On the Sum-of-Squares algorithm for bin packing

Abstract

In this article we present a theoretical analysis of the online Sum-of-Squares algorithm ( SS ) for bin packing along with several new variants. SS is applicable to any instance of bin packing in which the bin capacity B and item sizes s ( a ) are integral (or can be scaled to be so), and runs in time O ( nB ). It performs remarkably well from an average case point of view: For any discrete distribution in which the optimal expected waste is sublinear, SS also has sublinear expected waste. For any discrete distribution where the optimal expected waste is bounded, SS has expected waste at most O (log n ). We also discuss several interesting variants on SS , including a randomized O ( nB log B )-time online algorithm SS * whose expected behavior is essentially optimal for all discrete distributions. Algorithm SS * depends on a new linear-programming-based pseudopolynomial-time algorithm for solving the NP-hard problem of determining, given a discrete distribution F , just what is the growth rate for the optimal expected waste.