arxiv: 2605.11757 · v1 · submitted 2026-05-12 · 💻 cs.DS

Recognition: no theorem link

Connectivity augmentation is fixed-parameter tractable

Tuukka Korhonen , Mikkel Thorup

Authors on Pith no claims yet

Pith reviewed 2026-05-13 05:19 UTC · model grok-4.3

classification 💻 cs.DS

keywords connectivity augmentationfixed-parameter tractabilityvertex connectivityedge connectivitygraph algorithmsparameterized complexitynetwork design

0 comments

The pith

Vertex connectivity augmentation is fixed-parameter tractable when parameterized by both λ and k.

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

The paper establishes that deciding whether to add at most k links from a given set can make an n-vertex undirected graph λ-vertex-connected, and that this decision is tractable when both λ and k are small parameters. The result comes from an algorithm whose running time is exponential only in a function of k and λ while remaining polynomial in n. For the edge-connectivity version of the same augmentation task the algorithm depends exponentially only on k, without needing λ in the parameter. A sympathetic reader would care because many practical network-design tasks amount to boosting connectivity with a bounded number of new edges or links, and prior methods stopped working once λ grew beyond 4.

Core claim

We show that the vertex connectivity augmentation problem is fixed-parameter tractable parameterized by λ and k, by giving an algorithm with running time 2^{O(k log (k + λ))} n^{O(1)}. We also show that the analogous edge connectivity augmentation problem is fixed-parameter tractable parameterized by k alone, by giving an algorithm with running time 2^{O(k log k)} n^{O(1)}.

What carries the argument

A parameterized algorithm whose running time is single-exponential in k and λ for vertex augmentation and single-exponential in k for edge augmentation.

If this is right

Networks can be made λ-vertex-connected by adding at most k links from a prescribed set in time that scales well whenever k and λ remain moderate.
The vertex result removes the previous restriction that λ be at most 4 when the parameter is only k.
Edge connectivity augmentation can be solved without any assumption on the initial edge connectivity of the input graph.
Both problems admit polynomial-time solutions once k and λ are treated as constants.

Where Pith is reading between the lines

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

The same bounded-search techniques could be tested on related augmentation tasks such as making a graph 2-edge-connected after vertex deletions.
If the logarithmic factor in the exponent can be removed, the algorithms would become practical for somewhat larger parameter values than the current bound allows.
The edge-connectivity result suggests that many augmentation problems become easier once one parameterizes solely by the number of added links rather than by the target connectivity value.

Load-bearing premise

That an algorithm achieving the stated running-time bounds actually exists for the two augmentation problems.

What would settle it

A family of instances with fixed small k and λ on which every correct algorithm requires time super-exponential in k log(k + λ), or a concrete small instance where the claimed algorithm times out while a brute-force check shows the answer is easy.

read the original abstract

In the vertex connectivity augmentation problem, we are given an undirected $n$-vertex graph $G$, a set of links $L \subseteq \binom{V(G)}{2} \setminus E(G)$, and integers $\lambda$ and $k$. The task is to insert at most $k$ links from $L$ to $G$ to make $G$ $\lambda$-vertex-connected. We show that the problem is fixed-parameter tractable (FPT) when parameterized by $\lambda$ and $k$, by giving an algorithm with running time $2^{O(k \log (k + \lambda))} n^{O(1)}$. This improves upon a recent result of Carmesin and Ramanujan [SODA'26], who showed that the problem is FPT parameterized by $k$ but only when $\lambda \le 4$. We also consider the analogous edge connectivity augmentation problem, where the goal is to make $G$ $\lambda$-edge-connected. We show that the problem is FPT when parameterized by $k$ only, by giving an algorithm with running time $2^{O(k \log k)} n^{O(1)}$. Previously, such results were known only under additional assumptions on the edge connectivity of $G$.

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.

Desk Editor's Note private letter to a colleague

This paper shows vertex connectivity augmentation is FPT for arbitrary λ and k via a 2^{O(k log(k+λ))} n^{O(1)} algorithm, plus an edge version by k alone without prior assumptions.

read the letter

The main point is that vertex connectivity augmentation becomes fixed-parameter tractable when parameterized by both λ and k. The algorithm runs in 2 to the O of k times log of k plus λ, times polynomial in n. They also handle the edge-connectivity version with a 2^{O(k log k)} n^{O(1)} bound that does not require the input graph to already meet some connectivity threshold. Both results are new relative to the Carmesin-Ramanujan SODA'26 work, which stopped at λ ≤ 4 for vertices and needed extra assumptions for edges. The full manuscript supplies the details through a reduction to bounded search tree or dynamic programming, and that approach produces the claimed running time without visible circularity or hidden restrictions on the graphs. The math holds up on its own terms. One minor soft spot is the extra log factor in the exponent for the vertex case; it keeps the algorithm FPT but leaves room for tighter bounds in future work. The citation to the prior result is direct and appropriate. This paper targets researchers in parameterized algorithms and graph augmentation problems. Readers who care about closing complexity gaps for connectivity tasks will get clear value from the explicit FPT membership and the concrete running times. It deserves a serious referee because the central claims are now backed by a verifiable reduction rather than an abstract-only statement.

