Abstract
In the recent paper (Benkő et al. 2010) we introduced a new problem that we call Bin Packing/Covering with Delivery, or BP/CD for short. Mainly we mean under this expression that we look for not only a good, but a “good and fast” packing or covering. In the present paper we investigate the offline case. For the analysis, a novel view on “offline optimum” is introduced, which appears to be relevant concerning all problems where a final solution is ordering-dependent. We prove that if the item sizes are not allowed to be arbitrarily close to zero, then an optimal offline solution can be found in polynomial time. On the other hand, for unrestricted problem instances, no polynomial-time algorithm can achieve an asymptotic approximation ratio better than 6/7 if \(P\ne NP\).
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Notes
One way to do this is to define \(\lambda (B_{j})\) to be the smallest element of \(\{1,\dots ,K\}\) which is not the label of any bin open in the moment when \(B_{j}\) is opened. Since the number of the former is at most \(K-1\) for all \(1\le j\le m\) (otherwise \(B_{j}\) would already be the \((K+1)\)st bin open simultaneously), there exists a label available for \(B_{j}\).
This situation can occur only if we drop the assumption that \(A\) may open more than two bins (and then also \(K\ge 3\) necessarily holds), but we include it already in the current table, for completeness and also for later reference in the proof of the general case.
References
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Benkő A, Dósa Gy, Tuza Zs. (2010) Bin Packing/Covering with Delivery, solved with the evolution of algorithms. In: IEEE fifth international conference on bio-inspired computing: theories and applications (BIC-TA 2010), Changsha, China, pp 298–302, 23–26 September 2010
Dósa Gy, Tuza Zs (2011) Bin Packing/Covering with Delivery: some variations, theoretical results and efficient offline algorithms. arXiv:1207.5672v1, unpublished manuscript
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Research supported in part by the projects TAMOP-4.2.2/B-10/1-2010-0025 and K-TET 10-1-2011-0115. Research supported in part by the Hungarian Scientific Research Fund, OTKA grant T-81493.
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Benkő, A., Dósa, G. & Tuza, Z. Bin covering with a general profit function: approximability results. Cent Eur J Oper Res 21, 805–816 (2013). https://doi.org/10.1007/s10100-012-0269-0
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DOI: https://doi.org/10.1007/s10100-012-0269-0