Is it possible that the binary variables below be modeled as continuous variables?
\begin{alignat}2\max&\quad\sum _{{e\in E}}w(e_{j})\cdot y_{j}\\\text{s.t.}&\quad\sum {x_{i}}\leq k,\quad&(\text{no more than}\,\,k\,\,\text{sets are selected})\\&\quad{\sum _{e_{j}\in S_{i}}x_{i}\geq y_{j}},\quad&(\text{if}\,\,y_{j}>0\,\,\text{then at least one set}\,\,e_{j}\in S_{i}\,\,\text{is selected})\\&\quad{y_{j}\in \{0,1\}},\quad&(\text{if}\,\,y_{j}=1\,\,\text{then}\,\,e_{j}\,\,\text{is covered})\\&\quad{x_{i}\in \{0,1\}},\quad&(\text{if}\,\,x_{i}=1\,\,\text{then}\,\,S_{i}\,\,\text{is selected for the cover})\end{alignat}