1. Near-optimal Observation Selection using Submodular Functions
Andreas Krause, Carlos Guestrin
Carnegie Mellon University
AAAI, 2007
Presented by Haojun Chen

2. Introduction Observation selection with constraint Problems: NP-hard
Sensor placements with budget constraints (Krause et al. 2005a)
Multi-robot informative path planning (Singh et al. 2007)
Sensor placements for maximizing information at minimum communication cost (Krause et al. 2006) …
Problems: NP-hard
Heuristic approaches applied but no performance guarantees

3. Key Property: Diminishing Returns
A={s1,s2} s1 s2 s s1 s2 s3 s s4
B = {s1,s2,s3,s4}
Definition (Nemhauser et al. 1978)
A real value set function F on V is called submodular if for all
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4. Maximization of Submodular Functions
Optimization problem
Still NP-hard in general, but can get approximation guarantees
Approximation guarantees for three different constraints are reviewed in this paper
for some nonnegative budget B and ,where each
has a fixed positive cost

5. Cardinality and Budget Constraints
Unit cost case:
Approximation guarantees :
Greedy algorithm:
Start with
For i = 1 to k

6. Sensor Placements with Communication Constraints
Simple heuristic: Greedily optimize submodular utility function F(A)
Then add nodes to minimize communication cost C(A)
relay node 2 1
No communication possible!
C(A) = 1 2 2
1.5 1
F(A) = 3.5
C(A) = 3.5
relay node
F(A) = 0.2
Most informative
F(A) = 4
F(A) = 4
C(A) = 3 2
C(A) = 10
C(A) = 10
relay node
Second most informative 2 2
efficient communication!
Not very informative
Communication cost = Expected # of trials
(learned using Gaussian Processes)
Very informative,
High communication cost!
Want to find optimal tradeoff
between information and communication cost
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7. pSPIEL Algorithm
Padded Sensor Placements at Informative and cost-Effective Locations (pSPIEL):
Decompose sensing region into small, well-separated clusters
Solve cardinality constrained problem per cluster (greedy)
Combine solutions using k-minimum spanning tree (k-MST) algorithm 1 3 2 1 2 C1 C2 C4 C3
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8. Guarantees and Performance for pSPIEL
Approximation guarantee
Performance

9. Conclusions Many natural observation selection objectives: submodular
Key algorithmic problem: Constrained maximization of submodular functions
Efficient approximation algorithms with provable quality guarantees developed by exploiting submodularity

10. Reference
Chekuri, C., and Pal, M A recursive greedy algorithm for walks in directed graphs. In FOCS, 245–253
Krause, A., and Guestrin, C. 2005a. A note on the budgeted maximization of submodular functions. Technical report, CMUCALD
Krause, A.; Guestrin, C.; Gupta, A.; and Kleinberg, J Near-optimal sensor placements: Maximizing information while minimizing communication cost. In IPSN.
Nemhauser, G.; Wolsey, L.; and Fisher, M An analysis of the approximations for maximizing submodular set functions.Mathematical Programming 14:265–294.
Singh, A.; Krause, A.; Guestrin, C.; Kaiser, W.; and Batalin, M Efficient planning of informative paths for multiple robots.In IJCAI.