1. The Price of information in combinatorial optimization
(10th January, 2018)
Sahil Singla
(Carnegie Mellon University)

2. Motivation: Setting Up an Oil Drill
Set up One Oil Drill: Multiple potential sites
Have Estimates on their Values: Location, size, surveys
Conduct Inspections to Find Exact Value
Pay price per site
Which Sites Should you Inspect?
Want to Maximize
Max[value of inspected site] - Total inspection price
Similar Examples
Purchasing a company
Purchasing a house

3. Pandora’s Box How to generalize? Maximize Expected Utility
Given independent distributions on values: 𝑋 1 , 𝑋 2 , .. , 𝑋 𝑛
Given probing prices: 𝜋 1 , 𝜋 2 ,…, 𝜋 𝑛
Adaptively find 𝑃𝑟𝑜𝑏𝑒𝑑⊆{1,2,…,𝑛}
Maximize Expected Utility
max 𝑖∈𝑃𝑟𝑜𝑏𝑒𝑑 {𝑋 𝑖 } − 𝑖∈𝑃𝑟𝑜𝑏𝑒𝑑 𝜋 𝑖
Price of Information
Price of 1
per-box
X1~Unif(0,10)
X1= 4
X2~Exp(0.5)
X2=𝟔
X3~2
X3=𝟐
X4~Unif(0,10)
X4=9
Value = 𝟔
Price = 𝟑
Value = 𝟔
Price = 𝟐
Value = 𝟐
Price = 𝟏
Picked
How to generalize?
Weitzman’s optimal policy

4. Max-/Min-Weight Spanning Tree
Given distributions on edge values/costs: 𝑋 1 , 𝑋 2 , .. , 𝑋 𝑛
Given probing prices: 𝜋 1 , 𝜋 2 ,…, 𝜋 𝑛
Adaptively find 𝑃𝑟𝑜𝑏𝑒𝑑⊆{1,2,…,𝑛}
Utility Maximization Problem
Packing forest constraints: ℑ max 𝑆⊆𝑃𝑟𝑜𝑏𝑒𝑑 & 𝑆 ∈ ℑ { 𝑖∈𝑆 𝑋 𝑖 } − 𝑖∈𝑃𝑟𝑜𝑏𝑒𝑑 𝜋 𝑖
Disutility Minimization Problem
Covering spanning-tree constraints: ℑ′ min 𝑆⊆𝑃𝑟𝑜𝑏𝑒𝑑 & 𝑆 ∈ ℑ′ { 𝑖∈𝑆 𝑋 𝑖 } + 𝑖∈𝑃𝑟𝑜𝑏𝑒𝑑 𝜋 𝑖
Price of Information

5. Price of Information (PoI)
Given distributions on values/costs: 𝑋 1 , 𝑋 2 , .. , 𝑋 𝑛
Given probing prices: 𝜋 1 , 𝜋 2 ,…, 𝜋 𝑛
Adaptively find 𝑃𝑟𝑜𝑏𝑒𝑑⊆{1,2,…,𝑛}
Utility Maximization Problem
Packing constraints: ℑ max 𝑆⊆𝑃𝑟𝑜𝑏𝑒𝑑 & 𝑆 ∈ ℑ { 𝑖∈𝑆 𝑋 𝑖 } − 𝑖∈𝑃𝑟𝑜𝑏𝑒𝑑 𝜋 𝑖
Disutility Minimization Problem
Covering constraints: ℑ′ min 𝑆⊆𝑃𝑟𝑜𝑏𝑒𝑑 & 𝑆 ∈ ℑ′ { 𝑖∈𝑆 𝑋 𝑖 } + 𝑖∈𝑃𝑟𝑜𝑏𝑒𝑑 𝜋 𝑖
Price of Information
Examples
Max Matroid Basis
Maximum Matching
Knapsack
Examples
Min Matroid Basis
Vertex/Set Cover
Feedback Vertex Set
Facility Location
Prize-Collecting Steiner Tree
Semiadditive functions

6. Main Result
Frugal ≈ “Greedy”
Theorem 1: For any Packing/Covering Problem, an 𝜶–approx Frugal alg in the Free-Info World implies an 𝜶–approx strategy in the PoI World.
Think of Max Wt Matching with 𝛼=2
Problem
Approx Ratio
Max/Min Matroid Basis 1
Max Matching 2
Max 𝑘-system 𝑘
Max Knapsack
Min Vertex-/Set-Cover
min{f,log n}
Min Facility Location
1.861
Min Prize Collecting Steiner Tree 3
Feedback Vertex Set
O(log n)

