A performance guarantee for the greedy set-partitioning algorithm

Summary

Let S be a set of positive numbers and m an integer not less than 2. The problem is to partition S into m subsets such that the ratio of the largest subset sum to the smallest is as small as possible. Let ϱ _g (S) be the value of this ratio using the greedy or largest-first rule and ϱ ₀ (S) be the smallest possible value of this ratio, i.e., the optimal value. The authors prove that

$$\frac{{\rho _g \left( S \right)}}{{\rho _0 \left( S \right)}}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } {7 \mathord{\left/ {\vphantom {7 5}} \right. \kern-\nulldelimiterspace} 5}$$

, and that this is a best possible bound for all m.