Maximising Real-Valued Submodular Functions: Primal and Dual Heuristics for Location Problems

Recently it has been shown how various results on the behaviour of exact and approximate primal algorithms for certain discrete location problems can be obtained by studying the more general problem of maximising a submodular set function. Unfortunately, this more general model tells us little about an important aspect of the discrete location problems, their linear programming relaxations and the possible use of dual solutions in obtaining bounds. Here we examine a slightly more specialised model that again generalises the location problems of interest, but now also includes the continuous aspects of the problems missing from the earlier model, namely the problem: maxw (y): ∑ j=1 n a j y j ≤ b, 0 ≤ y j < 1 of maximising a real-valued nondecreasing piecewise linear, concave submodular function subject to a knapsack constraint. We show that a continuous greedy heuristic always attains at least (1 − e −1) × 100% of the optimal value, and that for the discrete problem an adapted greedy heuristic always attains 35% of the optimal value. Specialised to the capacitated and uncapacitated location problems these results permit us to strengthen earlier results, and to obtain new results on dual greedy heuristics.