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 PATH SET PACKING which runs in FPT time when parameterized by feedback edge number.

翻译：在路径集打包问题中，输入是一个无向图$G$、$G$中简单路径的集合$\calp$以及一个正整数$k$。该问题需要判断是否存在$\calp$中的$k$条边不相交路径。我们从自然参数和结构参数两个角度研究了路径集打包问题的参数化复杂性。我们证明该问题关于顶点覆盖数是$W[1]$困难的，关于路径宽度加上最大度加上解大小也是$W[1]$困难的。这些结果回答了COCOON 2018中提出的一个开放问题。在正面结果方面，我们提出了一个以反馈顶点数加上最大度为参数的FPT算法，并提出了一个以树宽度加上最大度加上$\calp$中路径的最大长度为参数的FPT算法。这些正面结果补充了路径集打包问题关于FPT算法所使用参数的任何子集的困难性。我们还给出了一个路径集打包问题的$4$近似算法，该算法在以反馈边数为参数时可在FPT时间内运行。