Approximation algorithms for multicut for special classes of graphs The multicut problem is the following.

Given a graph $G=(V,E)$ with edge costs $c_e$ for edge $e$ and a set of $k$ terminal pairs $\{(s_1,t_1),\ldots,(s_k,t_k)\}$, the objective is to find a set of edges $F \subseteq E$ of minimum cost, such that in the graph $H=(V,E \setminus F)$, no terminal pair $s_i,t_i$ are in the same connected component. In other words, deleting the edges in $F$ will remove all paths between any terminal pair $s_i,t_i$.

This problem is NP-hard. For general graphs, an $O(\log k)$-approximation is known. For trees, a 2-approximation is known. It is also known to be APX-hard. Is there any other results known for special classes of graphs?

I am particularly interested in the following graphs, but other classes of graphs will also be informative.


Paths
Cycles
$n \times n$ grids
Outerplanar graphs
Planar graphs
Series-parallel graphs


Any reference or survey paper will be highly appreciated.
Check the following survey:


Marie-Christine Costa, Lucas Létocart, and Frédéric Roupin, "Multicut and integral multiflow: a survey", 2005


You can also check my dissertation on the directed case.