Computing multiway cut within the given excess over the largest minimum isolating cut

Let $(G,T)$ be an instance of the (vertex) multiway cut problem where $G$ is a graph and $T$ is a set of terminals. For $t \in T$, a set of nonterminal vertices separating $t$ from $T \setminus {T}$ is called an \emph{isolating cut} of $t$. The largest among all the smallest isolating cuts is a natural lower bound for a multiway cut of $(G,T)$. Denote this lower bound by $m$ and let $k$ be an integer. In this paper we propose an $O(kn^{k+3})$ algorithm that computes a multiway cut of $(G,T)$ of size at most $m+k$ or reports that there is no such multiway cut. The core of the proposed algorithm is the following combinatorial result. Let $G$ be a graph and let $X,Y$ be two disjoint subsets of vertices of $G$. Let $m$ be the smallest size of a vertex $X-Y$ separator. Then, for the given integer $k$, the number of \emph{important} $X-Y$ separators \cite{MarxTCS} of size at most $m+k$ is at most $\sum_{i=0}^k{n \choose i}$.

💡 Research Summary

The paper addresses the classic Vertex Multiway Cut problem, where a graph G and a set of terminals T are given, and the goal is to delete a minimum number of non‑terminal vertices so that every pair of distinct terminals becomes disconnected. While the problem is NP‑hard, a large body of work has focused on fixed‑parameter tractable (FPT) algorithms parameterized by the size k of the cut. The most notable approach, due to Marx (2006), introduces the notion of “important separators” and yields an O(4^k·n^{O(1)}) algorithm.

Instead of parameterizing directly by the optimal cut size, the authors propose a more refined parameterization based on a natural lower bound. For each terminal t ∈ T, consider an isolating cut: a set of non‑terminal vertices that separates t from the rest of the terminals T \ {t}. Let m_t be the size of a minimum isolating cut for t, and define m = max_{t∈T} m_t. Clearly, any feasible multiway cut must have size at least m, because the most “difficult” terminal already requires m vertices to be isolated. The central computational question becomes: Given an integer k, can we find a multiway cut of size at most m + k? If the answer is negative, then no cut within that excess exists.

The technical heart of the paper is a new combinatorial bound on the number of important separators of bounded excess. Let X and Y be two disjoint vertex subsets of G, and let m be the size of a minimum X‑Y separator. An important X‑Y separator is a minimal separator that is inclusion‑wise minimal among all separators of the same size. The authors prove:

Theorem. For any integer k ≥ 0, the number of important X‑Y separators whose size does not exceed m + k is at most
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