7. OUTLINE Pandora’s Box and Price of Information Intuitive Examples
Bounding the Optimum Strategy
Using a Frugal Algorithm to Design a Strategy
Extensions and Conclusions

8. Optimal Adaptive Strategy
Assume Bernoulli Variables: 𝑿 𝒊 = 𝒗 𝒊 w.p. 𝒑 𝒊 𝟎 otherwise No
Yes
𝑋 1
𝑋 5
𝑋 2
𝑋 3
𝑋 4
𝑋 6
Adaptive
Decision Tree
Difficult to Find: Can be Exponential Sized
Want Simple Optimal/Approximate Strategy
Not comparing to hindsight optimum

9. Given Distributions : 𝑋 1 , 𝑋 2 , .. , 𝑋 𝑛
Given Probing Prices: 𝜋 1 , 𝜋 2 ,…, 𝜋 𝑛
Adaptively find 𝑃𝑟𝑜𝑏𝑒𝑑⊆{1,2,…,𝑛}
Maximize Expected Utility
max 𝑖∈𝑃𝑟𝑜𝑏𝑒𝑑 {𝑋 𝑖 } − 𝑖∈𝑃𝑟𝑜𝑏𝑒𝑑 𝜋 𝑖
Examples
Each has price 𝟏 𝑿 𝟏 , 𝑿 𝟐 ,…, 𝑿 𝟏𝟎𝟎𝟎 = 𝟐𝟎𝟎 w.p. 𝟎.𝟎𝟏 𝟎 otherwise
𝑿 𝟏 has price 𝟏𝟗𝟖 and others have price 𝟏 𝑿 𝟏 =𝟐𝟎𝟎 w.p. 𝟏 & 𝑿 𝟐 , 𝑿 𝟑 ,…, 𝑿 𝟏𝟎𝟎𝟎 = 𝟐𝟎𝟎 w.p. 𝟎.𝟎𝟏 𝟎 otherwise
Weitzman’s Index 𝝉 𝒊 is solution to 𝐸 𝑋 𝑖 − 𝜏 𝑖 + = 𝜋 𝑖
Open in decreasing index order
E[Opt] ≈𝟏𝟎𝟎
Simple Stopping rule: ∀𝒊: 𝐸 𝑀𝑎𝑟𝑔𝑖𝑛𝑎𝑙(𝑖) =𝐸 (𝑋 𝑖 −𝑐𝑢𝑟𝑟𝑒𝑛𝑡 + ]≤ 𝜋 𝑖
Naïve Greedy order can be bad: 𝑎𝑟𝑔𝑚𝑎 𝑥 𝑖 (𝐸 (𝑋 𝑖 −𝑐𝑢𝑟𝑟𝑒𝑛𝑡 + ]− 𝜋 𝑖 )
How to prove?

10. Proof Idea Theorem 1 (Utility Maximization):
Any 𝜶–approx Frugal alg in Free-Info World implies an 𝜶–approx strategy in the PoI World.
Three Steps:
E[Optimal Adap in PoI] ≤ E[Optimal Surrog in Free-Info]
Optimal Surrog in Free-Info ≤ 𝜶 × (Frugal Alg in Free-Info)
E[Frugal Alg in Free-Info] = E[Frugal Strategy in PoI]
⟹ E[Optimal Adap in PoI] ≤ 𝜶 × E[Frugal Strategy in PoI]

11. OUTLINE Pandora’s Box and Price of Information Intuitive Examples
Bounding the Optimum Strategy
Using a Frugal Algorithm to Design a Strategy
Extensions and Conclusions

12. The Surrogate Utility Maximization Problem
E[Optimal Adap in PoI] ≤ E[Optimal Surrog in Free-Info]
Optimal Surrog in Free-Info ≤ 𝜶 × (Frugal Alg in Free-Info)
E[Frugal Alg in Free-Info] = E[Frugal Strategy in PoI]
The Surrogate
Utility Maximization Problem
Index 𝜏 𝑖 is solution to 𝐸 𝑋 𝑖 − 𝜏 𝑖 + = 𝜋 𝑖
Surrogate 𝑌 𝑖 = min 𝑋 𝑖 , 𝜏 𝑖
Pandora’s Box: 𝐄[𝑂𝑝𝑡]≤𝐄[max{ 𝑌 1 , 𝑌 2 ,…, 𝑌 𝑛 }]
Packing Lemma: 𝐄[𝑂𝑝𝑡]≤𝐄[ max & 𝑆 ∈ ℑ ⁡{ 𝑖∈𝑆 𝑛 𝑌 𝑖 }]
Example: Single box 𝑋 1
𝐸[𝑂𝑝𝑡]=𝐸 𝑋 1 − 𝜋 1
𝐸 𝑌 1 = E[min 𝑋 1 , 𝜏 1 ]=𝐸[ 𝑋 1 − 𝑋 1 − 𝜏 ] =𝐸 𝑋 1 − 𝜋 1
Similar definitions for disutility minimization
Better than E[𝑂𝑝𝑡]≤𝐸[max{ 𝑋 1 , 𝑋 2 ,…, 𝑋 𝑛 }]

