Algorithms in C++
Fast Algorithm for Generating Ascending Compositions
Journal of Mathematical Modelling and Algorithms
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Let n be a positive integer and let g"1,...,g"n be real numbers. The following integer partition problem (IPP) is studied: find a partition of the integer n=@?"i"="1^ni@?@l"i such that @?"i"="1^ng"i@?@l"i is maximal. An extended variant of the IPP is the problem EIPP, where, as a secondary condition, the number @?"i"="1^n@l"i of items has to be minimal. The support of the partition is the index-set of all nonzero items, i.e., {i:@l"i0}. It is proved that there is always an optimal solution for the IPP (as well as for the EIPP) whose support contains at most @?log"2(n+1)@? elements and that this bound is sharp. An algorithm of time complexity O(n^2) for the determination of such an optimal solution is presented. Finally the following non-polynomial bounds for the maximum number M(n) of all optimal solutions for the EIPP are proved: 2.2324n^1^/^3@?lnM(n)@?1363n^1^/^3lnn as n-~.