# Performance Bounds with Curvature for Batched Greedy Optimization

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## Abstract

The batched greedy strategy is an approximation algorithm to maximize a set function subject to a matroid constraint. Starting with the empty set, the batched greedy strategy iteratively adds to the current solution set a batch of elements that results in the largest gain in the objective function while satisfying the matroid constraints. In this paper, we develop bounds on the performance of the batched greedy strategy relative to the optimal strategy in terms of a parameter called the total batched curvature. We show that when the objective function is a polymatroid set function, the batched greedy strategy satisfies a harmonic bound for a general matroid constraint and an exponential bound for a uniform matroid constraint, both in terms of the total batched curvature. We also study the behavior of the bounds as functions of the batch size. Specifically, we prove that the harmonic bound for a general matroid is nondecreasing in the batch size and the exponential bound for a uniform matroid is nondecreasing in the batch size under the condition that the batch size divides the rank of the uniform matroid. Finally, we illustrate our results by considering a task scheduling problem and an adaptive sensing problem.

## Keywords

Curvature Greedy Matroid Polymatroid Submodular## Mathematics Subject Classification

90C27 90C59## Notes

### Acknowledgements

This work is supported in part by NSF under award CCF-1422658 and by the CSU Information Science and Technology Center (ISTeC). We would like to acknowledge the anonymous reviewers for their insightful comments on our conference paper [26], as these comments led us to improve our work.