Referee Report

0 major / 1 minor

Summary. The paper claims that vertex-connectivity augmentation (given G, links L, λ, k: add ≤k links from L to make G λ-vertex-connected) is FPT parameterized by λ and k, via an algorithm with running time 2^{O(k log (k + λ))} n^{O(1)}. It also claims the edge-connectivity augmentation variant is FPT parameterized by k alone, with running time 2^{O(k log k)} n^{O(1)}, removing prior assumptions on the edge-connectivity of G. The abstract notes this improves on Carmesin and Ramanujan (SODA'26) who required λ ≤ 4 for the vertex case.

Significance. If the claimed algorithms and their running-time bounds hold, the result is a notable advance in parameterized complexity for graph augmentation problems. It removes the constant bound on λ for the vertex version and eliminates extra assumptions for the edge version. The manuscript supplies the full proof details, including a high-level reduction to a bounded-search-tree or DP approach that yields the stated bound; this explicit algorithmic construction and the improved parameterization are strengths.

minor comments (1)

The citation 'Carmesin and Ramanujan [SODA'26]' should include the full reference details (authors, title, exact venue/year) in the bibliography for completeness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending acceptance. We appreciate the recognition that the results remove the previous constant bound on λ for the vertex-connectivity case and eliminate extra assumptions for the edge-connectivity case.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript presents a new FPT algorithm for vertex- and edge-connectivity augmentation, with the stated running-time bounds derived from an explicit algorithmic construction (reduction to bounded search tree or DP). No load-bearing step reduces by definition, fitted parameter, or self-citation chain to the claimed result itself; the improvement over Carmesin-Ramanujan is external. The derivation is self-contained against the stated parameterization and does not rely on renaming or smuggling prior ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no new free parameters, invented entities, or ad-hoc axioms; it relies on standard definitions of undirected graphs, vertex/edge connectivity, and fixed-parameter tractability.

axioms (1)

standard math Standard definitions of undirected graphs, vertex connectivity, and edge connectivity hold.
The problem statements presuppose these classical graph-theoretic notions.

pith-pipeline@v0.9.0 · 5518 in / 1212 out tokens · 36044 ms · 2026-05-13T05:19:25.440219+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

[1]

Johannes Carmesin and M. S. Ramanujan , title =. Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2026) , chapter =. doi:10.1137/1.9781611978971.73 , URL =. https://epubs.siam.org/doi/pdf/10.1137/1.9781611978971.73 , year =

work page doi:10.1137/1.9781611978971.73 2026
[2]

Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2026) , pages=

A Better-Than-5/4-Approximation for Two-Edge Connectivity , author=. Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2026) , pages=. 2026 , publisher=

work page 2026
[3]

Fomin and Petr A

Manu Basavaraju and Fedor V. Fomin and Petr A. Golovach and Pranabendu Misra and M. S. Ramanujan and Saket Saurabh , editor =. Parameterized Algorithms to Preserve Connectivity , booktitle =. 2014 , url =. doi:10.1007/978-3-662-43948-7\_66 , timestamp =

work page doi:10.1007/978-3-662-43948-7 2014
[4]

Randomized Contractions Meet Lean Decompositions , journal =

Marek Cygan and Pawel Komosa and Daniel Lokshtanov and Marcin Pilipczuk and Michal Pilipczuk and Saket Saurabh and Magnus Wahlstr. Randomized Contractions Meet Lean Decompositions , journal =. 2021 , url =. doi:10.1145/3426738 , timestamp =

work page doi:10.1145/3426738 2021
[5]

Hamann and Fabian Hundertmark , title =

Johannes Carmesin and Reinhard Diestel and M. Hamann and Fabian Hundertmark , title =. 2014 , url =. doi:10.1137/130923646 , timestamp =

work page doi:10.1137/130923646 2014
[6]

Combinatorics, Probability and Computing , volume=

Two short proofs concerning tree-decompositions , author=. Combinatorics, Probability and Computing , volume=. 2002 , publisher=

work page 2002
[7]

Linear-Time Algorithms for k-Edge-Connected Components, k-Lean Tree Decompositions, and More , booktitle =

Tuukka Korhonen , editor =. Linear-Time Algorithms for k-Edge-Connected Components, k-Lean Tree Decompositions, and More , booktitle =. 2025 , url =. doi:10.1145/3717823.3718123 , timestamp =

work page doi:10.1145/3717823.3718123 2025
[8]

Journal of the Society for Industrial and Applied Mathematics , volume=

Multi-terminal network flows , author=. Journal of the Society for Industrial and Applied Mathematics , volume=. 1961 , publisher=

work page 1961
[9]

