Gaps for stochastic probing Submodular amp XOS Functions Sahil singla Carnegie mellon university Joint work with Anupam gupta and viswanath nagarajan 18 th ID: 524537
Download Presentation The PPT/PDF document "Adaptivity" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Adaptivity Gaps for stochastic probing(Submodular & XOS Functions)
Sahil singla Carnegie mellon universityJoint work with Anupam gupta and viswanath nagarajan
18
th
Jan,
2017Slide2
Stochastic probing
2Only 1 hour before shops close!Orienteering Constraint
Purchase
GiftsSlide3
Stochastic probing
3Only 1 hour before shops close!Orienteering Constraint0.8
0.5
0.5
0.25
0.4
Remark
:
If probabilities 1 & nodes distinct, then
Orienteering Problem
[Blum et al. FOCS’03]
30 min
15 min
25 min
15 min
20 min
30 min
20 min
20 minSlide4
Stochastic probing
4Input:Graph & Metric Probabilities:Independently PresentFunction(e.g., # Distinct Items)
Constraints
(
e.g.,
1 hour
Orienteering
)
OUTPUT:
Maximize the
Expected
Function Value
0.8
0.5
0.5
0.25
0.4
30 min
15 min
25 min
15 min
20 min
30 min
20 min
20 minSlide5
Simple strategy5
15 min
25 min
20 min
0.5(1+0.5+0)
+
0.5(0+0.5+0.8
) =1.4
0.5
0.5
0.8
0.8
0.5
0.5
0.25
0.4
30 min
15 min
25 min
15 min
20 min
30 min
20 min
20 min
0.5+0.5+0.4 =1.4
0.5
0.5
0.4
15 min
25 min
20 min
Can we Get Better?Slide6
Adaptivity gap
DefinitionRatio of best adaptive to best non-adaptive GAP :=
Example with
GAP
How
large
can the
GAP
be?
6
We Show
Small for many settings!Slide7
OUTLINEStochastic Probing & Adaptivity
GAPWhy Care About Adaptivity GAP?Proof Idea for Submodular FunctionsProof Idea for XOS Functions
Open Problems
7Slide8
best adaptive strategy8
Decision Tree
Every
Root-Leaf Path at most 1 Hour
Can Be
Exponential
Sized
!Slide9
Why Care about Adaptivity gap?9
BEST ADAPTIVEBEST NON-ADAPTIVE
ALGORITHM
Goal
: Bound Ratio
Adaptivity
GAP
Approx
ratio
.
GAP
GAP
:=
For what
Constraints
& Functions is the
GAP
small?Slide10
Constraints & Functions
ConstraintsDownward-Closed: If a set can be probed then also its subsets e.g. Knapsack, Matroid, OrienteeringFUNCTIONSSubmodular: If then
XOS
of
w
idth
:
Given
w
i
: V
R
+
for
S
V :
f(S
) = max
i
{
w
i
(S)
}
Subadditive
:
A,B
V: f(A
B)
f(A) + f(B
) Submodular
XOS Subadditive (when monotone)
Remark: XOS
approx
Subadditive
[Dobzinski-APPROX’07]
10Slide11
results11
Theorem 1
: Adaptivity Gap for
Constraints =
Downward-Closed
Function = Non-Negative
Submodular
is
.
Moreover, it
’
s
at most 3
when function also
Monotone
.
ADAP
NON-ADAP
ALGO
GAP
GAP
Theorem 2
: Adaptivity Gap for
Constraints
=
Downward-Closed
Function =
XOS
of
width W
is
.
e.g., # distinct items is monotone submodularSlide12
A Non-Adaptive algorithmQuasi-poly time
not possible unless NP
Quasi-poly
12
ADAP
NON-ADAP
ALGO
GAP
GAP
Theorem 3
: ALGO has
for
Constraints =
Orienteering
Function
= Number
of
Distinct
Items
Chekuri
and Pal – FOCS’05
Corollary
: Using Theorems 1 & 3 for
Constraints =
Orienteering
Function
= Number
of
Distinct
Items
ALGO is
approximation for ADAP.
