On-line and off-line approximation algorithms for vector covering problems
This paper deals with vector covering problems in d-dimensional space. The input to a vector covering problem consists of a set X of d-dimensional vectors in [0, 1]d. The goal is to partition X into a maximum number of parts, subject to the constraint that in every part the sum of all vectors is at least one in every coordinate. This problem is known to be NP-complete, and we are mainly interested in its on-line and off-line approximability.
For the on-line version, we construct approximation algorithms with worst case guarantee arbitrarily close to 1/(2d) in d ≥ 2 dimensions. This result contradicts a statement of Csirik and Frenk (1990) in  where it is claimed that for d ≥ 2, no on-line algorithm can have a worst case ratio better than zero. For the off-line version, we derive polynomial time approximation algorithms with worst case guarantee Ω(1/log d). For d=2, we present a very fast and very simple off-line approximation algorithm that has worst case ratio 1/2. Moreover, we show that a method from the areaof compact vector summation can be used to construct off-line approximation algorithms with worst case ratio 1/d for every d ≥ 2.
KeywordsApproximation algorithm worst case ratio packing problem covering problem on-line algorithm
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- 1.N. Alon, Transversal numbers of uniform hypergraphs, Graphs and Combinatorics 6, 1990, 1–4.Google Scholar
- 2.S.F. Assmann, “Problems in Discrete Applied Mathematics”, Doctoral Dissertation, Mathematics Department, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1983.Google Scholar
- 3.S.F. Assmann, D.S. Johnson, D. J. Kleitman, and J.Y.-T. Leung, On a dual version of the one-dimensional bin packing problem, J. Algorithms 5, 1984, 502–525.Google Scholar
- 4.J. Beck and T. Fiala, Integer-making Theorems, Disc. Appl. Math. 3, 1981, 1–8.Google Scholar
- 5.J. Csirik and J.B.G. Frenk, A dual version of bin packing, Algorithms Review 1, 1990, 87–95.Google Scholar
- 6.J. Csirik, J.B.G. Frenk, G. Galambos, and A.H.G. Rinnooy Kan, Probabilistic analysis of algorithms for dual bin packing problems, J. Algorithms 12, 1991, 189–203.Google Scholar
- 7.J. Csirik and V. Totik, On-line algorithms for a dual version of bin packing, Discr. Appl. Math. 21, 1988, 163–167.Google Scholar
- 8.W. Fernandez de la Vega and G.S. Lueker, Bin packing can be solved within 1 + ε in linear time, Combinatorica 1, 1981, 349–355.Google Scholar
- 9.T. Gaizer, An algorithm for the 2D dual bin packing problem, unpublished manuscript, University of Szeged, Hungary, 1989.Google Scholar
- 10.M.R. Garey, R.L. Graham, D.S. Johnson, and A.C. Yao, Resource constrained scheduling as generalized bin packing, J. Combinatorial Theory Sci. A 21, 1976, 257–298.Google Scholar
- 11.P. Raghavan, Probabilistic Construction of Deterministic Algorithms: Approximating Packing Integer Programs, Journal of Computer and System Sciences 37, 1988, 130–143.Google Scholar
- 12.M.B. Richey, Improved bounds for harmonic-based bin packing algorithms, Discrete Applied Mathematics 34, 1991, 203–227.Google Scholar
- 13.S.V. Sevastianov, Geometry in the theory of scheduling, Trudy Instituta Matematiki Sibirskogo Otdelenia Akademii Nauk SSSR 10, 1988, 226–261. (in Russian).Google Scholar
- 14.S.V. Sevastianov, On some geometric methods in scheduling theory: a survey, Discrete Applied Mathematics 55, 1994, 59–82.Google Scholar