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arxiv: 1907.02353 · v2 · pith:YLRHMI7Cnew · submitted 2019-07-04 · 💻 cs.CC · cs.DS

Fixed-parameter tractability of counting small minimum (S,T)-cuts

Pierre Berg\'e , Benjamin Mouscadet , Arpad Rimmel , Joanna Tomasik This is my paper

Pith reviewed 2026-05-25 08:32 UTC · model grok-4.3

classification 💻 cs.CC cs.DS
keywords fixed-parameter tractabilitycounting minimum cuts(S,T)-cutsdrainage transformationMenger's theoremparameterized complexityFPT algorithmsampling cuts
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The pith

An FPT algorithm counts all minimum edge (S,T)-cuts of size p in time 2^{O(p^2)} n^{O(1)}.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that counting minimum edge (S,T)-cuts in an undirected graph is fixed-parameter tractable when parameterized by cut size p. It first applies a drainage transformation to produce at most n successive minimum cuts Z_i. Every minimum cut must share at least one edge with one of these Z_i. This covering property plus Menger's theorem on disjoint paths yields an algorithm that enumerates and counts all such cuts. A sympathetic reader would care because the problem is #P-complete in the unrestricted case, so the result isolates tractability for small cuts.

Core claim

We design a fixed-parameter tractable algorithm which counts minimum edge (S,T)-cuts parameterized by their size p with running time 2^{O(p^2)}n^{O(1)}. The algorithm operates on a transformed graph instance. This transformation, called drainage, reveals a collection of at most n successive minimum (S,T)-cuts Z_i. We prove that any minimum (S,T)-cut X contains edges of at least one cut Z_i. This observation, together with Menger's theorem, allows us to build the algorithm counting all minimum (S,T)-cuts. It can be modified to obtain an FPT sampling of minimum edge (S,T)-cuts.

What carries the argument

The drainage transformation, which produces at most n successive minimum (S,T)-cuts Z_i such that every minimum (S,T)-cut shares an edge with at least one Z_i.

If this is right

  • The same procedure yields an FPT sampler for minimum edge (S,T)-cuts.
  • The running time bound 2^{O(p^2)} n^{O(1)} holds for any undirected graph and any disjoint vertex sets S and T.
  • The approach separates the parameterized complexity of the counting problem from its general #P-completeness.
  • Menger's theorem supplies the disjoint-path structure needed to count the cuts that intersect each Z_i.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The drainage covering property could be tested on directed graphs or on vertex-cut variants to see whether the FPT result carries over.
  • For concrete small values of p the exponential dependence on p squared may become practical once the polynomial factor is implemented.
  • The technique links counting to the enumeration of successive min-cuts in a flow network, suggesting possible reuse in other cut-enumeration tasks.

Load-bearing premise

The drainage transformation produces a collection of at most n successive minimum (S,T)-cuts Z_i such that every minimum (S,T)-cut contains at least one edge from some Z_i.

What would settle it

A concrete graph G with disjoint S and T together with one minimum (S,T)-cut that shares no edge with any of the Z_i produced by drainage on that instance.

Figures

Figures reproduced from arXiv: 1907.02353 by Arpad Rimmel, Benjamin Mouscadet, Joanna Tomasik, Pierre Berg\'e.

Figure 1
Figure 1. Figure 1: Illustration of Def. 2 for closest (S, T )-cuts: X2 is closest whereas X1 is not. As a minimum closest (S, T )-cut is also a minimum important (T, S)-cut on undirected graphs, there is a unique minimum closest (S, T )-cut according to Lemma 1. Since the graph is uncapacitated, computing the minimum closest (S, T )-cut is made in time O(mp), using p iterations of Ford-Fulkerson’s algo￾rithm [17]. 3 Framewor… view at source ↗
Figure 2
Figure 2. Figure 2: The drainage (cuts Zi, sets Ri and Si) for an instance containing graph G, sources S = {s1, s2, s3} and targets T = {t1, t2}. Here, R1 = S1 (in general, R1 ⊇ S1). Z3, respectively. Similarly, blue, red, green, and yellow vertices represent sets S1 = S, S2, S3, and S4. Reachable sets R1, R2, R3, and R4 are also appropriately colored. As the size of the minimum cut between S4 (yellow vertices) and T in graph… view at source ↗
Figure 3
Figure 3. Figure 3: Menger’s paths in graph G with sources S = {s1, s2, s3}, targets T = {t1, t2}. in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: An example of dam B2 and its dry instance D (I, B2) = (G ∗ (B2), S∗ (B2), T ∗ (B2)) Definition 7 (Dry instance). The dry instance induced by a dam Bi is an in￾stance D (I, Bi) = (G∗ (Bi), S∗ (Bi), T ∗ (Bi)) with graph G∗ (Bi) = (V ∗ (Bi), E∗ (Bi)). In particular, – set S ∗ (Bi) keeps vertices reachable from S “just before” dam Bi. Formally, it contains the tails of arcs in Bi: S ∗ (Bi) = {u : (u, v) ∈ Bi},… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the contradiction we arose for Case 2 in the proof of Lemma 5. reason, path Q must contain a vertex v ∈ A∗ [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of the segment of path Pe between S and be, traversing dams Bh and Bi. – Bi(X) is a minimum cut of instance D [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Illustration of Theorem 5: for any minimum cut with the front dam Bi, the tails of arcs in X\Bi belong to the yellow zone. is necessarily included into the reachable set of Zh(X)+1 = Bh(X)+1 ∪ Bh(X)+1 in graph G\R(Zh(X) , S). This is a contradiction to the construction of the drainage, as Zh(X)+1 is the unique minimum closest (Sh(X)+1, T )-cut in graph G\R(Zh(X) , S). Consequently, Bh(X)+1 is the minimum d… view at source ↗
Figure 8
Figure 8. Figure 8: Recursive calls used to compute values Cℓ (I). Proof of Theorem 6. The depth of instance I in the recursive tree is zero, we say ∆(I) = 0. For any closest dam Bh of Z(I), the depth of the dry instance of Bh is one: ∆ [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: An example of drainage when cuts are composed of vertices. With this definition, the drainage fulfils the properties given in Section 3 for the edge version, the most important of them being that any minimum vertex (S, T )-cut X admits a front dam Bi(X) . Moreover, there is a vertex version of the Menger’s theorem, stating that the size of the minimum (S, T )-cut is equal to the maximum number of vertex-di… view at source ↗
Figure 10
Figure 10. Figure 10: Illustration of the impossibility to prove Theorem 3 for vertex cuts However, Theorem 3 does not hold anymore: the set X\Bi(X) of a minimum (S, T )-cut X is not necessarily included in the dry instance of Bh(X) . We give an example in [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
read the original abstract

