# On-line variable sized covering

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## Abstract

We consider one-dimensional and multi-dimensional vector covering with variable sized bins. In the one-dimensional case, we consider variable sized bin covering with bounded item sizes. For every finite set of bins B, and upper bound 1/m on the size of items for some integer m, we define a ratio r(B,m). We prove this is the best possible competitive ratio for the set of bins B and the parameter m, by giving both an algorithm with competitive ratio r(B,m), and an upper bound of r(B,m) on the competitive ratio of any on-line deterministic or randomized algorithm. The ratio satisfies r(B,m) ≥ m/(m + 1), and equals this number if all bins are of size1. For multi-dimensional vector covering we consider the case where each bin is a binary d-dimensional vector. It is known that if B contains a single bin which is all 1, then the best competitive ratio is Θ(1/d). We show an upper bound of 1/2d(1−o(1)) for the general problem, and consider four special case variants. We show an algorithm with optimal competitive ratio 1/2 for the model where each bin in B is a unit vector. We consider the model where B consists of all unit prefix vectors. Each bin has i leftmost components of 1, and all other components are 0. We show that this model is harder than the case of unit vector bins, by giving an upper bound of O(1/ log d) on the competitive ratio of any deterministic or randomized algorithm. Next, we discuss the model where B contains all binary vectors. We show this model is easier than the model of one bin type which is all 1, by giving an algorithm of ratio Ω(1/ log d). The most interesting multi-dimensional case is d =2. Previous results give a 0.25-competitive algorithm for B = {(1, 1)}, and an upper bound of 0.4 on the competitive ratio of any algorithm. In this paper we consider all other models for d =2. For unit vectors, we give an optimal algorithm with competitive ratio 1/2. For unit prefix vectors we give an upper bound of 4/9 on the competitive ratio of any deterministic or randomized algorithm. For the model where B consists of all binary vectors, we design an algorithm with ratio larger than 0.4. These results show that all above relations between models hold for d = 2 as well.

Original language English Computing and Combinatorics - 7th Annual International Conference, COCOON 2001, Proceedings Jie Wang Springer Verlag 463-472 10 9783540424949 https://doi.org/10.1007/3-540-44679-6_52 Published - 2001 Yes 7th Annual International Conference on Computing and Combinatorics, COCOON 2001 - Guilin, ChinaDuration: 20 Aug 2001 → 23 Aug 2001

### Publication series

Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 2108 0302-9743 1611-3349

### Conference

Conference 7th Annual International Conference on Computing and Combinatorics, COCOON 2001 China Guilin 20/08/01 → 23/08/01