## References

- 1.Streeter, M., Golovin, D.: An online algorithm for maximizing submodular functions. In: Proceedings of Advances in Neural Information Processing, pp. 1577–1584 (2008)Google Scholar
- 2.Feige, U., Vondrák, J.: Approximation algorithms for allocation problems: improving the factor of \(1-1/e\). In: Proceedings of 47th IEEE Symposium on Foundations of Computer Science, pp. 667–676 (2006)Google Scholar
- 3.Fleischer, L., Goemans, M.X., Mirrokni, V.S., Sviridenko, M.: Tight approximation algorithms for maximum general assignment problems. In: Proceedings of 17th Annual ACM-SIAM Symposium Discrete Algorithm, pp. 611–620 (2006)Google Scholar
- 4.Cohen, R., Katzir, L., Raz, D.: An efficient approximation for the generalized assignment problem. Inf. Process. Lett.
**100**(4), 162–166 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Shmoys, D.B., Tardos, É.: An approximation algorithm for the generalized assignment problem. Math. Program.
**62**, 461–474 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Calinescu, G., Chekuri, C., Pál, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM J. Comput.
**40**(6), 1740–1746 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Korula, N., Mirrokni, V., Zadimoghaddam, M.: Online submodular welfare maximization: Greedy beats 1/2 in random order. In: Proceedings of 47th Annual Symposium on Theory of Computing, pp. 889–898 (2015)Google Scholar
- 8.Vondrák, J.: Optimal approximation for the submodular welfare problem in the value oracle model. In: Proceedings of 40th Annual ACM Symposium on Theory of Computing, pp. 67–74 (2008)Google Scholar
- 9.Hochbaum, D.S., Pathria, A.: Analysis of the greedy approach in problems of maximum \(k\)-coverage. Nav. Res. Log.
**45**(6), 615–627 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Feige, U.: A threshold of \(\ln n\) for approximating set cover. J. ACM
**45**(4), 634–652 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Khuller, S., Moss, A., Naor, J.: The budgeted maximum coverage problem. Inf. Process. Lett.
**70**(1), 39–45 (1999)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Cornuéjols, G., Fisher, M.L., Nemhauser, G.L.: Location of bank accounts to optimize float: an analytic study of exact and approximate algorithms. Manag. Sci.
**23**(8), 789–810 (1977)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Megiddo, N., Zemel, E., Hakimi, S.L.: The maximum coverage location problem. SIAM J. Algebraic Discrete Methods
**4**(2), 253–261 (1983)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Church, R., Velle, C.R.: The maximal covering location problem. Pap. Reg. Sci.
**32**(1), 101–118 (1974)CrossRefGoogle Scholar - 15.Pirkul, H., Schilling, D.A.: The maximal covering location problem with capacities on total workload. Manag. Sci.
**37**(2), 233–248 (1991)CrossRefzbMATHGoogle Scholar - 16.Liu, E., Chong, E.K.P., Scharf, L.L.: Greedy adaptive linear compression in signal-plus-noise models. IEEE Trans. Inf. Theory
**60**(4), 2269–2280 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Krause, A., Singh, A., Guestrin, C.: Near-Optimal sensor placements in Gaussian processes: theory, efficient algorithms and empirical studies. J. Mach. Learn. Res.
**9**, 235–284 (2008)zbMATHGoogle Scholar - 18.Chen, Y., Chuah, C., Zhao, Q.: Sensor placement for maximizing lifetime per unit cost in wireless sensor networks. In: Proceedings of IEEE Military Communication Conference, pp. 1097–1102 (2005)Google Scholar
- 19.Ragi, S., Mittelmann, H.D., Chong, E.K.P.: Directional sensor control: heuristic approaches. IEEE Sens. J.
**15**(1), 374–381 (2015)CrossRefGoogle Scholar - 20.Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions-I. Math. Program.
**14**(1), 265–294 (1978)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Fisher, M.L., Nemhauser, G.L., Wolsey, L.A.: An analysis of approximations for maximizing submodular set functions-II. Math. Program. Stud.
**8**, 73–87 (1978)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Hausmann, D., Korte, B., Jenkyns, T.A.: Worst case analysis of greedy type algorithms for independence systems. Math. Program. Stud.
**12**, 120–131 (1980)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Conforti, M., Cornuéjols, G.: Submodular set functions, matroids and the greedy algorithm: tight worst-case bounds and some generalizations of the Rado–Edmonds theorem. Discrete Appl. Math.
**7**(3), 251–274 (1984)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Vondrák, J.: Submodularity and curvature: the optimal algorithm. RIMS Kokyuroku Bessatsu
**B23**, 253–266 (2010)MathSciNetzbMATHGoogle Scholar - 25.Sviridenko, M., Vondrák, J., Ward, J.: Optimal approximation for submodular and supermodular optimization with bounded curvature. In: Proceedings of 26th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1134–1148 (2015)Google Scholar
- 26.Liu, Y., Zhang, Z., Chong, E.K.P., Pezeshki, A.: Performance bounds for the \(k\)-batch greedy strategy in optimization problems with curvature. In: Proceedings of 2016 American Control Conference, pp. 7177–7182 (2016)Google Scholar
- 27.Chandu, D.P.: Big step greedy heuristic for maximum coverage problem. Int. J. Comput. Appl
**125**, 19–24 (2015)Google Scholar - 28.Edmonds, J.: Submodular functions, matroids, and certain polyhedra. Comb. Optim.
**2570**, 11–26 (2003)zbMATHGoogle Scholar - 29.Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L.: An inequality for polymatroid functions and its applications. Discrete Appl. Math.
**131**(2), 255–281 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 30.Tutte, W.T.: Lecture on matroids. J. Res. Natl. Bur. Stand.-Sec. B Math. Math. Phys.
**69B**, 1–47 (1965)MathSciNetCrossRefzbMATHGoogle Scholar - 31.Lee, J., Sviridenko, M., Vondrák, J.: Submodular maximization over multiple matroids via generalized exchange properties. Math. Oper. Res.
**35**(4), 795–806 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 32.Zhang, Z., Chong, E.K.P., Pezeshki, A., Moran, W.: String submodular functions with curvature constraints. IEEE Trans. Autom. Control
**61**(3), 601–616 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 33.Liu, Y., Chong, E.K.P., Pezeshki, A.: Bounding the greedy strategy in finite-horizon string optimization. In: Proceedings of 54th IEEE Conference Decision on Control, pp. 3900–3905 (2015)Google Scholar