1. Monitoring rivers and lakes [IJCAI ‘07]
Need to monitor large spatial phenomena
Temperature, nutrient distribution, fluorescence, …
NIMS Kaiser et.al. (UCLA)
Color indicates actual temperature
Predicted temperature
Depth
Location across lake
Predict at unobserved locations 1 1

2. Monitoring water networks [J Wat Res Mgt 2008]
Contamination of drinking water could affect millions of people
Hach Sensor
Sensors
Simulator from EPA
Place sensors to detect contaminations
“Battle of the Water Sensor Networks” competition
~$14K
Where should we place sensors to quickly detect contamination?

3. Example reward functions
Entropy Reward[ P(X) ] = -H(X) = x P(x) log2 P(x)
Expected mean squared prediction error (EMSE)
Reward[ P(X) ] = -1/n s Var(Xs),
Many other objectives possible and useful… 3

4. Properties and Examples
Submodularity
Properties and Examples

5. Key observation: Diminishing returns
Placement B = {Y1,…, Y5}
Placement A = {Y1, Y2} Y2 Y1 Y‘
Adding Y’ will help a lot!
Adding Y’ doesn’t help much
New sensor Y’ 5

6. Set functions Finite set V = {1,2,…,n} Function F: 2V ! R
Will always assume F(;) = 0 (w.l.o.g.)
Assume black-box that can evaluate F for any input A
Approximate (noisy) evaluation of F is ok (e.g., [37]) 6

7. Submodular set functions
Submodularity: B A
For AµB, sB, F(A [ {s}) – F(A) ¸ F(B [ {s}) – F(B)

8. Set cover is submodular
A={S1,S2} S1 S2 S’
F(A[{S’})-F(A)
F(B[{S’})-F(B) S1 S2 S3 S’ S4
B = {S1,S2,S3,S4} 8

9. Example: Mutual information
Given random variables X1,…,Xn
F(A) = I(XA; XVnA) = H(XVnA) – H(XVnA |XA)
Lemma: Mutual information F(A) is submodular 9 9

10. Closedness properties
F1,…,Fm submodular functions on V and 1,…,m > 0
Then: F(A) = i i Fi(A) is submodular!
Submodularity closed under nonnegative linear combinations!
Extremely useful fact!!
F(A) submodular )  P() F(A) submodular!
Multicriterion optimization: F1,…,Fm submodular, i¸0 ) i i Fi(A) submodular 10

11. Monotonicity A set function is called monotonic if AµBµV ) F(A) · F(B)11

12. Approximate maximization
Greedy algorithm:
Start with A0 = ;
For i = 1 to k
si := argmaxs F(Ai-1 [ {s}) - F(Ai-1)
Ai := Ai-1 [ {si} 12 12

13. One reason submodularity is useful
Theorem [Nemhauser et al ‘78]
Greedy algorithm gives constant factor approximation
F(Agreedy) ¸ (1-1/e) F(Aopt)
Greedy algorithm gives near-optimal solution!
For information gain: Guarantees best possible unless P = NP! [Krause & Guestrin ’05]
~63% 13 13 13 13

14. Temperature data from sensor network
Performance of greedy
Optimal
Greedy
Temperature data from sensor network
Greedy empirically close to optimal. Why? 14

15. Example: Sensor Placement

16. Lazy greedy algorithm [Minoux ’78]
First iteration as usual
Keep an ordered list of marginal benefits i from previous iteration
Re-evaluate i only for top element
If i stays on top, use it, otherwise re-sort
Benefit s(A) a a a d b b b c c e d d c e e 16 16 16

17. Result of lazy evaluation1 2 3 4 5 6 7 8 9 10
100
200
300
Number of sensors selected
Running time (minutes)
Exhaustive search
(All subsets)
Naive
greedy
Fast greedy
Lower is better 17 17

18. Data dependent bounds [Minoux ’78]
Suppose A is candidate solution to
argmax F(A) s.t. |A| · k
and A* = {s1,…,sk} be an optimal solution
Then F(A*) · F(A [ A*)
= F(A)+i F(A[{s1,…,si})-F(A[ {s1,…,si-1})
· F(A) + i (F(A[{si})-F(A))
= F(A) + i si
For each s 2 VnA, let s = F(A[{s})-F(A)
Order such that 1 ¸ 2 ¸ … ¸ n
Then: F(A*) · F(A) + i=1k i 18

19. Bounds on optimal solution [Krause et al., J Wat Res Mgt ’08]5 10 15 20
0.2
0.4
0.6
0.8 1
1.2
1.4
Offline (Nemhauser) bound
Data-dependent bound
Greedy solution
Water networks data
Number of sensors placed
Sensing quality F(A)
Higher is better
Submodularity gives data-dependent bounds on the performance of any algorithm 19

20. The pSPIEL Algorithm [K, Guestrin, Gupta, Kleinberg IPSN ‘06]
pSPIEL: Efficient nonmyopic algorithm
(padded Sensor Placements at Informative and cost- Effective Locations)
Select starting and ending location s1 and sB
Decompose sensing region into small, well-separated clusters
Solve cardinality constrained problem per cluster (greedy)
Combine solutions using orienteering algorithm
Smooth resulting path 20 20

21. Locality
For two sets separated by at least distance r:

22. Example: Sensor Placement

23. pSPIEL

24. Approximation