Online unit clustering: Variations on a theme
Theoretical Computer Science
Online interval coloring with packing constraints
Theoretical Computer Science
Approximation schemes for packing splittable items with cardinality constraints
WAOA'07 Proceedings of the 5th international conference on Approximation and online algorithms
Class constrained bin packing revisited
Theoretical Computer Science
AFPTAS Results for Common Variants of Bin Packing: A New Method for Handling the Small Items
SIAM Journal on Optimization
SIGACT news online algorithms column 20: the power of harmony
ACM SIGACT News
Improved results for a memory allocation problem
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
Bin covering with cardinality constraints
Discrete Applied Mathematics
Cooperating to buy shoes in the real world: online cycle picking in directed graphs
Proceedings of the South African Institute for Computer Scientists and Information Technologists Conference
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We consider a one-dimensional storage system where each container can store a bounded amount of capacity as well as a bounded number of items $k\geq 2$. This defines the (standard) bin packing problem with cardinality constraints, which is an important version of bin packing. Following previous work on the unbounded space online problem, we establish the exact best competitive ratio for bounded space online algorithms for every value of $k$. This competitive ratio is a strictly increasing function of $k$ which tends to $\Pi_\infty+1\approx 2.69103$ for large $k$. Lee and Lee showed in 1985 [J. ACM, 32 (1985), pp. 562-572] that the best possible competitive ratio for online bounded space algorithms for the classical bin packing problem is the sum of a series, and tends to $\Pi_\infty$ as the allowed space (number of open bins) tends to infinity. We further design optimal online bounded space algorithms for variable sized bin packing, where each allowed bin size may have a distinct cardinality constraint, and for the resource augmentation model. All algorithms achieve the exact best possible competitive ratio possible for the given problem and use constant numbers of open bins. Finally, we introduce unbounded space online algorithms with smaller competitive ratios than the previously known best algorithms for small values of $k$, for the standard cardinality constrained problem. These are the first algorithms with competitive ratio below 2 for $k=4,5,6$.