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%Write your title here: 
\noindent{\bf\Large On the Parameterized Complexity of\\ Finding Paths with Shared Edges}

%List the authors as follows (you may underline the speaker, or just
%name the speaker, and mention your co-authors at the end, up to you):  
%\noindent M. Coauthor -- {\small Other university}
\noindent Till Fluschnik -- {\small TU Berlin}

%\noindent \underline{N.  Speaker} -- {\small Miscatonic University}



\vspace{2ex}

\noindent{\bf Abstract:} %Here write your abstract

%This paper is about ... Cite like that \cite{CCHPP}, \cite{Wol15}. 
We study the Minimum Shared Edges (MSE) problem on undirected graphs. 
Given an undirected graph, a source and a sink vertex, and two integers $p$ and $k$, the question is whether there are $p$ paths in the graph connecting the source with the sink that share at most $k$ edges. 
Herein, an edge is shared if it appears in at least two paths. 
Complementing an NP-hardness result for the directed variant, we show that MSE is NP-complete even on planar graphs.
Further, we show that MSE is W[2]-hard when parameterized by the number~$k$ of shared edges and W[1]-hard when parameterized by the treewidth. On the positive side, we show that MSE is fixed-parameter tractable with respect to the number $p$ of paths. 
For the latter result, we employ the so-called treewidth reduction technique due to Marx, O'Sullivan and Razgon \cite{MOR13}.

The talk is based on a joint work with Stefan Kratsch, Rolf Niedermeier and Manuel Sorge.

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% \bibitem{CCHPP}
% R.~H. Chitnis, M.~Cygan, M.~Hajiaghayi, M.~Pilipczuk, and M.~Pilipczuk.
% \newblock Designing {FPT} algorithms for cut problems using randomized
%   contractions.
% \newblock In {\em 53rd Annual {IEEE} Symposium on Foundations of Computer
%   Science (FOCS)}, pages 460--469, 2012.
% 
% \bibitem{Wol15}
% P.~Wollan.
% \newblock The structure of graphs not admitting a fixed immersion.
% \newblock {\em Journal of Combinatorial Theory, Series {B}}, 110:47--66, 2015.

\bibitem{MOR13}
D.~Marx, B.~O'Sullivan, and I.~Razgon.
\newblock Finding small separators in linear time via treewidth reduction.
\newblock {\em {ACM} Transactions on Algorithms}, 9(4):30, 2013.

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