Slide13
Prior workAssumes Simple Constraints (e.g., any k items)
Bound ADAP using LPCompare ALGO to LP[GN-IPCO’13], [DGV-FOCS’04],[ANS-WINE’08], [ASW-STACS’14]We Want General ConstraintsCannot always bound ADAP
using
LP
Assumes Simple Functions
Our
previous
work
for
Matroid
Rank
Fns
[GNS-SODA’16
]
13
ADAP
NON-ADAP
ALGO
LPSlide14
OUTLINEStochastic Probing & Adaptivity
GAPWhy Care About Adaptivity GAP?Proof Idea for Submodular FunctionsProof Idea for XOS Functions
Open Problems
14Slide15
TWO IDEAS1. Random Root-Leaf Path GoodOnly show existence
2. Stem-By-Stem InductionStem Lemma15
ADAP
NON-ADAP
ALGO
GAP
GAP
0.7
0.3
0.5
0.5
0.3
0.2
0.8
0.7
NO
YES
Theorem 1
: Adaptivity Gap for
Constraints =
Orienteering
Function
= Number
of
Distinct
Items
is
at most 3
.Slide16
Only Existence
of a “Good” NA-PathAssume Best ADAP Strategy KnownRandom Path with ADAP ProbabilitiesExample:Red
path
prob
=
Here
ADAP
gets
Here
NA
gets
Show: E[
Random Path
]
ADAP
Since
Best
Path
E[
Random Path
]
Adaptivity
GAP
Random
root-leaf
path
16
0.7
0.3
0.5
0.5
0.3
0.2
0.8
0.7
NO
YESSlide17
Stem is the All-No Path
ADAP
= Distinct items that appear before
i
NA
Stem Lemma
Stem-by-stem
induction
17
i
0.7
0.3
0.5
0.5
0.3
0.2
0.8
0.7
0.2
0.8
0.7
0.3
0.5
0.5
0.2
0.8
NO
YESSlide18
Stem lemma
LEMMA: Let = Distinct items that appear before i
. Then
i.e.
PROOF
:
=
Now
i.e.
i.e.
i.e.
18
0.7
0.5
0.2
0.3
0.5
0.8
NO
YESSlide19
OUTLINEStochastic Probing & Adaptivity
GAPWhy Care About Adaptivity GAP?Proof Idea for Submodular FunctionsProof Idea for XOS Functions
Open Problems
19Slide20
PROOF IDEA FOR XOS
Can assume coefficients in wi are ``small’’ and we can truncate tree s.t. no-path has ``very large’’ value.Control variance of wi using Freedman’s concentration
inequality, and take
Union Bound
over all
w
i
Remark
: Some of these ideas also used in our previous work [GNS-SODA’16]
Can We Get
20
Theorem 2
: Adaptivity Gap for
Constraints
=
Downward-Closed
Function =
XOS
of
width W
is
.
For
,
f(S
)
:=
max
i
{
w
i
(S) }
Slide21
OUTLINEStochastic Probing & Adaptivity
GAPWhy Care About Adaptivity GAP?Proof Idea for Submodular FunctionsProof Idea for XOS Functions
Open
Problems
21Slide22
Open Problems22
Question 1
: Is
Adaptivity
Gap for
Constraints
= Downward-Closed
Function = Non-Negative
Monotone
Submodular
at most
e/(e-1)
?
Question 2
: Is
Adaptivity
Gap for
Constraints
= Downward-Closed
Function = Non-Negative Monotone
Subadditive
at most
?
What
about
Max
Indep
Set
with
Stoch
Vertices?
Recollect
:
Proving
for XOS functions
suffice [Dobzinski-APPROX’07]Slide23
Lower Bound for XOSPROOF
:Complete k-ary tree, where kk = W = n/2Each edge active w.p. 1/kFunction = MaxRoot-Leaf Path {# active edges}Constraint = At most k2 probes
Now,
ADAP
=
NA
=
23
Theorem
: Adaptivity Gap for
Constraints
=
Downward-Closed
Function =
XOS
of
width W
is
.
Slide24
summaryStochastic Probing Problem
Captures many natural problemsAdap GAP Small for Submodular & low-width XOS FnsFocus on simpler non-adaptive algosRandom root-leaf path & stem-by-stem inductionFreedman’s concentration inequalityOpen Problemse/(e-1) for Monotone Submodular Functions?p
olylog
(n) for Subadditive Functions?
24
Questions
?
ADAP
NON-ADAP
ALGO
GAP
GAP
Slide25
referencesM. Adamczyk, M.
Sviridenko, and J. Ward. `Submodular Stochastic Probing on Matroids’. STACS’14.A. Asadpour, H. Nazerzadeh, and A. Saberi. `Maximizing Stochastic Monotone Submodular Functions’. WINE’08.A. Blum, S. Chawla, D. Karger, T. Lane, A. Meyerson, M.
Minkoff
. `
Approximation Algorithms for Orienteering and Discounted-Reward
TSP
’.
FOCS’03
.
C.
Chekuri
and M. Pal. `
A Recursive Greedy Algorithm for Walks in Directed Graphs
’.
FOCS’05
.
B.C. Dean, M.X.
Goemans
, and J.
Vondrak. `Approxmiating
the Stochastic Knapsack Problem: The Benefit of Adaptivity’. FOCS’04.S. Dobzinski. `Two randomized mechanisms for combinatorial auctions
’.
APPROX’07A. Gupta and V.
Nagarajan. `A Stochastic Probing Problem with Applications’.
IPCO’13.A. Gupta, V.
Nagarajan, and S. Singla. `Algorithms and
Adaptivity Gaps for Stochastic Probing’. SODA’16.
A. Gupta, V. Nagarajan, and S. Singla
. `Adaptivity Gaps for Stochastic Probing: Submodular and XOS Functions
’. SODA’17.
25