Min-Sum Bin Packing
We study min-sum bin packing (MSBP). This is a bin packing problem, where the cost of an item is the index of the bin into which it is packed. The problem is equivalent to a batch scheduling problem we define, where the total completion time is to be minimized. The problem is NP-hard in the strong sense. We show that it is not harder than this by designing a polynomial time approximation scheme for it. We also show that several natural algorithms which are based on well-known bin packing heuristics (such as First Fit Decreasing) fail to achieve an asymptotic finite approximation ratio, whereas Next Fit Increasing has an absolute approximation ratio of at most 2, and an asymptotic approximation ratio of at most 1.6188. We design a new heuristic that applies Next Fit Increasing on the relatively small items and adds the larger items using First Fit Decreasing, and show that its asymptotic approximation ratio is at most 1.5604.
KeywordsBin packing Approximation algorithms Analysis of algorithms
- Coffman E G, Garey M R, Johnson D S (1997) Approximation algorithms. In: Hochbaum D (ed) Approximation algorithms for bin packing: a survey. PWS Publishing Company, BostonGoogle Scholar
- Csirik J, Woeginger GJ (1998) On-line packing and covering problems. In: Fiat A, Woeginger GJ (eds) Online algorithms: the state of the art, pp 147–177Google Scholar
- Epstein L, Levin A (2007) Minimum weighted sum bin packing. In: Proceedings of the 5th international workshop on approximation and online algorithms (WAOA’2007), pp. 218–231Google Scholar
- Munagala K, Babu S, Motwani R, Widom J (2005) The pipelined set cover problem. In: Proceedings of the 10th international conference on database theory (ICDT’2005), pp 83–98Google Scholar
- Ullman JD (1971) The performance of a memory allocation algorithm. Technical Report 100, Princeton University, PrincetonGoogle Scholar