Given an undirected graph $G$, a collection ${(s_1,t_1), dots, (s_{k},t_{k})}$ of pairs of vertices, and an integer ${{p}}$, the Edge Multicut problem asks if there is a set $S$ of at most ${{p}}$ edges such that the removal of $S$ disconnects every $s_i$ from the corresponding $t_i$. Vertex Multicut is the analogous problem where $S$ is a set of at most ${{p}}$ vertices. Our main result is that both problems can be solved in time $2^{O({{p}}^3)}cdot n^{O(1)}$, i.e., fixed-parameter tractable parameterized by the size ${{p}}$ of the cutset in the solution. By contrast, it is unlikely that an algorithm with running time of the form $f({{p}})cdot n^{O(1)}$ exists for the directed version of the problem, as we show it to be W[1]-hard parameterized by the size of the cutset.
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机译：给定一个无向图$ G $，一个成对的顶点集合$ {（s_1，t_1）， dots，（s_ {k}，t_ {k}）} $和一个整数$ {{p}}在“ $”中，Edge Multicut问题询问是否存在一组$ S $最多$ {{p}} $条边，以使$ S $的删除会使每个$ s_i $与对应的$ t_i $断开连接。顶点多重切割是一个类似的问题，其中$ S $是一组最多$ {{p}} $个顶点。我们的主要结果是，这两个问题都可以在$ 2 ^ {O（{{p}} ^ 3）} cdot n ^ {O（1）} $的时间得到解决，即由大小$ {解决方案中割据的{p}} $。相比之下，针对该问题的定向版本，不太可能存在运行时间为$ f（{{{p}}） cdot n ^ {O（1）} $形式的算法，因为我们证明它是W [1] -hard由割集的大小参数化。
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