Optimal on-line algorithms for variable-sized bin covering

doi:10.1016/S0167-6377(99)00023-1

Operations Research Letters

Volume 25, Issue 1, August 1999, Pages 47-50

https://doi.org/10.1016/S0167-6377(99)00023-1 Get rights and content

Abstract

We deal with the variable-sized bin covering problem: Given a list L of items in (0,1] and a finite collection $B$ of feasible bin sizes, the goal is to select a set of bins with sizes in $B$ and to cover them with the items in L such that the total size of the covered bins is maximized. In the on-line version of this problem, the items must be assigned to bins one by one without previewing future items. This note presents a complete solution to the on-line problem: For every collection $B$ of bin sizes, we give an on-line approximation algorithm with a worst-case ratio $r(B)$ , and we prove that no on-line algorithm can perform better in the worst case. The value $r(B)$ mainly depends on the largest gap between consecutive bin sizes.

Introduction

Problem statement.

In the variable-sized bin covering problem, we are given a finite collection $B$ of bin sizes and a list L=〈a₁,a₂,…,a_n〉 of items in (0,1]. We say that a bin is covered by a subset L′ of the items if the total size of the items in L′ is greater or equal to the size of this bin. A feasible bin cover for L and $B$ is an assignment of the items in L to a set of bins with sizes in $B$ such that every bin is covered. The goal is to find a feasible bin cover that maximizes the sum of the sizes of the used bins. This covering problem arises in a wide variety of contexts. It has many applications in business and in industry, from packing peach slices into tin cans so that each tin can contains at least its advertised net weight, to such complex problems as breaking up monopolies into smaller companies, each of which is large enough to be viable. Since the problem of finding an optimal bin cover is NP-hard, research has concentrated on approximation algorithms that find near-optimal bin covers.

Approximation algorithms and on-line algorithms.

Let $OPT (L, B)$ and $A(L, B)$ denote, respectively, the total size of the bin cover produced by an optimum algorithm and the total size of the bin cover produced by an approximation algorithm A for an input list L and a collection $B$ of bin sizes. The asymptotic worst-case ratio $R_{A,B}^{∞}$ of algorithm A for the collection $B$ is defined as $R_{A,B}^{∞} = lim s→∞ inf L {A(L, B)/ OPT (L, B) | OPT (L, B)⩾s}$ (cf. [8]). Clearly, $0⩽R_{A,B}^{∞} ⩽1$ . The asymptotic worst-case ratio is the usual measure for the quality of an approximation algorithm for covering problems: the larger the ratio, the better the approximation algorithm.

Now assume an environment where the list L of items arrives one by one. When item a_i arrives, it must immediately and irrevocably be assigned to its bin, and the next item a_i+1 becomes only known after item a_i has been assigned. Such an environment is called on-line, and an approximation algorithm that is able to work in an on-line environment is called an on-line algorithm. In contrast to this type of algorithm are the off-line algorithms that may assume knowledge of the entire problem instance before producing any output.

Known results.

Bin covering was first studied in the 1980s by Assmann [2] and Assmann et al. [3], who investigated the case where all bins are of size 1. They gave a polynomial-time approximation algorithm with asymptotic worst-case ratio $34$ for the off-line version, and an algorithm with asymptotic worst-case ratio $12$ for the on-line version. Csirik and Totik [6] showed that there is no on-line algorithm with asymptotic worst-case ratio better than $12$ . This completely settles the on-line case where all bin are of size 1. For related results on probabilistic aspects of the problem see [5], for a higher-dimensional variant see [1], for a short survey on covering problems see [4], and for a survey on on-line packing and covering problems see [7].

Our results and organization of the paper.

We present a complete solution for the on-line variable-sized bin covering problem: for every collection $B$ of bin sizes, we determine the best possible asymptotic worst-case ratio $r(B)$ over all on-line algorithms. The precise values $r(B)$ are defined in Section 2. In Section 3 we design and analyze an on-line algorithm for variable-sized bin covering whose asymptotic worst case ratio reaches $r(B)$ , and in Section 4 we prove that no on-line algorithm can beat the bound $r(B)$ .

Section snippets

Some definitions and statement of the main result

Throughout the paper, L=〈a₁,a₂,…,a_n〉 is a list of items in (0,1] and $B$ is the set of feasible bin sizes. We assume that the bins are normalized such that the largest bin has a size of 1. Denote by k the number of bin sizes in $B$ that are strictly greater than $12$ , and let $1=b_{1} >b_{2} >⋯>b_{k} > 12$ be an enumeration of these bin sizes. Define $q(B)≐ max {b_{j} /b_{j+1} : 1⩽j⩽k−1}∪{2b_{k}}.$ Observe that $q(B)>1$ . Finally, define $r(B)≐1/q(B)$ .

Theorem 1

For every collection $B$ of bin sizes, there exists an on-line approximation algorithm A

The approximation algorithm

In this section, we present an on-line algorithm A for bin covering. Our construction is based on the ideas of the ‘harmonic’ algorithm of Lee and Lee [9] for on-line bin packing.

Define m to be the smallest non-negative integer that fulfills 1+q2^−m−1⩽q. Define a partition of the interval (0,1] into k(m+1) intervals $I_{j,l}$ with 1⩽j⩽k and $0⩽ l ⩽m$ as follows.

•
For 1⩽j⩽k−1 and $0⩽ l ⩽m$ , let $I_{j,l} =(b_{j+1} /2^{l},b_{j} /2^{l}]$ .
•
For $0⩽ l ⩽m−1$ , let $I_{k,l} =(b_{1} /2^{l+1},b_{k} /2^{l}]$ .
•
Let $I_{k,m} =(0,b_{k} /2^{m}]$ .

Moreover, we assign to each interval $I_{}$

The non-approximability result

For technical reasons, we will set $b_{k+1} = 12$ throughout this section, and we will allow the on-line algorithm to use bins of size $12$ even in case $12 ∉ B$ holds. The optimum off-line solution, however, will not be allowed to use bins of size $12$ . Since we are heading for a non-approximability result, these assumptions will not cause any problems. As a consequence, there does exist an index z, 1⩽z⩽k, for which b_z/b_z+1=q holds; cf. Eq. (3).

Now let us suppose that there does exist an on-line

Acknowledgements

We would like to thank János Csirik for bringing the on-line variable-sized bin covering problem to our attention.

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¹: Supported by the START program Y43-MAT of the Austrian Ministry of Science.

²: Supported by the Science Foundation under the State Education Committee of China.

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