Abstract: Motivated by various network improvement models,we study the general edge capacity augmentation problem on spanning trees in networks (GECAT).The polynomial equivalence between the GECAT problem and the constrained minimum spanning tree problem (CST) is presented,which indicates the GECAT problem is NP-hard.By transforming the GECAT problem into an instance of the CST problem,a polynomial-time approximation scheme (PTAS) is designed to solve the GECAT problem.Two special versions of the GECAT problem are studied:the minimum number of edges to be augmented on spanning trees (MNEAT) and the minimum-cost edge capacity augmentation on minimum spanning trees (MCEAT).A strongly polynomial-time algorithm is designed for each version.For the MNEAT problem,the T-exchange method is used on the spanning trees.For the MCEAT problem,the lexicographical order strategy is utilized.