Minkowski's convex body theorem and integer programming
Mathematics of Operations Research
Fast approximation algorithms for fractional packing and covering problems
Mathematics of Operations Research
Non-averaging subsets and non-vanishing transversals
Journal of Combinatorial Theory Series A
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
An efficient approximation scheme for the one-dimensional bin-packing problem
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Carathéodory bounds for integer cones
Operations Research Letters
Bounding the running time of algorithms for scheduling and packing problems
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
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As Bin Packing is NP-hard already for k=2 bins, it is unlikely to be solvable in polynomial time even if the number of bins is a fixed constant. However, if the sizes of the items are polynomially bounded integers, then the problem can be solved in time n^O^(^k^) for an input of length n by dynamic programming. We show, by proving the W[1]-hardness of Unary Bin Packing (where the sizes are given in unary encoding), that this running time cannot be improved to f(k)@?n^O^(^1^) for any function f(k) (under standard complexity assumptions). On the other hand, we provide an algorithm for Bin Packing that obtains in time 2^O^(^k^l^o^g^^^2^k^)+O(n) a solution with additive error at most 1, i.e., either finds a packing into k+1 bins or decides that k bins do not suffice.