In Path Set Packing, the input is an undirected graph $G$, a collection $\calp$ of simple paths in $G$, and a positive integer $k$. The problem is to decide whether there exist $k$ edge-disjoint paths in $\calp$. We study the parameterized complexity of Path Set Packing with respect to both natural and structural parameters. We show that the problem is $W[1]$-hard with respect to vertex cover number, and $W[1]$-hard respect to pathwidth plus maximum degree plus solution size. These results answer an open question raised in COCOON 2018. On the positive side, we present an FPT algorithm parameterized by feedback vertex number plus maximum degree, and present an FPT algorithm parameterized by treewidth plus maximum degree plus maximum length of a path in $\calp$. These positive results complement the hardness of Path Set Packing with respect to any subset of the parameters used in the FPT algorithms. We also give a $4$-approximation algorithm for maximum path set packing problem which runs in FPT time when parameterized by feedback edge number.

翻译：在路径集打包问题中，输入为一个无向图 $G$、$G$ 中简单路径的集合 $\calp$ 以及一个正整数 $k$。该问题需要判断 $\calp$ 中是否存在 $k$ 条边不相交的路径。我们研究了路径集打包问题相对于自然参数与结构参数的参数化复杂度。我们证明该问题相对于顶点覆盖数是 $W[1]$-难的，并且相对于路径宽度、最大度与解规模之和也是 $W[1]$-难的。这些结果回答了 COCOON 2018 会议上提出的一个开放性问题。在积极方面，我们提出了一个以反馈顶点数加最大度为参数的 FPT 算法，以及一个以树宽、最大度与 $\calp$ 中路径最大长度之和为参数的 FPT 算法。这些积极结果与路径集打包问题在 FPT 算法所用参数的任意子集上的困难性形成了互补。我们还给出了最大路径集打包问题的一个 $4$-近似算法，该算法在以反馈边数为参数时具有 FPT 时间复杂度。