The Davenport constant of a group determines how large a subset can be without containing a set of elements which sum to zero. Its determination is an example of a zero-sum problem. In general, finite abelian groups \(G\) are considered. The Davenport constant \(D(G)\) is the smallest integer \(d\) such that every sequence of elements of length \(d\) contains a non-empty subsequence with sum equal to the zero element of \(G\). The constant \(E(G)\) is the least positive integer \(k\) such that every sequence of length \(k\) contains a zero-sum subsequence of length \(|G|\). For a finite cyclic group \(G\), one has the well-known constants \(\eta (G)=D(G)\) and \(s(G)=E(G)\). It was proved by \textit{W. D. Gao} [J. Number Theory 58, No. 1, 100--103 (1996; Zbl 0892.11005)] that for any finite abelian group \(G\), \( E(G)=|G|+D(G)-1\). It was conjectured by \textit{W. D. Gao} et al. [Integers 7, No. 1, Paper A21, 22 p., electronic only (2007; Zbl 1201.11030)] that \(s(G)=\eta (G)+\exp(G)-1\). Among several existing generalizations, the paper deals with sums of sequences weighted by \(\{\pm 1\}\). It is proved that \(s_{\pm }(C_{n}\oplus C_{n})>\eta _{\pm }(C_{n}\oplus C_{n})+\exp(C_{n}\oplus C_{n})-1\), for any odd integer \(n>7\). However, it is also proved that \(s_{\pm }(G)=\eta _{\pm }(G)+\exp(G)-1\), for any abelian group \(G\) of order 8 and 16.