::<math>x_i \in \{0,1\}</math> (if <math>x_i=1</math> then <math>S_i</math> is selected for the cover).
A greedy algorithm will no longer produce solutions with a performance guarantee. Namely, the worst case behavior of this algorithm might be very far from the optimal solution. The approximation algorithm is extended by the following way. First, after finding a solution usingfind the greedybest algorithmcover, returnthat thedoes betternot ofviolate the greedybudget, algorithm'samong solutioncovers andof thecardinality set<math>1, of2, largest..., weightk-1</math>. Call this algorithmcover the<math>H_1</math>. modifiedNext, greedyfind algorithm.all Second,covers startingof withcardinality all<math>k</math> possiblethat familiesdo ofnot setsviolate ofthe sizesbudget. fromUsing onethese tocovers (atof least)cardinality three,<math>k</math> augmentas thesestarting solutionspoints, withapply the modified greedy algorithm. Third, returnmaintaining the best outcover found so far. Call this cover <math>H_2</math>. At the end of allthe augmentedprocess, solutionsthe approximate best cover will be either <math>H_1</math> or <math>H_2</math>. This algorithm achieves an approximation ratio of <math>1- 1/e</math> for values of <math>k \geq 3</math>. This is the best possible approximation ratio unless <math>NP \subseteq DTIME(n^{O(\log\log n)})</math>.<ref>Khuller, S., Moss, A., and Naor, J. 1999. [http://dx.doi.org/10.1016/S0020-0190(99)00031-9 The budgeted maximum coverage problem]. ''Inf. Process. Lett''. 70, 1 (Apr. 1999), 39-45.</ref>
