Seeking maximum capacity utilization for 0-1 Multidimensional Knapsack

Question

The typical objective function of 0-1 Multidimensional Knapsack is to maximize the value of all items in the knapsack. A good algorithm was provided in the Stackexchange link here: 0-1 Multidimensional Knapsack .

However, what if my objective function is to pack as many items in the knapsack as possible? All pieces have equal values. The Stackexchange post ( Knapsack problem with all profits equal to 1 ) claims that a one dimensional knapsack with equal values can be solved by Greedy Algorithm. Is that true? I thought the 01 knapsack problem is NP-hard therefore the greedy algorithm may not give the optimum solution.

So my questions are two-part? 1) can optimum solution be given by greedy algorithm in this case? 01 knapsack with equal values 2) how to implement a multi-dimensional greedy algorithm? the vi/wi is a value divided by a vector...

Answer 1

1.) The knapsack problem IS an NP-Hard problem. So in short, no, you cannot solve it optimally using a greedy algorithm. Instead heuristic approaches exist that can get you very close.

2.) In the case of equal profit knapsack, this will likely to degrade to a simple bin packing problem. In this context if all choices are equal in terms of profit, you will likely need to focus on a different aspect of the problem which is probably something like "size." If that is the case you will want to pick the smallest item you can each time - in which case a greedy algorithm is probably sufficient and can be achieved by simply looking through your choices and choosing the smallest element.

It should be noted that a linear search can add an annoying amount of overhead to your program if its being repeated a great deal of times. Instead you may want to consider using a MIN-Heap as this will make the smallest value available at a lower computational cost.