A performance guarantee for the greedy set-partitioning algorithm

SummaryLet 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 ϱg(S) be the value of this ratio using the greedy or largest-first rule and ϱ0(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.