A Dense Hierarchy of Sublinear Time Approximation Schemes for Bin Packing

TL;DR

This paper introduces a new randomized approximation scheme for bin packing that operates in sublinear time, providing near-optimal solutions efficiently and establishing bounds on the algorithm's complexity.

Contribution

It presents a novel sublinear time randomized approximation scheme for bin packing and proves lower bounds on the necessary time for such approximations.

Findings

Developed an $O(n(\log n)(\log\log n)/ ext{sum of sizes})$ time approximation scheme.

Proved lower bounds showing no faster algorithms can achieve similar approximation ratios.

Established complexity bounds for specific classes of bin packing problems based on total item size.

Abstract

The bin packing problem is to find the minimum number of bins of size one to pack a list of items with sizes $a_1,..., a_n$ in $(0,1]$. Using uniform sampling, which selects a random element from the input list each time, we develop a randomized $O({n(\log n)(\log\log n)\over \sum_{i=1}^n a_i}+({1\over \epsilon})^{O({1\over\epsilon})})$ time $(1+\epsilon)$-approximation scheme for the bin packing problem. We show that every randomized algorithm with uniform random sampling needs $\Omega({n\over \sum_{i=1}^n a_i})$ time to give an $(1+\epsilon)$-approximation. For each function $s(n): N\rightarrow N$, define $\sum(s(n))$ to be the set of all bin packing problems with the sum of item sizes equal to $s(n)$. For a constant $b\in (0,1)$, every problem in $\sum(n^{b})$ has an $O(n^{1-b}(\log n)(\log\log n)+({1\over \epsilon})^{O({1\over\epsilon})})$ time $(1+\epsilon)$-approximation for an arbitrary constant $\epsilon$. On the other hand, there is no $o(n^{1-b})$ time $(1+\epsilon)$-approximation scheme for the bin packing problems in $\sum(n^{b})$ for some constant $\epsilon>0$.