13. Packing Lemma Lemma: 𝐄[𝑂𝑝𝑡]≤𝐄[ max & 𝑆 ∈ ℑ ⁡{ 𝑖∈𝑆 𝑌 𝑖 }]
E[Optimal Adap in PoI] ≤ E[Optimal Surrog in Free-Info]
Optimal Surrog in Free-Info ≤ 𝜶 × (Frugal Alg in Free-Info)
E[Frugal Alg in Free-Info] = E[Frugal Strategy in PoI]
Packing Lemma
Lemma: 𝐄[𝑂𝑝𝑡]≤𝐄[ max & 𝑆 ∈ ℑ ⁡{ 𝑖∈𝑆 𝑌 𝑖 }]
Proof: Let 𝑨 𝒊 and 𝟏 𝒊 denote Opt picking and Opt probing, resp.
𝐄 𝑂𝑝𝑡 = 𝐄 𝑖 ( 𝑨 𝒊 𝑋 𝑖 − 𝟏 𝒊 𝜋 𝑖 ) = 𝐄 𝑖 ( 𝑨 𝒊 𝑋 𝑖 − 𝟏 𝒊 𝑋 𝑖 − 𝜏 𝑖 ) ≤ 𝐄 𝑖 ( 𝑨 𝒊 𝑋 𝑖 − 𝑨 𝒊 𝑋 𝑖 − 𝜏 𝑖 ) =𝐄[ 𝑖 ( 𝑨 𝒊 𝑌 𝑖 )] ≤ 𝐸[ max & 𝑆 ∈ ℑ ⁡{ 𝑖∈𝑆 𝑌 𝑖 }]
𝐄 𝑋 𝑖 − 𝜏 𝑖 + = 𝜋 𝑖
𝑌 𝑖 = min 𝑋 𝑖 , 𝜏 𝑖
since 𝑨 𝒊 ≤ 𝟏 𝒊
Q.E.D.
Note: Proof similar to [Kleinberg et al. EC’16]

14. OUTLINE Pandora’s Box and Price of Information Intuitive Examples
Bounding the Optimum Strategy
Using a Frugal Algorithm to Design a Strategy
Extensions and Conclusions

15. Frugal Alg An ALG in Free-Info World is Frugal Packing if Examples
E[Optimal Adap in PoI] ≤ E[Optimal Surrog in Free-Info]
Optimal Surrog in Free-Info ≤ 𝜶 × (Frugal Alg in Free-Info)
E[Frugal Alg in Free-Info] = E[Frugal Strategy in PoI]
Frugal Alg
An ALG in Free-Info World is Frugal Packing if
∃ marginal-value function 𝑔 𝒀 𝑀 , 𝑖, 𝑌 𝑖 ≥0 increasing in 𝑌 𝑖
In an iteration, let j have largest marginal (“best”)
If j can be picked then pick j irrevocably: 𝑀=𝑀∪𝑗
Else, discard j forever
Think of Max Wt Matching: 𝑔( 𝒀 𝑀 ,𝑖, 𝑌 𝑖 )= 𝑌 𝑖
Examples
Matroids or Matching: 𝑔( 𝒀 𝑀 ,𝑖, 𝑌 𝑖 )= 𝑌 𝑖
Set cover: 𝑔( 𝒀 𝑀 ,𝑖, 𝑌 𝑖 )= 1 𝑌 𝑖 ( ∪ 𝑗∈𝑀∪𝑖 𝑆 𝑗 − ∪ 𝑗∈𝑀 𝑆 𝑗 )
Observations
Marginal 𝑔 𝒀 𝑀 , 𝑖, 𝑌 𝑖 is independent of unseen elements
Captures primal-dual algos without “cleanup” phase

16. Frugal Strategy Strategy Proof Idea
E[Optimal Adap in PoI] ≤ E[Optimal Surrog in Free-Info]
Optimal Surrog in Free-Info ≤ 𝜶 × (Frugal Alg in Free-Info)
E[Frugal Alg in Free-Info] = E[Frugal Strategy in PoI]
Frugal Strategy
Strategy
Use index for unprobed elements to compute marginal
In an iteration, let j have largest marginal (“best”)
If j cannot be picked, discard j forever
Else, if j is unprobed then probe it
otherwise, j is already probed and pick it
Proof Idea
Couple both worlds: Same expected change in value
Although no price in Free-Info, they get lower value
𝑔( 𝒀 𝑀 ,𝑖, 𝜏 𝑖 )
𝑌 𝑖 = min 𝑋 𝑖 , 𝜏 𝑖

