Greedy algorithms for on-line set-covering and related problems

We study the following on-line model for set-covering: elements of a ground set of size n arrive one-by-one and with any such element ci, arrives also the name of some set Si0 containing ci and covering the most of the uncovered ground set-elements (obviously, these elements have not been yet revealed). For this model we analyze a simple greedy algorithm consisting of taking Si0 into the cover, only if ci is not already covered. We prove that the competitive ratio of this algorithm is √ n and that it is asymptotically optimal for the model dealt, since no on-line algorithm can do better than √ n/2. We next show that this model can also be used for solving minimum dominating set with competitive ratio bounded above by the square root of the size of the input graph. We finally deal with the maximum budget saving problem. Here, an initial budget is allotted that is destined to cover the cost of an algorithm for solving set-covering. The objective is to maximize the savings on the initial budget. We show that when this budget is at least equal to √ n times the size of the optimal (off-line) solution of the instance under consideration, then the natural greedy off-line algorithm is asymptotically optimal.