# Is there an FPT or XP algorithm known for this version of $k$-edge disjoint paths problem?

The shortest $$k$$-edge disjoint paths problem is defined as follows:

Input: An undirected graph $$G=(V,E)$$ and $$k$$ pairs of vertices $$(s_1,t_1),\ldots,(s_k,t_k)$$.

Question: Find (if exist) $$k$$-pairwise edge-disjoint paths $$P_1,\ldots,P_k$$ where $$P_i$$ goes from $$s_i$$ to $$t_i$$, such that $$\sum\limits_{i=1}^k|P_i|$$ is minimized, where $$|P_i|$$ denotes the number of edges in path $$P_i$$.

Is there a fixed-parameter tractable or an XP algorithm known for the problem, with parameter $$k$$ ?

PS: This is a related question: For which values of $k$ is minimum length undirected $k$-disjoint-paths in $\mathcal{P}$?.

• IIRC I did some literature search for this some time ago and it seems to be open even for $k=3$. May 22 at 7:04