Given an undirected complete weighted graph $G=(V,E)$ with nonnegative weight function obeying the triangle inequality, a set $\{C_1,C_2,$ $\ldots,C_k\}$ of cycles is called a \textit{cycle cover} if $V \subseteq \bigcup_{i=1}^k V(C_i)$ and its cost is given by the maximum weight of the cycles. The Minimum Cycle Cover Problem (MCCP) aims to find a cycle cover of cost at most $\lambda$ with the minimum number of cycles. We propose new LP relaxations for MCCP as well as its variants, called the Minimum Path Cover Problem (MPCP) and the Minimum Tree Cover Problem, where the cycles are replaced by paths or trees. Moreover, we give new LP relaxations for a special case of the rooted version of MCCP/MPCP. We show that these LP relaxations have significantly better integrality gaps than the previous relaxations.