Abstract
The problem of maximizing non-negative monotone submodular functions under a certain constraint has been intensively studied in the last decade. In this paper, we address the problem for functions defined over the integer lattice.
Suppose that a non-negative monotone submodular function \(f:\mathbb {Z}_+^n \rightarrow \mathbb {R}_+\) is given via an evaluation oracle. Assume further that f satisfies the diminishing return property, which is not an immediate consequence of the submodularity when the domain is the integer lattice. Then, we show polynomial-time \((1-1/e-\epsilon )\)-approximation algorithm for cardinality constraints, polymatroid constraints, and knapsack constraints. For a cardinality constraint, we also show a \((1-1/e-\epsilon )\)-approximation algorithm with slightly worse time complexity that does not rely on the diminishing return property.
Our algorithms for a polymatroid constraint and a knapsack constraint first extend the domain of the objective function to the Euclidean space and then run the continuous greedy algorithm. We give two different kinds of continuous extensions, one is for polymatroid constraints and the other is for knapsack constraints, which might be of independent interest.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Note that f is DR-submodular if and only if it is lattice submodular and satisfies the coordinate-wise concave condition: \(f(\varvec{x}+\chi _i) - f(\varvec{x}) \ge f(\varvec{x}+2\chi _i) - f(\varvec{x}+\chi _i)\) for any \(\varvec{x}\) and \(i \in E\) (see [26, Lemma 2.3]).
References
Alon, N., Gamzu, I., Tennenholtz, M.: Optimizing budget allocation among channels and influencers. In: Proceedings of WWW, pp. 381–388 (2012)
Badanidiyuru, A., Vondrák, J.: Fast algorithms for maximizing submodular functions. In: Proceedings of SODA, pp. 1497–1514 (2013)
Buchbinder, N., Feldman, M., Naor, J., Schwartz, R.: A tight linear time \((1/2)\)-approximation for unconstrained submodular maximization. In: Proceedings of FOCS, pp. 649–658 (2012)
Buchbinder, N., Feldman, M.: Deterministic algorithms for submodular maximization problems. In: Proceedings of SODA, pp. 392–403 (2016)
Buchbinder, N., Feldman, M., Naor, J.S., Schwartz, R.: Submodular maximization with cardinality constraints. In: Proceedings of SODA, pp. 1433–1452 (2014)
Calinescu, G., Chekuri, C., Pál, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM J. Comput. 40, 1740–1766 (2011)
Chekuri, C., Vondrák, J., Zenklusen, R.: Dependent randomized rounding via exchange properties of combinatorial structures. In: Proceedings of FOCS, pp. 575–584 (2010)
Cunningham, W.H.: Testing membership in matroid polyhedra. J. Comb. Theor. Ser. B 188, 161–188 (1984)
Demaine, E.D., Hajiaghayi, M., Mahini, H., Malec, D.L., Raghavan, S., Sawant, A., Zadimoghadam, M.: How to influence people with partial incentives. In: Proceedings of WWW, pp. 937–948 (2014)
Feige, U.: A threshold of \(\ln n\) for approximating set cover. J. ACM 45, 634–652 (1998)
Feige, U., Mirrokni, V.S., Vondrak, J.: Maximizing non-monotone submodular functions. SIAM J. Comput. 40(4), 1133–1153 (2011)
Fujishige, S.: Submodular Functions and Optimization, 2nd edn. Elsevier, New York (2005)
Gottschalk, C., Peis, B.: Submodular function maximization on the bounded integer lattice. ArXiv preprint (2015)
Iwata, S.: Submodular function minimization. Math. Program. 112(1), 45–64 (2007)
Iwata, S., Tanigawa, S., Yoshida, Y.: Bisubmodular function maximization and extensions. Mathematical Engineering Technical Reports (2013)
Kapralov, M., Post, I., Vondrak, J.: Online submodular welfare maximization: Greedy is optimal. In: Proceedings of SODA, pp. 1216–1225 (2012)
Kempe, D., Kleinberg, J., Tardos, E.: Maximizing the spread of influence through a social network. In: Proceedings of KDD, pp. 137–146 (2003)
Krause, A., Golovin, D.: Submodular function maximization. In: Tractability: Practical Approaches to Hard Problems, pp. 71–104. Cambridge University Press (2014)
Lin, H., Bilmes, J.: Multi-document summarization via budgeted maximization of submodular functions. In: Proceedings of NAACL, pp. 912–920 (2010)
Lin, H., Bilmes, J.: A class of submodular functions for document summarization. In: Proceedings of NAACL, pp. 510–520 (2011)
Murota, K.: Discrete Convex Analysis. SIAM, Philadelphia (2003)
Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions - II. Math. Program. Studies 8, 73–87 (1978)
Shioura, A.: On the pipage rounding algorithm for submodular function maximization – a view from discrete convex analysis–. Discrete Math. Algorithms Appl. 1(1), 1–23 (2009)
Singh, A., Guillory, A., Bilmes, J.: On bisubmodular maximization. In: Proceedings of AISTATS, pp. 1055–1063 (2012)
Soma, T., Kakimura, N., Inaba, K., Kawarabayashi, K.: Optimal budget allocation: theoretical guarantee and efficient algorithm. In: Proceedings of ICML (2014)
Soma, T., Yoshida, Y.: A generalization of submodular cover via the diminishing return property on the integer lattice. In: Proceedings of NIPS (2015)
Sviridenko, M.: A note on maximizing a submodular set function subject to a knapsack constraint. Oper. Res. Lett. 32(1), 41–43 (2004)
Ward, J., Živný, S.: Maximizing bisubmodular and \(k\)-submodular functions. In: Proceedings of SODA, pp. 1468–1481 (2014)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Soma, T., Yoshida, Y. (2016). Maximizing Monotone Submodular Functions over the Integer Lattice. In: Louveaux, Q., Skutella, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 2016. Lecture Notes in Computer Science(), vol 9682. Springer, Cham. https://doi.org/10.1007/978-3-319-33461-5_27
Download citation
DOI: https://doi.org/10.1007/978-3-319-33461-5_27
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-33460-8
Online ISBN: 978-3-319-33461-5
eBook Packages: Computer ScienceComputer Science (R0)