Parametric packing of selfish items and the subset sum algorithm

TL;DR

This paper determines the exact approximation ratio of the subset sum heuristic for bin packing, generalizes it to parametric sizes, and analyzes the game-theoretic Price of Anarchy for these settings.

Contribution

It establishes the exact approximation ratio of the subset sum algorithm and extends the analysis to parametric bin packing and its game-theoretic Price of Anarchy.

Findings

Exact approximation ratio of the subset sum heuristic is determined.

Provides tight bounds for the Strong Price of Anarchy for all   .

Abstract

The subset sum algorithm is a natural heuristic for the classical Bin Packing problem: In each iteration, the algorithm finds among the unpacked items, a maximum size set of items that fits into a new bin. More than 35 years after its first mention in the literature, establishing the worst-case performance of this heuristic remains, surprisingly, an open problem. Due to their simplicity and intuitive appeal, greedy algorithms are the heuristics of choice of many practitioners. Therefore, better understanding simple greedy heuristics is, in general, an interesting topic in its own right. Very recently, Epstein and Kleiman (Proc. ESA 2008) provided another incentive to study the subset sum algorithm by showing that the Strong Price of Anarchy of the game theoretic version of the bin-packing problem is precisely the approximation ratio of this heuristic. In this paper we establish the exact approximation ratio of the subset sum algorithm, thus settling a long standing open problem. We generalize this result to the parametric variant of the bin packing problem where item sizes lie on the interval (0,\alpha] for some \alpha \leq 1, yielding tight bounds for the Strong Price of Anarchy for all \alpha \leq 1. Finally, we study the pure Price of Anarchy of the parametric Bin Packing game for which we show nearly tight upper and lower bounds for all \alpha \leq 1.