Greedily Excluding Algorithm for Submodular Maximization

This paper develops an approximation algorithm for maximization of monotone submodular reward function with a cardinality constraint. The main focus of our development is to enhances the level of guaranteed optimality, computational complexity or both compared with the well-known existing greedy based approaches. The proposed algorithm is called the greedily excluding algorithm. In this algorithm, an element is removed from the ground set to find the solution set and this allows the algorithm to improve both the optimality level and computational complexity for a problem with a high cardinality constraint. The characteristics of the proposed algorithm are mathematically analysed and compared with those of well-known greedy-based algorithms. A case study is also carried out to demonstrate the performance of the approximation algorithms developed.

[1]  Gérard Cornuéjols,et al.  Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the Rado-Edmonds theorem , 1984, Discret. Appl. Math..

[2]  Andreas Krause,et al.  Near-optimal Observation Selection using Submodular Functions , 2007, AAAI.

[3]  John Lygeros,et al.  On Submodularity and Controllability in Complex Dynamical Networks , 2014, IEEE Transactions on Control of Network Systems.

[4]  Andreas Krause,et al.  Lazier Than Lazy Greedy , 2014, AAAI.

[5]  Jan Vondrák,et al.  Fast algorithms for maximizing submodular functions , 2014, SODA.

[6]  George J. Pappas,et al.  Sensor placement for optimal Kalman filtering: Fundamental limits, submodularity, and algorithms , 2015, 2016 American Control Conference (ACC).

[7]  Haris Vikalo,et al.  Greedy sensor selection: Leveraging submodularity , 2010, 49th IEEE Conference on Decision and Control (CDC).

[8]  M. L. Fisher,et al.  An analysis of approximations for maximizing submodular set functions—I , 1978, Math. Program..

[9]  Andreas Krause,et al.  Near-Optimal Sensor Placements in Gaussian Processes: Theory, Efficient Algorithms and Empirical Studies , 2008, J. Mach. Learn. Res..

[10]  Antonios Tsourdos,et al.  Decentralised submodular multi-robot Task Allocation , 2015, 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).

[11]  Joseph Naor,et al.  Submodular Maximization with Cardinality Constraints , 2014, SODA.

[12]  C. Guestrin,et al.  Near-optimal sensor placements: maximizing information while minimizing communication cost , 2006, 2006 5th International Conference on Information Processing in Sensor Networks.

[13]  Atte Moilanen,et al.  New performance guarantees for the greedy maximization of submodular set functions , 2015, Optim. Lett..

[14]  Andreas Krause,et al.  Efficient Sensor Placement Optimization for Securing Large Water Distribution Networks , 2008 .