# An analysis of approximations for maximizing submodular set functions—I

- 3.4k Downloads
- 1k Citations

## Abstract

Let*N* be a finite set and*z* be a real-valued function defined on the set of subsets of*N* that satisfies z(S)+z(T)≥z(S⋃T)+z(S⋂T) for all*S, T* in*N.* Such a function is called submodular. We consider the problem max_{S⊂N}{a(S):|S|≤K,z(S) submodular}.

Several hard combinatorial optimization problems can be posed in this framework. For example, the problem of finding a maximum weight independent set in a matroid, when the elements of the matroid are colored and the elements of the independent set can have no more than*K* colors, is in this class. The uncapacitated location problem is a special case of this matroid optimization problem.

We analyze greedy and local improvement heuristics and a linear programming relaxation for this problem. Our results are worst case bounds on the quality of the approximations. For example, when*z(S)* is nondecreasing and*z*(0) = 0, we show that a “greedy” heuristic always produces a solution whose value is at least 1 −*[(K − 1)/K]*^{ K } times the optimal value. This bound can be achieved for each*K* and has a limiting value of (e − 1)/e, where e is the base of the natural logarithm.

## Key words

Heuristics Greedy Algorithm Interchange Algorithm Linear Programming Matroid Optimization Submodular Set Functions## Preview

Unable to display preview. Download preview PDF.

## References

- [1]D.A. Babayev, “Comments on the note of Frieze”,
*Mathematical Programming*7 (1974) 249–252.Google Scholar - [2]G. Cornuejols, M.L. Fisher and G.L. Nemhauser, “Location of bank accounts to optimize float: An analytic study of exact and approximate algorithms”,
*Management Science*23 (1977) 789–810.Google Scholar - [3]J. Edmonds, “Matroid partition”, in: G.B. Dantzig and A.M. Veinott, eds.,
*Mathematics of the decision sciences*, A.M.S. Lectures in Applied Mathematics 11 (Am. Math. Soc., Providence, RI, 1968) pp. 333–345.Google Scholar - [4]J. Edmonds, “Submodular functions, matroids and certain polyhedra”, in: R. Guy, ed.,
*Combinatorial structures and their applications*(Gordon and Breach, New York, 1971) pp. 69–87.Google Scholar - [5]J. Edmonds, “Matroids and the greedy algorithm”,
*Mathematical Programming*1 (1971) 127–136.Google Scholar - [6]A.M. Frieze, “A cost function property for plant location problems”,
*Mathematical Programming*7 (1974) 245–248.Google Scholar - [7]L.S. Shapley, “Complements and substitutes in the optimal assignment problem”,
*Naval Research Logistics Quarterly*9 (1962) 45–48.Google Scholar - [8]L.S. Shapley, “Cores of convex games”,
*International Journal of Game Theory*1 (1971) 11–26.Google Scholar - [9]K. Spielberg, “Plant location with generalized search origin”,
*Management Science*16 (1969) 165–178.Google Scholar - [10]D.R. Woodall, “Application of polymatroids and linear programming to transversals and graphs”, presented at the 1973 British Combinatorial Conference (Aberystwyth).Google Scholar