  In this paper we give an $\widetilde{O}((nm)^{2/3}\log C)$ time algorithm for
computing min-cost flow (or min-cost circulation) in unit capacity planar
multigraphs where edge costs are integers bounded by $C$. For planar
multigraphs, this improves upon the best known algorithms for general graphs:
the $\widetilde{O}(m^{10/7}\log C)$ time algorithm of Cohen et al. [SODA 2017],
the $O(m^{3/2}\log(nC))$ time algorithm of Gabow and Tarjan [SIAM J. Comput.
1989] and the $\widetilde{O}(\sqrt{n}m \log C)$ time algorithm of Lee and
Sidford [FOCS 2014]. In particular, our result constitutes the first known
fully combinatorial algorithm that breaks the $\widetilde{O}(m^{3/2})$ time
barrier for min-cost flow problem in planar graphs.
  To obtain our result we first give a very simple successive shortest paths
based scaling algorithm for unit-capacity min-cost flow problem that does not
explicitly operate on dual variables. This algorithm also runs in
$\widetilde{O}(m^{3/2}\log{C})$ time for general graphs, and, to the best of
our knowledge, it has not been described before. We subsequently show how to
implement this algorithm faster on planar graphs using well-established tools:
$r$-divisions and efficient algorithms for computing (shortest) paths in
so-called dense distance graphs.
