On the Sum-of-Squares Algorithm for Bin Packing

TL;DR

This paper provides a theoretical analysis of the Sum of Squares bin packing algorithm, demonstrating its efficiency and optimality in various distribution scenarios, and introduces new variants including a randomized online algorithm.

Contribution

It offers a comprehensive theoretical analysis of the Sum of Squares algorithm and introduces new variants, including an almost optimal randomized online algorithm.

Findings

SS performs well with sublinear expected waste for certain distributions.

SS has expected waste at most O(log n) when optimal waste is bounded.

A new LP-based algorithm determines the growth rate of optimal expected waste.

Abstract

In this paper we present a theoretical analysis of the deterministic on-line {\em Sum of Squares} algorithm ($SS$) for bin packing introduced and studied experimentally in \cite{CJK99}, 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 $O(\log n)$. In addition, we discuss several interesting variants on $SS$, including a randomized $O(nB\log B)$-time on-line algorithm $SS^*$, based on $SS$, whose expected behavior is essentially optimal for all discrete distributions. Algorithm $SS^*$ also 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. This article is a greatly expanded version of the conference paper \cite{sumsq2000}.