CDAM: Computational, Discrete and Applicable Mathematics@LSE
	CDAM Research Report, LSE-CDAM-2007-33 December 2007

A Sublinear-Time Approximation Scheme for Bin Packing

Tugkan Batu, Petra Berenbrink, and Christian Sohler

We present a sublinear-time asymptotic approximation scheme for the bin packing problem; that is, for any ε>0, we present an algorithm A_ε that has sampling access to the input instance and outputs a value k such that C_opt≤ k≤ (1+ε) C_opt+1, where C_opt is the cost of an optimal solution. It is clear that uniform sampling by itself will not allow a sublinear-time algorithm in this setting; a small number of items might constitute most of the total weight and uniform samples will not hit them. In this work we use weighted samples, where item i is sampled with probability proportional to its weight: that is, with probability w_i/∑_i w_i. In the presence of weighted samples, the approximation algorithm runs in O(n^1/2 poly(log(n)/ε)) + g(1/ε) time, where g(x) is an exponential function of x. When both weighted and uniform sampling are allowed, O(n^1/3 poly(log(n)/ε)) + g(1/ε) time suffices. In addition to an approximate value to C_opt, our algorithm can also output a constant-size ``template'' of a packing that can later be used to find a near-optimal packing in linear time.

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