Journal of Combinatorial Optimization

, Volume 31, Issue 1, pp 29–43

# Approximation for maximizing monotone non-decreasing set functions with a greedy method

Article

## Abstract

We study the problem of maximizing a monotone non-decreasing function $$f$$ subject to a matroid constraint. Fisher, Nemhauser and Wolsey have shown that, if $$f$$ is submodular, the greedy algorithm will find a solution with value at least $$\frac{1}{2}$$ of the optimal value under a general matroid constraint and at least $$1-\frac{1}{e}$$ of the optimal value under a uniform matroid $$(\mathcal {M} = (X,\mathcal {I})$$, $$\mathcal {I} = \{ S \subseteq X: |S| \le k\}$$) constraint. In this paper, we show that the greedy algorithm can find a solution with value at least $$\frac{1}{1+\mu }$$ of the optimum value for a general monotone non-decreasing function with a general matroid constraint, where $$\mu = \alpha$$, if $$0 \le \alpha \le 1$$; $$\mu = \frac{\alpha ^K(1-\alpha ^K)}{K(1-\alpha )}$$ if $$\alpha > 1$$; here $$\alpha$$ is a constant representing the “elemental curvature” of $$f$$, and $$K$$ is the cardinality of the largest maximal independent sets. We also show that the greedy algorithm can achieve a $$1 - (\frac{\alpha + \cdots + \alpha ^{k-1}}{1+\alpha + \cdots + \alpha ^{k-1}})^k$$ approximation under a uniform matroid constraint. Under this unified $$\alpha$$-classification, submodular functions arise as the special case $$0 \le \alpha \le 1$$.

## Keywords

Monotone submodular set function Matroid Approximation algorithm

## Notes

### Acknowledgments

This study was supported in part by the NSFC (No.61135001) and the AFOSR grant (FA2386-13-1-4080).

## References

1. Ageev A, Sviridenko M (2004) Pipage rounding: a new method of constructing algorithms with proven performance guarantee. J Comb Optim 8(3):307–328
2. Alimonti P (1994) New local search approximation techniques for maximum generalized satisfiability problems. In: Proceedings of the 2nd Italian conference on algorithms and complexity, pp 40–53Google Scholar
3. Badanidiyuru A, Dobzinski S, Oren S (2011) Optimization with demand oracles. In: Proceedings of the 13th ACM conference on electronic commerce, pp 110–127Google Scholar
4. Buchbinder N, Feldman M, Naor J, Schwartz R (2012) A tight linear time $$(1/2)$$-approximation for unconstrained submodular maximization. In: 53rd annual IEEE symposium on foundations of computer scienceGoogle Scholar
5. Calinescu G, Chekuri C, Pál M, Vondrák J (2011) Maximizing a submodular set function subject to a matroid constraint. SIAM J Comput 40(6):1740–1766
6. Chakrabarty D, Goel G (2008), On the approximability of budgeted allocations and improved lower bounds for submodular welfare maximization and GAP. In: Proceedings of the 49th annual IEEE symposium on foundations of computer science, pp 687–696Google Scholar
7. Chekuri C, Vondrák J, Zenklusen R (2011), Submodular function maximization via the multilinear relaxation and contention resolution schemes. In: Proceedings of the 43rd ACM symposium on theory of computing, pp 783–792Google Scholar
8. Conforti M, Cornuéjols G (1984) 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
9. Dobzinski S, Schapira M (2006) An improved approximation algorithm for combinatorial auctions with submodular bidders. In: Proceedings of the seventeenth annual ACM-SIAM symposium on discrete algorithm, pp 1064–1073Google Scholar
10. Feige U (1998) A threshold of ln n for approximation set cover. J ACM 45(4):634–652Google Scholar
11. Feige U, Vondrák J (2006), Approximation algorithms for allocation problems: improveing the factor of $$1-\frac{1}{e}$$. In: Proceedings of 47th annual IEEE symposium on foundations of computer science, pp 667–676Google Scholar
12. Feige U, Vondrák J (1998) The submodular welfare problem with demand queries. Theory Comput 6:247–290
13. Filmus Y, Ward J (2012) A tight combinatorial algorithm for submodular maximization subject to a matroid constraint. In: Proceedings of 53rd annual IEEE symposium on foundations of computer scienceGoogle Scholar
14. Fisher ML, Nemhauser GL, Wolsey LA (1978) An analysis of approximations for maximizing submodular set functions - II. Math Progr Study 8:73–87
15. Kempe D, Kleinberg J, Tardos E (2005) Influential nodes in a diffusion model for social networks. In: Proceedings of 32nd international colloquium on automata, languages and programming, Lisboa, PortugalGoogle Scholar
16. Korte B, Hausmann D (1998) An analysis of the greedy heuritic for independence systems. Ann Discret Math 2:65–74
17. Kulik A, Shachnai H, Tamir T (2009) Maximizing submodular set functions subject to multiple linear constraints. In: Proceedings of the 20th annual ACM-SIAM symposium on discrete algorithms, pp 545–554Google Scholar
18. Lee J, Mirrokni V S, Nagarajan V, Sviridenko Maxim (2009) Non-monotone submodular maximization under matroid and knapsack constraints. In: Proceedings of the 41st annual ACM symposium on theory of computing, pp 323–332Google Scholar
19. Lee J, Sviridenko M, Vondrák (2010) Submodular maximization over multiple matroids via generalized exchange properties. Math Oper Res 35(4):795–806
20. Lloyd SP, Witsenhausen HS (1986) Weapons allocation is NP-complete. In: Proceedings of the 1986 summer conference on simulation, RenoGoogle Scholar
21. Lu J, Suda T (2003) Coverage-aware self-scheduling in sensor networks. In: Proceedings of IEEE 18th annual workshop on computer communications, Laguna NiguelGoogle Scholar
22. Nemhauser GL, Wolsey LA, Fisher ML (1978) An analysis of approximations for maximizing submodular set functions-I. Math Progr 14(1):265–294
23. Nembauser GL, Wolsey LA (1978) Best algorithms for approximating the maximum of a submodular set function. Math Oper Res 3(3):177–188
24. Sviridenko M (2004) A note on maximizing a submodular set function subject to a knapsack constraint. Oper Res Lett 32(1):41–43
25. Vondrák J (2008), Optimal approximation for the submodular welfare problem in the value oracle model. In: Proceedings of the 40th annual ACM symposium on theory of computing, pp 67–74Google Scholar
26. Vondrák J (2010) Submodularity and curvature: the optimal algorithm. RIMS Kokyuroku Bessatsu B23:253–266

## Authors and Affiliations

• Zengfu Wang
• 1
• 3
• Bill Moran
• 2
• Xuezhi Wang
• 2
• Quan Pan
• 1
1. 1.School of AutomationNorthwestern Polytechnical UniversityXi’an China
2. 2.Department of Electrical and Electronic EngineeringThe University of MelbourneMelbourneAustralia
3. 3.Department of Electrical and Electronic EngineeringThe University of MelbourneMelbourneAustralia