 We define a new continuous relaxation
f1/2*(x) := min𝑆
⊆ 𝑉
{ ET~S/2[ f(T) + ∑i∈V∖S
 Show
Max-corr
≤
O(1)⋅ f1/2*(x) ≤
O(1)⋅ F(x)
OPT
13
Corr Gap
ONLINE CRS
F(x)
ALG
OCRS
• GOAL: Get ALG close to F(x)
• NOTE:
 x is in the matroid polytope
 Each element i active w.p. at least xi
 For submodular functions, F(x/4) ≥ ¼ F(x)
• On average, can we pick each item i w.p. ≥
𝒙𝒊
?
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
Each 𝐰𝐢 ∈ 𝟎, 𝟏
{0,1}-XOS
ADDITIVE
𝐕
16
SUBADDITIVE TO {0,1}-XOS
• SUBADDITIVE
TO
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 ⊆ Ai .
• 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 ⋅ 𝐥𝐨𝐠 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
21
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
22
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
• 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