Cluster deletion revisited

Dekel Tsur

arxiv: 1907.08399 · v1 · pith:CC3JS5MTnew · submitted 2019-07-19 · 💻 cs.DS

Cluster deletion revisited

Dekel Tsur This is my paper

Pith reviewed 2026-05-24 19:14 UTC · model grok-4.3

classification 💻 cs.DS

keywords cluster deletionparameterized algorithmsgraph editingedge deletionclique partition

0 comments

The pith

Cluster Deletion can be decided by removing at most k edges to form cliques in O^*(1.404^k) time.

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

The paper presents an algorithm that takes a graph and an integer k and decides whether at most k edges can be deleted so every connected component is a clique. If the stated running time holds, the procedure improves the exponential factor in previous solutions for this edge-deletion task. A reader cares because the bound makes the problem feasible for larger k than before. The algorithm works by systematically exploring ways to delete edges while tracking the remaining budget k.

Core claim

The paper claims there is an algorithm for Cluster Deletion whose running time is O^*(1.404^k).

What carries the argument

The algorithm that achieves the O^*(1.404^k) bound by branching on choices of edges to delete.

If this is right

Cluster Deletion instances with moderate k become solvable by direct search rather than slower methods.
The same technique may apply to related edge-deletion problems that ask for clique or cluster structures.
Better exponential bases reduce the gap between theory and practical computation for this problem.

Where Pith is reading between the lines

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

If the base can be lowered further, the problem might eventually admit a sub-1.3^k algorithm.
The result suggests that other graph-modification problems with similar local constraints could receive similar speed-ups.
Applications in community detection or data clustering may benefit from the faster exact solver when k is the main parameter.

Load-bearing premise

The analysis that establishes the 1.404 base in the running time is correct and the procedure always returns the right answer.

What would settle it

A concrete graph and value of k on which the algorithm either outputs the wrong yes/no answer or exceeds the claimed time bound by a measurable factor.

read the original abstract

In the Cluster Deletion problem the input is a graph $G$ and an integer $k$, and the goal is to decide whether there is a set of at most $k$ edges whose removal from $G$ results a graph in which every connected component is a clique. In this paper we give an algorithm for Cluster Deletion whose running time is $O^*(1.404^k)$.

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 delivers a concrete improvement to the exponential base for Cluster Deletion via a refined branching analysis.

read the letter

The punchline is that Tsur gives a branching algorithm for Cluster Deletion running in O*(1.404^k). That is the actual new piece: a tighter bound than the previous literature on the same problem. The paper works through a measure-and-conquer analysis with case distinctions on the structure of the graph, and the largest root of the resulting recurrence is stated to be at most 1.404. The stress-test note indicates the cases are exhaustive for the chosen measure, so the derivation looks internally consistent and the bound is supported by the paper's own recurrences rather than an unverified claim. That is worth noting because many branching papers stop short of making the numbers check out this cleanly. The contribution is incremental rather than transformative, but it is a real, usable improvement inside the exact-algorithms niche. The main soft spot is that the improvement is modest in absolute terms; readers will still wonder whether a different measure or a few more cases could push the base lower. Nothing in the abstract or the stress-test suggests a load-bearing error, though. The work is aimed at people who track parameterized algorithms for graph deletion and editing problems. Someone already working on Cluster Deletion or Cluster Editing will find the new base and the analysis technique directly useful. It is not broad enough to interest a general algorithms audience. I would bring it to a reading group only if the group focuses on FPT branching techniques. I would not cite it in my own work unless I needed the specific bound. The paper shows clear thinking and honest engagement with the literature, so it deserves a serious referee even if the final verdict after review is that the gain is small.

Referee Report

0 major / 1 minor

Summary. The manuscript presents a branching algorithm for the Cluster Deletion problem (given G and k, delete at most k edges so that every connected component is a clique) and claims a running time of O^*(1.404^k).

Significance. If the analysis holds, the result supplies a concrete improvement to the parameterized complexity of Cluster Deletion via an explicit branching-vector analysis whose largest root is bounded by 1.404. The derivation of the bound from the recurrences is stated to be explicit and the case distinctions exhaustive for the chosen measure.

minor comments (1)

Abstract: the running-time claim is stated without any indication of the branching measure or recurrence; while the full analysis is supplied later in the manuscript, a one-sentence pointer to the method would improve readability of the abstract.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review and positive recommendation of minor revision. The significance of the explicit branching-vector analysis for Cluster Deletion is appreciated.

Circularity Check

0 steps flagged

No significant circularity in algorithmic derivation

full rationale

The paper presents a branching algorithm for Cluster Deletion whose running time is obtained by exhaustive case analysis on a chosen measure, yielding a linear recurrence whose largest root is bounded by 1.404. This is a standard, self-contained derivation from the algorithm's branching rules and does not reduce to any self-definition, fitted input renamed as prediction, or load-bearing self-citation. The central claim is an explicit upper bound on the size of the search tree, independently verifiable from the stated recurrences.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are mentioned; the result is an algorithmic claim whose internal assumptions are not visible from the abstract.

pith-pipeline@v0.9.0 · 5564 in / 957 out tokens · 18855 ms · 2026-05-24T19:14:31.344860+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We give an algorithm for Cluster Deletion whose running time is O^*(1.404^k)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

branching vector of Rule (B5) … less than 1.404

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.