  Partial set cover problem and set multi-cover problem are two generalizations
of set cover problem. In this paper, we consider the partial set multi-cover
problem which is a combination of them: given an element set $E$, a collection
of sets $\mathcal S\subseteq 2^E$, a total covering ratio $q$ which is a
constant between 0 and 1, each set $S\in\mathcal S$ is associated with a cost
$c_S$, each element $e\in E$ is associated with a covering requirement $r_e$,
the goal is to find a minimum cost sub-collection $\mathcal S'\subseteq\mathcal
S$ to fully cover at least $q|E|$ elements, where element $e$ is fully covered
if it belongs to at least $r_e$ sets of $\mathcal S'$. Denote by
$r_{\max}=\max\{r_e\colon e\in E\}$ the maximum covering requirement. We
present an $(O(\frac{r_{\max}\log^2n}{\varepsilon}),1-\varepsilon)$-bicriteria
approximation algorithm, that is, the output of our algorithm has cost at most
$O(\frac{r_{\max}\log^2 n}{\varepsilon})$ times of the optimal value while the
number of fully covered elements is at least $(1-\varepsilon)q|E|$.
