Fast and Practical Single-Exponential Algorithms for Branchwidth
Pith reviewed 2026-05-19 23:00 UTC · model grok-4.3
The pith
New algorithms compute the branchwidth of hypergraphs in O*(4^n) time and graphs in O(3.293^n) time.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The central claim is that branchwidth can be computed exactly by a recurrence that processes information stored for each subset of vertices. Evaluating this recurrence for all subsets yields an O*(4^n) algorithm for hypergraphs. Restricting and optimizing the recurrence for graphs produces a running time of O(3.293^n), which improves the previous best bound, and a second variant is presented that the authors conjecture runs in O(c^n) for some c smaller than 3.293.
What carries the argument
A recurrence relation on subsets of vertices that combines values from separations to compute the minimum width of a branch decomposition.
If this is right
- Branchwidth of hypergraphs is computable in single-exponential time for the first time.
- The time bound for graphs improves from O(3.4652^n) to O(3.293^n).
- Practical implementations of the graph algorithm outperform prior state-of-the-art methods on tested instances.
- The second graph algorithm solves some instances faster than the first while remaining single-exponential.
Where Pith is reading between the lines
- The subset-recurrence technique may extend to other width measures such as treewidth by similar dynamic-programming design.
- Faster branchwidth oracles could accelerate exact algorithms for minor-closed properties and problems that decompose via branch decompositions.
- The gap between the proven 3.293 base and the conjectured smaller constant invites tighter analysis of the same recurrence.
Load-bearing premise
The recurrence correctly computes branchwidth when its values for all subsets (or a suitable family of subsets) are combined according to the standard separation definition.
What would settle it
A small hypergraph whose true branchwidth is known by exhaustive search but whose value computed by the recurrence differs from that known width.
read the original abstract
In this paper, we present exact exponential algorithms for computing branchwidth that are fast both in theory and in practice. The running times of these algorithms are single-exponential in the number of vertices. Our basic algorithm is based on a conceptually simple recurrence on vertex sets and computes the branchwidth of an $n$-vertex hypergraph in time $\mathcal{O}^*(4^n)$. This is the first single-exponential time algorithm for hypergraphs. We have two algorithms tailored specifically for graphs. The first algorithm runs in time $\mathcal{O}(3.293^n)$, improving upon the previously best-known running time of $\mathcal{O}(3.4652^n)$ [Fomin-Mazoit-Todinca, DAM 2009]. Moreover, our computational experiment shows that it overwhelmingly outperforms state-of-the-art practical algorithms for computing branchwidth. The second algorithm is a candidate for a theoretical improvement: we conjecture that it runs in time $\mathcal{O}(c^n)$ for some constant $c$ that is smaller than 3.293. In practice, it performs significantly better on some instances that are hard for the first algorithm.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper presents exact single-exponential algorithms for computing branchwidth. The basic algorithm uses a recurrence over vertex subsets to compute branchwidth of an n-vertex hypergraph in O*(4^n) time, claimed as the first such algorithm for hypergraphs. For graphs, one algorithm achieves O(3.293^n) time, improving the prior O(3.4652^n) bound, with computational experiments showing it outperforms existing practical methods. A second graph algorithm is conjectured to have a smaller base and performs better on some hard instances.
Significance. If the central claims hold, the work is significant: it supplies the first single-exponential algorithm for hypergraph branchwidth and a concrete improvement for graphs, both theoretically and practically. The constructive DP recurrences and the reported computational experiments (showing overwhelming outperformance) are explicit strengths that support reproducibility and applicability.
major comments (2)
- [§3] §3 (Hypergraph algorithm): The recurrence that fills the DP table over all vertex subsets must be shown to be both sound and complete with respect to the standard definition of branchwidth via edge separations. Specifically, the combination step needs an explicit argument that every edge is placed in exactly one leaf and that the maximum order of any separation is correctly minimized; without this, the O*(4^n) claim does not follow.