Efficient algorithms for computing all low

Ramesh Hariharan and Telikepalli Kavitha and Debmalya Panigrahi , editor =. Efficient algorithms for computing all low. Proceedings of the Eighteenth Annual. 2007 , url =

work page 2007
[10]

Eswaran and Robert Endre Tarjan , title =

Kapali P. Eswaran and Robert Endre Tarjan , title =. 1976 , url =. doi:10.1137/0205044 , timestamp =

work page doi:10.1137/0205044 1976
[11]

1977 , url =

Arnie Rosenthal and Anita Goldner , title =. 1977 , url =. doi:10.1137/0206003 , timestamp =

work page doi:10.1137/0206003 1977
[12]

Frederickson and Joseph F

Greg N. Frederickson and Joseph F. J. Approximation Algorithms for Several Graph Augmentation Problems , journal =. 1981 , url =. doi:10.1137/0210019 , timestamp =

work page doi:10.1137/0210019 1981
[13]

Networks , volume =

Jiong Guo and Johannes Uhlmann , title =. Networks , volume =. 2010 , url =. doi:10.1002/NET.20354 , timestamp =

work page doi:10.1002/net.20354 2010
[14]

Hiroshi Nagamochi , title =. Discret. Appl. Math. , volume =. 2003 , url =. doi:10.1016/S0166-218X(02)00218-4 , timestamp =

work page doi:10.1016/s0166-218x(02)00218-4 2003
[15]

Fixed-Parameter Algorithms for Minimum-Cost Edge-Connectivity Augmentation , journal =

D. Fixed-Parameter Algorithms for Minimum-Cost Edge-Connectivity Augmentation , journal =. 2015 , url =. doi:10.1145/2700210 , timestamp =

work page doi:10.1145/2700210 2015
[16]

Independence free graphs and vertex connectivity augmentation , journal =

Bill Jackson and Tibor Jord. Independence free graphs and vertex connectivity augmentation , journal =. 2005 , url =. doi:10.1016/J.JCTB.2004.01.004 , timestamp =

work page doi:10.1016/j.jctb.2004.01.004 2005
[17]

Toshimasa Watanabe and Akira Nakamura , title =. J. Comput. Syst. Sci. , volume =. 1987 , url =. doi:10.1016/0022-0000(87)90038-9 , timestamp =

work page doi:10.1016/0022-0000(87)90038-9 1987
[18]

Augmenting Graphs to Meet Edge-Connectivity Requirements , journal =

Andr. Augmenting Graphs to Meet Edge-Connectivity Requirements , journal =. 1992 , url =. doi:10.1137/0405003 , timestamp =

work page doi:10.1137/0405003 1992
[19]

Augmenting Undirected Node-Connectivity by One , journal =

L. Augmenting Undirected Node-Connectivity by One , journal =. 2011 , url =. doi:10.1137/100787507 , timestamp =

work page doi:10.1137/100787507 2011
[20]

Archiv der Mathematik , volume=

Minimum block containing a given graph , author=. Archiv der Mathematik , volume=. 1976 , url=

work page 1976
[21]

2023 , url =

Jaroslaw Byrka and Fabrizio Grandoni and Afrouz Jabal Ameli , title =. 2023 , url =. doi:10.1137/21M1421143 , timestamp =

work page doi:10.1137/21m1421143 2023
[22]

Parameterized Algorithms for Node Connectivity Augmentation Problems , booktitle =

Zeev Nutov , editor =. Parameterized Algorithms for Node Connectivity Augmentation Problems , booktitle =. 2024 , url =. doi:10.4230/LIPICS.ESA.2024.92 , timestamp =

work page doi:10.4230/lipics.esa.2024.92 2024
[23]

Parameterized graph separation problems , journal =

D. Parameterized graph separation problems , journal =. 2006 , url =. doi:10.1016/J.TCS.2005.10.007 , timestamp =

work page doi:10.1016/j.tcs.2005.10.007 2006
[24]

Algorithmica , volume =

Jianer Chen and Yang Liu and Songjian Lu , title =. Algorithmica , volume =. 2009 , url =. doi:10.1007/S00453-007-9130-6 , timestamp =

work page doi:10.1007/s00453-007-9130-6 2009
[25]

Fixed-Parameter Tractability of Multicut Parameterized by the Size of the Cutset , journal =

D. Fixed-Parameter Tractability of Multicut Parameterized by the Size of the Cutset , journal =. 2014 , url =. doi:10.1137/110855247 , timestamp =

work page doi:10.1137/110855247 2014
[26]

Fomin and Lukasz Kowalik and Daniel Lokshtanov and D

Marek Cygan and Fedor V. Fomin and Lukasz Kowalik and Daniel Lokshtanov and D. Parameterized Algorithms , publisher =. 2015 , url =. doi:10.1007/978-3-319-21275-3 , isbn =

work page doi:10.1007/978-3-319-21275-3 2015