A Tighter Lower Bound for Optimal Bin Packing (original) (raw)

In this paper, we present an efficient algorithm to compute a tighter lower bound for the one-dimensional bin packing problem. The time complexity of the algorithm is O(n log n). We have simulated the algorithm on randomly generated bin packing problems with item sizes drawn uniformly from (a; b], 0 a ! b B. If our lower bound is used, on average, the error of BFD is less than 2%. For a + b B, the error is less than 0.003%. Key words: bin packing, lower bound, best fit decreasing, harmonic partition, matching. i 1 Introduction For NP-complete problems, it may not be possible to find optimal solutions in polynomial time. The quality of an approximation algorithm A is often measured by its asymptotic performance ratio [3] or worst-case performance ratio [6]. For an instance L of a minimization problem Pi, let S (L) be the optimal solution and A(L) be the solution obtained by using algorithm A. If a quantity implicitly depends on the problem instance L, then we drop L from our...

A tight lower bound for optimal bin packing

Operations research letters, 1995

In this paper, we present an O (n log n) algorithm to compute a tight lower bound for the one-dimensional bin packing problem. We have simulated the algorithm on randomly generated bin packing problems with item sizes drawn uniformly from (a, b), where 0⩽ a< b⩽ B and B ...

An All-Around Near-Optimal Solution for the Classic Bin Packing Problem

In this paper we present the first algorithm with optimal average-case and close-to-best known worst-case performance for the classic on-line problem of bin packing. It has long been observed that known bin packing algorithms with optimal average-case performance were not optimal in the worst-case sense. In particular First Fit and Best Fit had optimal average-case ratio of 1 but a worst-case competitive ratio of 1.7. The wasted space of First Fit and Best Fit for a uniform random sequence of length nnn is expected to be Theta(n2/3)\Theta(n^{2/3})Theta(n2/3) and Theta(sqrtnlog3/4n)\Theta(\sqrt{n} \log ^{3/4} n)Theta(sqrtnlog3/4n), respectively. The competitive ratio can be improved to 1.691 using the Harmonic algorithm; further variations of this algorithm can push down the competitive ratio to 1.588. However, Harmonic and its variations have poor performance on average; in particular, Harmonic has average-case ratio of around 1.27. In this paper, first we introduce a simple algorithm which we term Harmonic Match. This algorithm performs...

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