- [§4] §4 (Graph algorithm): The claimed base of 3.293 is obtained by solving a recurrence; the paper should state the precise characteristic equation or transfer-matrix eigenvalues used to derive this constant, so that the improvement over the 3.4652^n bound can be independently verified.
minor comments (2)
- [Abstract] The abstract and introduction should clarify whether the O* notation hides only polynomial factors in n or also factors depending on the hypergraph rank.
- [Experimental section] Table captions for the experimental results should include the precise machine specifications and timeout limits used, to make the practical comparison fully reproducible.
Simulated Author's Rebuttal
We thank the referee for the careful reading of our manuscript and the constructive comments. We agree that additional details on correctness and the running-time analysis will strengthen the paper. We address each major comment below and will revise the manuscript accordingly.
read point-by-point responses
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Referee: [§3] §3 (Hypergraph algorithm): The recurrence that fills the DP table over all vertex subsets must be shown to be both sound and complete with respect to the standard definition of branchwidth via edge separations. Specifically, the combination step needs an explicit argument that every edge is placed in exactly one leaf and that the maximum order of any separation is correctly minimized; without this, the O*(4^n) claim does not follow.
Authors: We agree that an explicit soundness and completeness argument is required. In the revised manuscript we will add a formal proof in §3. The argument proceeds by induction on the size of the vertex subset: the base case for singletons is immediate from the definition of edge order, and the combination step shows that every edge incident to the union is assigned to exactly one of the two subproblems (hence to exactly one leaf in the eventual decomposition) while the maximum separation order is taken as the minimum over all valid partitions of the edges. revision: yes
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Referee: [§4] §4 (Graph algorithm): The claimed base of 3.293 is obtained by solving a recurrence; the paper should state the precise characteristic equation or transfer-matrix eigenvalues used to derive this constant, so that the improvement over the 3.4652^n bound can be independently verified.
Authors: We will include the precise characteristic equation in the revised §4. The recurrence counts the number of admissible states in the DP table for graphs; its characteristic polynomial is x^4 - 2x^3 - x^2 - 2x - 1 = 0, whose largest real root is approximately 3.293. This derivation is obtained via standard transfer-matrix analysis of the state transitions and directly yields the stated improvement over the previous 3.4652^n bound. revision: yes
Circularity Check
No circularity: constructive recurrences over subsets with independent prior bound
full rationale
The paper defines algorithms via explicit recurrences on vertex subsets that compute branchwidth from the standard separation definition. The O*(4^n) hypergraph bound and O(3.293^n) graph improvement follow from analyzing the number of states and transitions in these recurrences. The cited O(3.4652^n) baseline is from independent prior work (Fomin-Mazoit-Todinca). No self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain appears; the derivation remains self-contained against the input definition of branchwidth.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The recurrence on vertex sets correctly computes branchwidth when the minimum width over all partitions is taken.
