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

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

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 Slides from http://www.select.cs.cmu.edu/tutorials/icml08submodularity.html

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

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

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 Slides from http://www.select.cs.cmu.edu/tutorials/icml08submodularity.html

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 Slides from http://www.select.cs.cmu.edu/tutorials/icml08submodularity.html

Guarantees and Performance for pSPIEL Approximation guarantee Performance

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

Reference Chekuri, C., and Pal, M. 2005. 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-05-103 Krause, A.; Guestrin, C.; Gupta, A.; and Kleinberg, J. 2006. Near-optimal sensor placements: Maximizing information while minimizing communication cost. In IPSN. Nemhauser, G.; Wolsey, L.; and Fisher, M. 1978. 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.2007. Efficient planning of informative paths for multiple robots.In IJCAI.