Finding small separators in linear time via treewidth reduction

We present a method for reducing the treewidth of a graph while preserving all of its minimal $s-t$ separators up to a certain fixed size $k$. This technique allows us to solve $s-t$ Cut and Multicut problems with various additional restrictions (e.g., the vertices being removed from the graph form an independent set or induce a connected graph) in linear time for every fixed number $k$ of removed vertices. Our results have applications for problems that are not directly defined by separators, but the known solution methods depend on some variant of separation. for example, we can solve similarly restricted generalizations of Bipartization (delete at most $k$ vertices from $G$ to make it bipartite) in almost linear time for every fixed number $k$ of removed vertices. These results answer a number of open questions in the area of parameterized complexity. Furthermore, our technique turns out to be relevant for $(H,C,K)$- and $(H,C,\le K)$-coloring problems as well, which are cardinality constrained variants of the classical $H$-coloring problem. We make progress in the classification of the parameterized complexity of these problems by identifying new cases that can be solved in almost linear time for every fixed cardinality bound.

💡 Research Summary

The paper introduces a powerful technique for reducing the treewidth of a graph while preserving all inclusion‑minimal s‑t separators of size at most k. Given a graph G, two terminals s and t, and an integer k, the authors construct in O(f(k)·(|V|+|E|)) time a new graph G* whose treewidth is bounded by a function g(k) and such that every minimal s‑t separator of size ≤ k in G appears as a union of the neighbourhoods of a carefully chosen laminar family of vertex sets in G*. The construction proceeds in two main steps. First, using an uncrossing argument they compute a linear‑time sequence X₁⊂X₂⊂…⊂X_q where each N(X_i) is a minimum s‑t cut of size ℓ (the size of a minimum cut). The differences X_{i+1}\X_i contain all vertices that belong to any minimum cut, and the total work is O(ℓ·(|V|+|E|)). Second, they apply a “torso” operation that contracts each N(X_i) into a single bag, thereby obtaining G*. By analyzing brambles they show that if G* had treewidth larger than g(k) then the original graph would contain a bramble of order > k, contradicting the existence of only small separators. Hence the treewidth of G* is indeed bounded by a function of k.

With a bounded‑treewidth graph in hand, any property expressible in Monadic Second‑Order logic (MSO) can be decided in linear time by Courcelle’s theorem. The authors exploit this to handle a wide variety of cut‑type problems that impose additional combinatorial constraints on the solution set. The key insight is that many natural restrictions—being an independent set, belonging to a hereditary graph class, inducing a connected subgraph, or respecting edge‑label constraints—are MSO‑definable. Consequently, after the treewidth reduction, each problem reduces to evaluating an MSO formula on G*, which yields an O(f(k)·n) algorithm (or O(f(k)·n·α(n,n)) when the underlying linear‑time Courcelle implementation incurs an inverse‑Ackermann factor).

The paper systematically applies this framework to obtain linear‑time FPT algorithms for:

• Minimum Stable s‑t Cut – the cut set must be an independent set. The result extends to any hereditary class G, giving a “G‑Cut” problem solvable in linear time for fixed k.
• Multicut and Multicut‑Uncut – simultaneously separate ℓ terminal pairs while optionally forbidding separation of another set of pairs. Both become linear‑time FPT when ℓ is fixed.
• Connected s‑t Cut – the removal set must induce a connected subgraph. Although connectivity is not hereditary, the authors devise a problem‑specific augmentation that still fits the treewidth‑reduction pipeline.
• Edge‑Included Vertex Cut – delete vertices whose incident edges are removed; this answers an open question from 2007.
• Stable Bipartization and Exact Stable Bipartization – delete an independent set (or exactly k independent vertices) to make the graph bipartite. The exact‑size variant is surprising because finding an independent set of exact size k is W