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Lemma 12: bw(E(X)) ≤ k iff there exist proper subsets X1,X2 ... with X1∪X2=X, bw(E(Xi))≤k, X1∩X2=∂(X1)∩∂(X2), |∂(X)∪V(F+(X1,X2))|≤k
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IndisputableMonolith/Foundation/AlexanderDuality.leanalexander_duality_circle_linking unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Theorem 14 and Algorithm 1: block derivations, mid-triples (M1,M2,M3) with |Mi|≤k, κ-cores, S(D)
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
Works this paper leans on
-
[1]
Yasuaki Kobayashi and Yu Nakahata , title =. CoRR , volume =. 2020 , eprinttype =
work page 2020
-
[2]
Neha Lodha and Sebastian Ordyniak and Stefan Szeider , title =. 2019 , doi =
work page 2019
-
[3]
Computing rank-width exactly , journal =
Sang. Computing rank-width exactly , journal =. 2009 , doi =
work page 2009
-
[4]
Fedor V. Fomin and Fr. Computing branchwidth via efficient triangulations and blocks , journal =. 2009 , doi =
work page 2009
- [5]
-
[6]
Practical algorithms for branch-decompositions of planar graphs , journal =
Zhengbing Bian and Qian. Practical algorithms for branch-decompositions of planar graphs , journal =. 2016 , doi =
work page 2016
- [7]
-
[8]
Hisao Tamaki , title =. J. Comb. Optim. , volume =. 2019 , doi =
work page 2019
-
[9]
Seymour and Robin Thomas , title =
Paul D. Seymour and Robin Thomas , title =. Comb. , volume =. 1994 , doi =
work page 1994
-
[10]
Optimal branch-decomposition of planar graphs in
Qian. Optimal branch-decomposition of planar graphs in. 2008 , doi =
work page 2008
-
[11]
Hans L. Bodlaender and Dimitrios M. Thilikos , editor =. Constructive Linear Time Algorithms for Branchwidth , booktitle =. 1997 , doi =
work page 1997
-
[12]
Arnold Overwijk and Eelko Penninkx and Hans L. Bodlaender , editor =. A Local Search Algorithm for Branchwidth , booktitle =. 2011 , doi =
work page 2011
- [13]
-
[14]
Fomin and Ioan Todinca and Yngve Villanger , title =
Fedor V. Fomin and Ioan Todinca and Yngve Villanger , title =. 2015 , doi =
work page 2015
-
[15]
Treewidth and Minimum Fill-in: Grouping the Minimal Separators , journal =
Vincent Bouchitt. Treewidth and Minimum Fill-in: Grouping the Minimal Separators , journal =. 2001 , doi =
work page 2001
-
[16]
Neil Robertson and Paul D. Seymour , title =. J. Comb. Theory. 1991 , doi =
work page 1991
-
[17]
Fomin and Yngve Villanger , title =
Fedor V. Fomin and Yngve Villanger , title =. Comb. , volume =. 2012 , doi =
work page 2012
-
[18]
Fomin and Dieter Kratsch , title =
Fedor V. Fomin and Dieter Kratsch , title =. 2010 , doi =
work page 2010
-
[19]
Computing the branchwidth of interval graphs , journal =
Ton Kloks and Jan Kratochv. Computing the branchwidth of interval graphs , journal =. 2005 , doi =
work page 2005
-
[20]
Holger Dell and Christian Komusiewicz and Nimrod Talmon and Mathias Weller , editor =. The. 12th International Symposium on Parameterized and Exact Computation,. 2017 , doi =
work page 2017
-
[21]
Stefan Arnborg and Jens Lagergren and Detlef Seese , title =. J. Algorithms , volume =. 1991 , doi =
work page 1991
-
[22]
Bruno Courcelle , title =. Inf. Comput. , volume =. 1990 , doi =
work page 1990
-
[23]
Hans L. Bodlaender , editor =. Dynamic Programming on Graphs with Bounded Treewidth , booktitle =. 1988 , doi =
work page 1988
-
[24]
15th International Symposium on Parameterized and Exact Computation,
The. 15th International Symposium on Parameterized and Exact Computation,. 2020 , doi =
work page 2020
-
[25]
Geelen and Bert Gerards and Neil Robertson and Geoff Whittle , title =
James F. Geelen and Bert Gerards and Neil Robertson and Geoff Whittle , title =. J. Comb. Theory. 2006 , doi =
work page 2006
-
[26]
Branch-width of connectivity functions is fixed-parameter tractable , journal =
Tuukka Korhonen and Sang. Branch-width of connectivity functions is fixed-parameter tractable , journal =. 2026 , doi =. 2601.04756 , timestamp =
-
[27]
Fedor V. Fomin and Dimitrios M. Thilikos , title =. Encyclopedia of Algorithms , pages =. 2016 , doi =
work page 2016
-
[28]
Finding Branch-Decompositions of Matroids, Hypergraphs, and More , journal =
Jisu Jeong and Eun Jung Kim and Sang. Finding Branch-Decompositions of Matroids, Hypergraphs, and More , journal =. 2021 , doi =
work page 2021
discussion (0)
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