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Absolute o(log m) Error in Approximating Random Set Covering: An Average Case Analysis
 INFORMATION PROCESSING LETTERS 94(4
, 2005
"... This work concerns average case analysis of simple solutions for random set covering (SC) instances. Simple solutions are constructed via an O(nm) algorithm. At first an analytical upper bound on the expected solution size is provided. The bound in combination with previous results yields an absolut ..."
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This work concerns average case analysis of simple solutions for random set covering (SC) instances. Simple solutions are constructed via an O(nm) algorithm. At first an analytical upper bound on the expected solution size is provided. The bound in combination with previous results yields an absolute asymptotic approximation result of o(log m) order. An upper bound on the variance of simple solution values is calculated. Sensitivity analysis performed on simple solutions for random SC instances shows that they are highly robust, in the sense of maintaining their feasibility against augmentation of the input data with additional random constraints.
Greedy Algorithms for onLine SetCovering and Related Problems
, 2006
"... We study the following online model for setcovering: elements of a ground set of size n arrive onebyone and with any such element c i , arrives also the name of some set S i 0 containing c i and covering the most of the uncovered ground setelements (obviously, these elements have not been yet r ..."
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We study the following online model for setcovering: elements of a ground set of size n arrive onebyone and with any such element c i , arrives also the name of some set S i 0 containing c i and covering the most of the uncovered ground setelements (obviously, these elements have not been yet revealed). For this model we analyze a simple greedy algorithm consisting of taking S i 0 into the cover, only if c i 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 online algorithm can do 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 setcovering. 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 (o#line) solution of the instance under consideration, then the natural greedy o#line algorithm is asymptotically optimal.