Before revision:  At a high level, the algorithm maintains \ell solutions and greedily updates them in a simultaneous fashion , rather than a sequential one . SimultaneousGreedys achieves the tightest known approximation guarantees for both k-extendible systems and the more general k-systems, which are (k+1)^2/k = k + O(1) and (1 + k+2)^2 = k + O(k), respectively.
After revision:  At a high level, the algorithm maintains \ell solutions and greedily updates them in a simultaneous fashion . SimultaneousGreedys achieves the tightest known approximation guarantees for both k-extendible systems and the more general k-systems, which are (k+1)^2/k = k + O(1) and (1 + k+2)^2 = k + O(k), respectively.