A Short Note on Relevant Cuts

The set of relevant cuts in a graph is the union of all minimum weight bases of the cut space. A cut is relevant if and only if it is the a minimum weight cut between two distinct vertices. Moreover, we give a characterization in terms of Picard-Queyranne Directed Acyclic Graphs that can be used to accelerate the enumeration of the relevant cuts. Finally, we perform an experimental evaluation by comparing with state-of-the-art algorithms.

💡 Research Summary

The paper introduces the notion of “relevant cuts” in an undirected weighted graph G = (V,E,w) and establishes a precise equivalence between these cuts and minimum‑weight s‑t cuts for some pair of distinct vertices. The authors begin by recalling the cut space D(G), a vector space over GF(2) of dimension |V| − 1, whose elements are edge sets that separate the vertex set into two non‑empty parts. A cut basis is any set of |V| − 1 cuts that spans this space; a minimum cut basis is a basis whose total weight is minimal. Classical results guarantee that a Gomory‑Hu tree T* exists for any weighted graph and that the cuts induced by its edges form a minimum cut basis.

The central contribution is a matroid‑theoretic characterization: a cut D is “relevant” if it belongs to at least one minimum cut basis. Lemma 3.2 shows that a cut is relevant precisely when it cannot be expressed as the symmetric‑difference (⊕) of strictly shorter cuts. Lemma 3.3 further restricts relevant cuts to bonds (cuts that contain no proper sub‑cut). The main theorem (Theorem 3.4) proves the bidirectional statement: a cut is relevant if and only if it is a minimum‑weight s‑t cut for some vertex pair (s,t). The forward direction uses a greedy construction of a minimum cut basis ordered by non‑decreasing weight; the first cut that separates s and t must be inserted, guaranteeing its relevance. The reverse direction leverages the representation of any cut as a linear combination of the Gomory‑Hu tree cuts (Equation 3.1). If a cut is not a minimum s‑t cut for any pair, each contributing tree edge has strictly smaller weight, so the cut can be expressed as a sum of shorter cuts, contradicting relevance.

Algorithmically, the paper exploits the Picard‑Queyranne directed acyclic graphs (PQ‑DAGs) Q_{s,t}, which compactly encode all minimum s‑t cuts via their closed vertex sets. By pre‑computing a PQ‑DAG for each edge of the Gomory‑Hu tree, any Q_{s,t} can be assembled in linear time along the unique tree path between s and t, as shown in prior work