Subsequence sums: Direct and inverse problems

Abstract

Let A=(a0,…,a0︸r0copies,a1,…,a1︸r1copies,…,ak−1,…,ak−1︸rk−1copies) be a finite sequence of integers with k distinct terms, denoted alternatively by (a0,a1,…,ak−1)r¯, where a0<a1<⋯<ak−1, r¯=(r0,r1,…,rk−1), ri≥1 for i=0,1,…,k−1. The sum of all the terms of a subsequence of length at least one of the sequence A is said to be a subsequence sum of A. The set of all subsequence sums of A is denoted by S(r¯,A). The direct problem for subsequence sums is to find the lower bound for |S(r¯,A)| in terms of the number of distinct terms in the sequence A. The inverse problem for subsequence sums is to determine the structure of the finite sequence A of integers for which |S(r¯,A)| is minimal. In this paper, both the problems are solved and some well-known results for subset sum problem are obtained as corollaries of the results for subsequence sum problem.