The parameterized complexity of counting minimum cuts stands as a natural question because Ball and Provan showed its #P-completeness. For any undirected graph $G=(V,E)$ and two disjoint sets of its vertices $S,T$, we design a fixed-parameter tractable algorithm which counts minimum edge $(S,T)$-cuts parameterized by their size $p$. Our algorithm operates on a transformed graph instance. This transformation, called drainage, reveals a collection of at most $n=\left| V \right|$ successive minimum $(S,T)$-cuts $Z_i$. We prove that any minimum $(S,T)$-cut $X$ contains edges of at least one cut $Z_i$. This observation, together with Menger's theorem, allows us to build the algorithm counting all minimum $(S,T)$-cuts with running time $2^{O(p^2)}n^{O(1)}$. Initially dedicated to counting minimum cuts, it can be modified to obtain an FPT sampling of minimum edge $(S,T)$-cuts.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to design an FPT algorithm counting minimum edge (S,T)-cuts of size p with running time 2^{O(p^2)}n^{O(1)}. It applies a 'drainage' transformation yielding at most n successive minimum (S,T)-cuts Z_i such that every minimum cut X shares an edge with at least one Z_i; Menger's theorem then reduces the count to an FPT enumeration procedure. The same framework is adaptable to sampling.

Significance. If the drainage covering property is correct, the result is significant: it supplies the first FPT algorithm for a #P-complete counting problem (Ball-Provan) when parameterized by cut size p. The drainage transformation and its claimed covering property constitute the core technical contribution; the reduction itself introduces no fitted parameters.

major comments (1)
  1. [Abstract, paragraph beginning 'This transformation, called drainage...'] Abstract, paragraph beginning 'This transformation, called drainage...': the covering property (every minimum (S,T)-cut X contains edges of at least one Z_i) is stated as proved and is load-bearing for the reduction to the claimed 2^{O(p^2)}n^{O(1)} enumeration; however, the abstract supplies neither a proof sketch, section reference, nor example verification of the property. If the property fails on any graph family, the algorithm undercounts and the running-time claim does not hold.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed reading and for highlighting the presentation of the covering property. We respond to the single major comment below and are prepared to revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract, paragraph beginning 'This transformation, called drainage...'] Abstract, paragraph beginning 'This transformation, called drainage...': the covering property (every minimum (S,T)-cut X contains edges of at least one Z_i) is stated as proved and is load-bearing for the reduction to the claimed 2^{O(p^2)}n^{O(1)} enumeration; however, the abstract supplies neither a proof sketch, section reference, nor example verification of the property. If the property fails on any graph family, the algorithm undercounts and the running-time claim does not hold.

    Authors: The covering property is proved in full in Section 3 (Theorem 3.2 together with Lemmas 3.3–3.5). The abstract is a concise summary and does not normally contain proof sketches; however, we agree that the absence of a section reference reduces clarity. We will revise the abstract to read: “We prove that any minimum (S,T)-cut X contains edges of at least one cut Z_i (see Section 3).” An illustrative example appears in Figure 1. The proof itself relies on the drainage construction and an inductive argument on the number of successive min-cuts; we believe it is correct, but would welcome any counter-example the referee may have identified. revision: yes

Circularity Check

0 steps flagged

No circularity; key covering property is internally proved

full rationale

The paper states it proves the drainage covering property ('We prove that any minimum (S,T)-cut X contains edges of at least one cut Z_i') and then invokes Menger's theorem to obtain the FPT counting algorithm. No self-definitional steps, no parameters fitted then renamed as predictions, and no load-bearing self-citations appear in the provided derivation chain. The result is constructed from an explicit graph transformation plus an external theorem, rendering the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on Menger's theorem (standard graph theory) and the correctness of the newly introduced drainage transformation; no free parameters or invented entities are introduced.

axioms (1)
  • standard math Menger's theorem on the maximum number of edge-disjoint paths
    Invoked to build the counting procedure from the successive cuts.

pith-pipeline@v0.9.0 · 5714 in / 1110 out tokens · 18566 ms · 2026-05-25T08:32:45.089992+00:00 · methodology

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