Bin Packing via Discrepancy of Permutations

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(logn) additional bins. The three-permutations-conjecture of Beck is the following. Given any 3 permutations on n symbols, one can color the symbols red and blue, such that in any interval of any of those permutations, the number of red and blue symbols differs only by a constant. Beck's conjecture is well known in the field of discrepancy theory. We establish a surprising connection between bin packing and Beck's conjecture: If the latter holds true, then the additive integrality gap of the 3partition linear programming relaxation is bounded by a constant. This result indicates that improving approximability results for bin packing requires a better un...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Friedrich Eisenbrand, Dömötör Pálvölgyi, Thomas Rothvoß
• Keywords: Bin Packing, CORR 2010, Education, Famous Karmarkar-karp Algorithm, Suitable Lp Relaxation