O((logn)²) Time Online Approximation Schemes for Bin Packing and Subset Sum Problems

Abstract

Given a set S = {b ₁, ⋯ , b _n } of integers and an integer s, the subset sum problem is to decide if there is a subset S′ of S such that the sum of elements in S′ is exactly equal to s. We present an online approximation scheme for this problem. It updates in O(logn) time and gives a (1 + ε)-approximation solution in $O((\log n+{1\over \epsilon^2}{(\log{1\over\epsilon})^{O(1)}})\log n)$ time. The online approximation for target s is to find a subset of the items that have been received. The bin packing problem is to find the minimum number of bins of size one to pack a list of items a ₁, ⋯ , a _n of size in [0,1]. Let function bp(L) be the minimum number of bins to pack all items in the list L. We present an online approximate algorithm for the function bp(L) in the bin packing problem, where L is the list of the items that have been received. It updates in O(logn) updating time and gives a (1 + ε)-approximation solution app(L) for bp(L) in $O((\log n)^2+({1\over \epsilon})^{O({1\over\epsilon})})$ time to satisfy app(L) ≤ (1 + ε)bp(L) + 1.

This research is supported by NSF Career Award 0845376.