17. Applications Problem Approx Ratio Max/Min Matroid Basis 1 Max Matching2
Max 𝑘-system 𝑘
Max Knapsack
Min Vertex-/Set-Cover
min{f,log n}
Min Facility Location
1.861
Min Prize Collecting Steiner Tree 3
Feedback Vertex Set
O(log n)

18. OUTLINE Pandora’s Box and Price of Information Intuitive Examples
Bounding the Optimum Strategy
Using a Frugal Algorithm to Design a Strategy
Extensions and Conclusions

19. General Functions Disutility Minimization
Covering constraints ℑ′: min 𝑆⊆𝑃𝑟𝑜𝑏𝑒𝑑 & 𝑆 ∈ ℑ′ { 𝑖∈𝑆 𝑋 𝑖 } + 𝑖∈𝑃𝑟𝑜𝑏𝑒𝑑 𝜋 𝑖
More generally, min 𝑆⊆𝑃𝑟𝑜𝑏𝑒𝑑 & 𝑆 ∈ ℑ′ {𝑐𝑜𝑠𝑡(𝑆, 𝑋 )} + 𝑖∈𝑃𝑟𝑜𝑏𝑒𝑑 𝜋 𝑖
Semiadditive Functions: 𝑐𝑜𝑠𝑡 𝑆, 𝑋 = 𝑖∈𝑆 𝑋 𝑖 +ℎ(𝑆) 1
Facility location: ℎ 𝑆 = 𝑗∈𝑐𝑙𝑖𝑒𝑛𝑡𝑠 min 𝑖∈𝑆 𝑑(𝑖,𝑗)
Examples
Min Matroid Basis
Vertex/Set Cover
Feedback Vertex Set
Examples
Facility Location
Prize-Collecting Steiner Tree
ℎ: 2 𝑉 → 𝑅 ≥0
independent of 𝑋

20. Constrained Utility Max
Consider Pandora’s Box
Besides Prices, Allowed to Probe at most k items
How to Adaptively probe a set 𝑃𝑟𝑜𝑏𝑒𝑑, s.t. 𝑃𝑟𝑜𝑏𝑒𝑑 ≤𝑘
Maximize Expected Utility
max 𝑖∈𝑃𝑟𝑜𝑏𝑒𝑑 {𝑋 𝑖 } − 𝑖∈𝑃𝑟𝑜𝑏𝑒𝑑 𝜋 𝑖
Use Small Adaptivity Gap for Stoch Probing [GNS’17]
Find NA Soln: Submod max over knapsack
Convert NA soln to PoI strategy: Use Pandora’s Box
Other applications: Set Probing Problem
More generally, any packing constraint

21. Open Problems Problem 1: Any Interesting Algorithms Beyond Frugal?
Min s-t cut in the PoI-World
Shortest s-t Path in the PoI-World
Problem 2: Any Hardness Results for Max-Matching in the PoI-World?
Problem 3: Frugal Algorithms with Better Approx Factors?
Feedback Vertex Set
Max Matching
Problem 4:
How to Learn and Optimize from Samples when Probability Distributions Not Known?

22. Summary Questions? What is Price of Information
Pandora’s Box
Many packing/covering problems
Frugal Algorithms Suffice
OPT bounded by a random surrogate in Free-Info world
Run Frugal algorithm in PoI world using index
Extensions
Constrained Utility Max and Set Probing
Markov Chains by generalizing Gittins index policies
Open Problems
Beyond Frugal algos and prove Hardness results
Questions?

23. References
I. Dumitriu, P. Tetali, and P. Winkler. `On playing golf with two balls’. SIDMA’03.
A. Gupta, H. Jiang, Z. Scully, and S. Singla. ` The Markovian Price of Information ' . In Preparation.
A. Gupta, V. Nagarajan, and S. Singla. `Adaptivity Gaps for Stochastic Probing: Submodular and XOS Functions' . SODA’17.
A. Gupta, V. Nagarajan, and S. Singla. `Algorithms and Adaptivity Gaps for Stochastic Probing' . SODA’16.
R. Kleinberg, B. Waggoner, and E. Glen Weyl. `Descending Price Optimally Coordinates Search' . EC’16.
S. Singla. `The Price of Information in Combinatorial Optimization'. SODA’18.
M. L. Weitzman. `Optimal Search for the Best Alternative’. Econometrica’79.