## Abstract

For NP-complete problems, it may not be possible to find an optimal solution in polynomial time. However, efficient approximation algorithms for many NP-complete problems do exist. A good approach is to find tighter upper and lower bounds on the optimal solutions. As the gap between these two bounds approaches zero, optimality is reached. For minimization problems, any feasible solution serves as an upper bound. In general, researchers attempt to find heuristic algorithms to narrow the gap by decreasing the upper bound. Another approach is to narrow the gap by increasing the lower bound. To determine a good lower bound on a problem requires careful examination of the problem's characteristics. In this paper, we present an efficient algorithm for calculating a lower bound on the number of bins needed in a bin-packing problem. It is obvious that the sum of the sizes of all objects is a lower bound. In addition to this, we consider objects that cannot share their bins with other objects. We consider a subproblem class which contains only objects whose sizes are greater than L/4, where L is the size of a bin, given the harmonic partition. An O(n1ogn) algorithm finds a lower bound for the subproblem, which in turn is a lower bound for the original problem. Notably, our approach also leads to a polynomial time algorithm for finding optimal solutions of some special cases of bin-packing. Simulation data show that our lower bound provides an estimate of the optimal number of bins required which is equal to or better than the sum. The approximate error rate is normally less than 1% if our bound is used. The improvement can be as high as 79% when compared to the sum.

## Keywords

Heuristic Performance Ratio, Bin-Packing, Lower Bound, Best-Fit Decreasing, Harmonic Partition, Perfect-k-Way Merge, Matching

## Date of this Version

July 1993