Discrepancy of set-systems and matrices
European Journal of Combinatorics
Handbook of combinatorics (vol. 2)
Balancing vectors and Gaussian measures of n-dimensional convex bodies
Random Structures & Algorithms
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
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
Constructive Algorithms for Discrepancy Minimization
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Bin packing via discrepancy of permutations
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Tight hardness results for minimizing discrepancy
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
An OPT+1 algorithm for the cutting stock problem with constant number of object lengths
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
The tight bound of first fit decreasing bin-packing algorithm is FFD(I) ≤ 11/9OPT(I) + 6/9
ESCAPE'07 Proceedings of the First international conference on Combinatorics, Algorithms, Probabilistic and Experimental Methodologies
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A well-studied special case of bin packing is the 3-partition problem, where n items of size 1/4 have to be packed in a minimum number of bins of capacity one. The famous Karmarkar-Karp algorithm transforms a fractional solution of a suitable LP relaxation for this problem into an integral solution that requires at most O(log n) additional bins. The three-permutations-problem of Beck is the following. Given any three permutations on n symbols, color the symbols red and blue, such that in any interval of any of those permutations, the number of red and blue symbols is roughly the same. The necessary difference is called the discrepancy. We establish a surprising connection between bin packing and Beck’s problem: The additive integrality gap of the 3-partition linear programming relaxation can be bounded by the discrepancy of three permutations. This connection yields an alternative method to establish an O(log n) bound on the additive integrality gap of the 3-partition. Conversely, making use of a recent example of three permutations, for which a discrepancy of Ω(log n) is necessary, we prove the following: The O(log2 n) upper bound on the additive gap for bin packing with arbitrary item sizes cannot be improved by any technique that is based on rounding up items. This lower bound holds for a large class of algorithms including the Karmarkar-Karp procedure.