Computational Complexity 28 JAN, 2012

Fixed-parameter tractability of multicut in directed acyclic graphs

Fixed-parameter tractability of multicut in directed acyclic graphs

The MULTICUT problem, given a graph G, a set of terminal pairs T={(s_i,t_i) | 1 <= i <= r} and an integer p, asks whether one can find a cutset consisting of at most p non-terminal vertices that separates all the terminal pairs, i.e., after removing the cutset, t_i is not reachable from s_i for each 1 <= i <= r. The fixed-parameter tractability of MULTICUT in undirected graphs, parameterized by the size of the cutset only, has been recently proven by Marx and Razgon (STOC'11) and, independently, by Bousquet et al. (STOC'11), after resisting attacks as a long-standing open problem. In this paper we prove that MULTICUT is fixed-parameter tractable on directed acyclic graphs, when parameterized both by the size of the cutset and the number of terminal pairs. We complement this result by showing that this is implausible for parameterization by the size of the cutset only, as this version of the problem remains W[1]-hard.

💡 Research Summary

The paper studies the MULTICUT problem on directed acyclic graphs (DAGs). An instance consists of a graph G, a collection of terminal pairs T = {(s_i, t_i) | 1 ≤ i ≤ r}, and an integer p. The task is to delete at most p non‑terminal vertices so that for every pair (s_i, t_i) there is no directed s_i‑to‑t_i path after deletion. While the undirected version became FPT when parameterized solely by p (Marx‑Razgon, Bousquet et al., STOC 2011), the directed case remained open because cycles and asymmetry destroy many of the combinatorial tools used for the undirected setting.

The authors focus on DAGs, a natural restriction where every vertex can be placed in a topological order and all directed paths are monotone with respect to that order. They consider a two‑parameterization: the cutset size p and the number of terminal pairs r. Their main result is that DAG‑MULTICUT is fixed‑parameter tractable (FPT) with respect to the combined parameter (p + r). The algorithm proceeds in several conceptual steps:

  1. Topological preprocessing – The DAG is topologically sorted, which yields a linear ordering of vertices. This ordering guarantees that any s_i‑t_i path respects the order, limiting the ways different paths can intersect.

  2. Important vertices – For each terminal pair, the authors define a set of “important vertices”. They prove that if a cut of size ≤ p exists for a given pair, then there is a cut that uses only vertices from this important set, and the size of the set is bounded by O(p). This mirrors the “important separators” technique from the undirected case but is adapted to the directed, acyclic context.

  3. Divide‑and‑conquer on terminal pairs – The set T is split roughly in half, producing two sub‑instances. The algorithm is applied recursively to each sub‑instance. To keep track of vertices already chosen for the global cut, a “mask” data structure records the current budget consumption and prevents double counting.

  4. Dynamic programming within each recursion level – For a fixed sub‑instance, the algorithm enumerates all subsets of important vertices (2^p possibilities) and checks, via standard reachability tests, whether the chosen subset indeed separates all pairs in that sub‑instance. Because the number of important vertices per pair is O(p), the DP runs in O(2^p·poly(n)) time per level.

The recursion depth is O(log r), so the total running time is f(p, r)·poly(n) for some computable function f, establishing FPT with respect to (p, r). The paper provides a rigorous correctness proof, showing that the recursive division never loses a feasible solution and that the mask correctly propagates the remaining budget.

In contrast, the authors show that parameterizing solely by p is unlikely to yield an FPT algorithm. They give a parameter‑preserving reduction from the W