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arxiv: 2607.01770 · v1 · pith:KI7JXBOQnew · submitted 2026-07-02 · 💻 cs.DS

Faster Parameterized Broadcasting

\'Edouard Bonnet , Carl Feghali , Manolis Vasilakis This is my paper

Pith reviewed 2026-07-03 04:20 UTC · model grok-4.3

classification 💻 cs.DS
keywords Telephone BroadcastParameterized algorithmsFixed-parameter tractabilityb-MatchingVertex coverVertex integrityDistance to clique
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The pith

Telephone Broadcast admits faster FPT algorithms with single-exponential dependence on vertex cover via a Turing reduction to edge-weighted b-Matching.

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

The paper establishes improved fixed-parameter tractable algorithms for the Telephone Broadcast problem on graphs. It replaces prior running times of 2 to the O of vc cubed for vertex cover number, double-exponential for vertex integrity, and 2 to the O of k squared for distance to clique with new bounds of 2 to the O of vc log vc, 2 to the O of vi squared log vi, and 2 to the O of k log k respectively. These speedups rest on a Turing reduction that converts broadcast instances into edge-weighted b-Matching instances whose solutions recover the minimum broadcast time. A sympathetic reader would care because the new bounds make the problem computationally feasible on graphs whose vertex covers or similar measures are moderate in size. The reduction is presented as the main technical ingredient enabling the improvements.

Core claim

The authors prove that Telephone Broadcast can be solved in time 2 to the O of vc log vc times n to the O of 1 when parameterized by vertex cover number vc, in time 2 to the O of vi squared log vi times n to the O of 1 when parameterized by vertex integrity vi, and in time 2 to the O of k log k times n to the O of 1 when parameterized by distance to clique k. These bounds improve the previous 2 to the O of vc cubed, double-exponential, and 2 to the O of k squared algorithms. The central mechanism is a Turing reduction to edge-weighted b-Matching that preserves the relevant parameters up to logarithmic factors.

What carries the argument

Turing reduction to edge-weighted b-Matching, which converts the broadcast instance so that an optimal solution to the matching problem yields the minimum number of broadcast rounds.

If this is right

  • Telephone Broadcast becomes fixed-parameter tractable with single-exponential dependence on vertex cover number.
  • The same reduction technique yields the stated improved bounds for vertex integrity and distance to clique.
  • Any future improvement to algorithms for edge-weighted b-Matching immediately transfers to faster broadcast algorithms under these parameters.
  • The problem remains NP-hard on certain graphs of small pathwidth and treedepth even though it is FPT under the listed parameters.

Where Pith is reading between the lines

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

  • The reduction approach may apply to other message-dissemination problems that involve choosing neighbors sequentially.
  • Graphs with small vertex cover could now be handled in practice for broadcast scheduling if the matching solver is efficient.
  • The technique suggests looking for similar Turing reductions from dissemination tasks to matching or flow problems in parameterized settings.

Load-bearing premise

The Turing reduction from Telephone Broadcast to edge-weighted b-Matching is correct and maps parameters to parameters of comparable size.

What would settle it

A concrete graph together with a source vertex where the broadcast time obtained from the b-Matching instance differs from the true minimum number of rounds needed.

read the original abstract

Given a connected graph $G$ and a source $s \in V(G)$, what is the smallest number of rounds necessary for all vertices of $G$ to receive a message initially only held by $s$, where at each round every informed vertex passes the message to one of its neighbors? This problem is called Telephone Broadcast and is suprisingly hard: it remains NP-hard on cycles intersecting at a single shared vertex, in particular, graphs of pathwidth 2 with a linear feedback vertex set of size 1, as well as on graphs with treedepth at most 6 [Egami et al.; MFCS '25]. Vertex cover number, vertex integrity, and distance to clique are among the few parameters for which Telephone Broadcast is fixed-parameter tractable. There is a $2^{\mathcal{O}(\mathrm{vc}^3)} n^{\mathcal{O}(1)}$-time algorithm parameterized by vertex cover number $\mathrm{vc}$ [Fomin, Fraigniaud, Golovach; TCS '24], a double-exponential algorithm parameterized by vertex integrity $\mathrm{vi}$, and a $2^{\mathcal{O}(k^2)} n^{\mathcal{O}(1)}$-time algorithm parameterized by distance to clique $k$ [Egami et al.; MFCS '25]. In this paper, we give improved parameterized algorithms for Telephone Broadcast with running times $2^{\mathcal{O}(\mathrm{vc} \log \mathrm{vc})} n^{\mathcal{O}(1)}$, $2^{\mathcal{O}(\mathrm{vi}^2 \log \mathrm{vi})} n^{\mathcal{O}(1)}$, and $2^{\mathcal{O}(k \log k)} n^{\mathcal{O}(1)}$. The main ingredient that makes our algorithms faster is a Turing reduction to edge-weighted $b$-Matching.

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

0 major / 1 minor

Summary. The paper claims improved FPT algorithms for Telephone Broadcast: 2^{O(vc log vc)} n^{O(1)} parameterized by vertex cover, 2^{O(vi^2 log vi)} n^{O(1)} by vertex integrity, and 2^{O(k log k)} n^{O(1)} by distance to clique. The central technical contribution is a Turing reduction from Telephone Broadcast to edge-weighted b-Matching that enables these single-exponential bounds, improving on prior 2^{O(vc^3)}, double-exponential, and 2^{O(k^2)} algorithms.

Significance. If the reduction is correct and parameter-preserving, the results constitute a meaningful advance in parameterized complexity for dissemination problems, replacing higher-degree or double-exponential dependence with logarithmic factors. The Turing reduction to b-Matching is a reusable technique that could apply to related broadcast or gossiping problems on sparse graphs.

minor comments (1)
  1. Abstract: 'suprisingly' is a typo for 'surprisingly'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of our results and for noting the potential significance of the Turing reduction to b-Matching. The recommendation is listed as uncertain, but the report contains no major comments or specific technical questions. We are prepared to address any points that may arise during further review.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The abstract describes a Turing reduction to edge-weighted b-Matching as the key technical step enabling single-exponential FPT bounds for Telephone Broadcast under vc, vi, and distance-to-clique parameters. No equations, ansatzes, fitted parameters, or self-citations appear in the provided text that would reduce the claimed running times or correctness to the inputs by construction. Prior results are cited from distinct author groups (Fomin et al., Egami et al.) and serve only as baseline comparisons. The algorithmic construction is presented as independent of the target bounds.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the reduction itself is the novel component but cannot be audited for hidden assumptions without the full text.

pith-pipeline@v0.9.1-grok · 5866 in / 1088 out tokens · 26803 ms · 2026-07-03T04:20:08.414478+00:00 · methodology

discussion (0)

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Reference graph

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