# An FPT Algorithm for Planar Multicuts with Sources and Sinks on the Outer Face

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## Abstract

Given a list of *k* source–sink pairs in an edge-weighted graph *G*, the *minimum multicut problem* consists in selecting a set of edges of minimum total weight in *G*, such that removing these edges leaves no path from each source to its corresponding sink. To the best of our knowledge, no non-trivial FPT result for special cases of this problem, which is **APX**-hard in general graphs for any fixed \(k \ge 3\), is known with respect to *k* only. When the graph *G* is planar, this problem is known to be polynomial-time solvable if \(k = O(1)\), but cannot be FPT with respect to *k* under the *Exponential Time Hypothesis*. In this paper, we show that, if *G* is planar and in addition all sources and sinks lie on the outer face, then this problem does admit an FPT algorithm when parameterized by *k* (although it remains **APX**-hard when *k* is part of the input, even in stars). To do this, we provide a new characterization of optimal solutions in this case, and then use it to design a “divide-and-conquer” approach: namely, some edges that are part of any such solution actually define an optimal solution for a polynomial-time solvable multiterminal variant of the problem on some of the sources and sinks (which can be identified thanks to a reduced enumeration phase). Removing these edges from the graph cuts it into several smaller instances, which can then be solved recursively.

## Keywords

Multicuts Planar graphs FPT algorithms## References

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