Document

# Generalized Budgeted Submodular Set Function Maximization

## File

LIPIcs.MFCS.2018.31.pdf
• Filesize: 470 kB
• 14 pages

## Cite As

Francesco Cellinese, Gianlorenzo D'Angelo, Gianpiero Monaco, and Yllka Velaj. Generalized Budgeted Submodular Set Function Maximization. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.MFCS.2018.31

## Abstract

In this paper we consider a generalization of the well-known budgeted maximum coverage problem. We are given a ground set of elements and a set of bins. The goal is to find a subset of elements along with an associated set of bins, such that the overall cost is at most a given budget, and the profit is maximized. Each bin has its own cost and the cost of each element depends on its associated bin. The profit is measured by a monotone submodular function over the elements. We first present an algorithm that guarantees an approximation factor of 1/2(1-1/e^alpha), where alpha <= 1 is the approximation factor of an algorithm for a sub-problem. We give two polynomial-time algorithms to solve this sub-problem. The first one gives us alpha=1- epsilon if the costs satisfies a specific condition, which is fulfilled in several relevant cases, including the unitary costs case and the problem of maximizing a monotone submodular function under a knapsack constraint. The second one guarantees alpha=1-1/e-epsilon for the general case. The gap between our approximation guarantees and the known inapproximability bounds is 1/2. We extend our algorithm to a bi-criterion approximation algorithm in which we are allowed to spend an extra budget up to a factor beta >= 1 to guarantee a 1/2(1-1/e^(alpha beta))-approximation. If we set beta=1/(alpha)ln (1/(2 epsilon)), the algorithm achieves an approximation factor of 1/2-epsilon, for any arbitrarily small epsilon>0.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Approximation algorithms analysis
• Theory of computation → Packing and covering problems
##### Keywords
• Submodular set function
• Approximation algorithms
• Budgeted Maximum Coverage

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Ashwinkumar Badanidiyuru, Christos H. Papadimitriou, Aviad Rubinstein, Lior Seeman, and Yaron Singer. Locally adaptive optimization: Adaptive seeding for monotone submodular functions. In 27th ACM-SIAM Symp. on Disc. Alg., (SODA), pages 414-429, 2016.
2. Niv Buchbinder, Moran Feldman, Joseph Naor, and Roy Schwartz. A tight linear time (1/2)-approximation for unconstrained submodular maximization. In 53rd IEEE Symp. on Foundations of Computer Science, FOCS, pages 649-658, 2012.
3. Gruia Călinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM J. Comput., 40(6):1740-1766, 2011.
4. Ioannis Caragiannis. Wavelength management in WDM rings to maximize the number of connections. SIAM J. Discrete Math., 23(2):959-978, 2009.
5. Ioannis Caragiannis and Gianpiero Monaco. A 6/5-approximation algorithm for the maximum 3-cover problem. J. Comb. Optim., 25(1):60-77, 2013.
6. Chandra Chekuri and Amit Kumar. Maximum coverage problem with group budget constraints and applications. In 7th Intl. Work. on Approximation Algorithms for Combinatorial Optimization Problems, APPROX, pages 72-83, 2004.
7. Reuven Cohen and Liran Katzir. The generalized maximum coverage problem. Information Processing Letters, 108(1):15-22, 2008.
8. Gianlorenzo D'Angelo, Lorenzo Severini, and Yllka Velaj. Influence maximization in the independent cascade model. In Proceedings of the 17th Italian Conference on Theoretical Computer Science (ICTCS2016), volume 1720, pages 269-274. CEUR-WS.org, 2016.
9. Gianlorenzo D'Angelo, Lorenzo Severini, and Yllka Velaj. Selecting nodes and buying links to maximize the information diffusion in a network. In 42st Intl. Symp. on Mathematical Foundations of Computer Science, MFCS, volume 83 of LIPIcs, pages 75:1-75:14, 2017.
10. Gianlorenzo D'Angelo, Lorenzo Severini, and Yllka Velaj. Recommending links through influence maximization. Theoretical Computer Science, 2018. URL: http://dx.doi.org/10.1016/j.tcs.2018.01.017.
11. Uriel Feige. A threshold of ln n for approximating set cover. Journal of ACM, 45(4):634-652, 1998.
12. Uriel Feige, Vahab S. Mirrokni, and Jan Vondrák. Maximizing non-monotone submodular functions. In 48th IEEE Symp. on Foundations of Computer Science (FOCS), pages 461-471, 2007.
13. Yuval Filmus and Justin Ward. Monotone submodular maximization over a matroid via non-oblivious local search. SIAM J. Comput., 43(2):514-542, 2014.
14. D.S. Hochbaum. Approximation Algorithms for NPHard Problems. PWS Publishing Company,, Boston, MA, USA, 1997.
15. Rishabh K. Iyer and Jeff A. Bilmes. Submodular optimization with submodular cover and submodular knapsack constraints. In 27th Annual Conference on Neural Information Processing Systems (NIPS), pages 2436-2444, 2013.
16. V Kann. Maximum bounded 3-dimensional matching is max snp-comple. Information Processing Letters, 37:27-35, 1991.
17. David Kempe, Jon M. Kleinberg, and Éva Tardos. Maximizing the spread of influence through a social network. Theory of Computing, 11:105-147, 2015.
18. Samir Khuller, Anna Moss, and Joseph Seffi Naor. The budgeted maximum coverage problem. Information Processing Letters, 70(1):39-45, 1999.
19. Jon Lee, Maxim Sviridenko, and Jan Vondrák. Submodular maximization over multiple matroids via generalized exchange properties. Math. Oper. Res., 35(4):795-806, 2010.
20. G.L. Nemhauser, L.A. Wolsey, and M.L. Fisher. An analysis of approximations for maximizing submodular set functions-I. Mathematical Programming, 14(1):265-294, 1978.
21. Aviad Rubinstein, Lior Seeman, and Yaron Singer. Approximability of adaptive seeding under knapsack constraints. In 16th ACM Conf. on Economics and Computation, pages 797-814. ACM, 2015.
22. Lior Seeman and Yaron Singer. Adaptive seeding in social networks. In IEEE 54th Symp. on Foundations of Computer Science (FOCS), pages 459-468. IEEE, 2013.
23. Maxim Sviridenko. A note on maximizing a submodular set function subject to a knapsack constraint. Operation Research Letters, 32(1):41-43, 2004.
24. Irving van Heuven van Staereling, Bart de Keijzer, and Guido Schäfer. The Ground-Set-Cost Budgeted Maximum Coverage Problem. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016), volume 58 of LIPIcs, pages 50:1-50:13, 2016.
25. Jan Vondrák, Chandra Chekuri, and Rico Zenklusen. Submodular function maximization via the multilinear relaxation and contention resolution schemes. In 43rd ACM Symp. on Theory of Computing, STOC, pages 783-792, 2011.
26. Justin Ward. A (k+3)/2-approximation algorithm for monotone submodular k-set packing and general k-exchange systems. In 29th Intl. Symp.on Theoretical Aspects of Computer Science, STACS, pages 42-53, 2012.
X

Feedback for Dagstuhl Publishing