Before revision: } n\to[n] %%% that minimizes \sum_{i=1^n |p_ix-b_{\pi(i)}|^z. (ii) Points-to-Lines alignment: Given n }  lines \ell_1,\cdots,\ell_n on the plane  , compute the matching \pi:[n]\to[n] and alignment (rotation matrix R and a translation vector t) that minimize the sum of Euclidean distances  ^n \mathrm{dist}(Rp_i-t,\ell_{\pi(i)})^z } between each point to its corresponding line.
After revision: } n\to[n] %%% that minimizes \sum_{i=1^n |p_ix-b_{\pi(i)}|^z. (ii) Points-to-Lines alignment: Given n } lines \ell_1,\cdots,\ell_n on the plane , compute the matching \pi:[n]\to[n] and alignment (rotation matrix R and a translation vector t) that minimize the sum of Euclidean distances  \sum_{i=1^n \mathrm{dist}(Rp_i-t,\ell_{\pi(i)})^z } between each point to its corresponding line.