Packing a bin online to maximize the total number of items

A bin of capacity 1 and a nite sequence of items of\ud sizes a1; a2; : : : are considered, where the items are given one by one\ud without information about the future. An online algorithm A must\ud irrevocably decide whether or not to put an item into the bin whenever\ud it is presented. The goal is to maximize the number of items collected.\ud A is f-competitive for some function f if n() f(nA()) holds for all\ud sequences , where n is the (theoretical) optimum and nA the number\ud of items collected by A.\ud A necessary condition on f for the existence of an f-competitive\ud (possibly randomized) online algorithm is given. On the other hand,\ud this condition is seen to guarantee the existence of a deterministic online\ud algorithm that is "almost" f-competitive in a well-dened sense