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arxiv: 1906.09595 · v1 · pith:NVW52BLUnew · submitted 2019-06-23 · 💻 cs.DS

Dynamic Maximal Independent Set

Morteza Monemizadeh This is my paper

Pith reviewed 2026-05-25 17:56 UTC · model grok-4.3

classification 💻 cs.DS
keywords dynamic algorithmsmaximal independent setgraph streamsamortized update timeedge insertions deletions
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The pith

A dynamic algorithm maintains a maximal independent set under edge insertions and deletions with amortized update time O(log³ n).

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

The paper proposes a dynamic algorithm that processes a stream of edge insertions and deletions on a graph with fixed vertex set V of size n. It maintains a maximal independent set of the current graph at every step. The algorithm achieves an amortized update time of O(log cubed n) per edge change. A sympathetic reader cares because this bound is polylogarithmic and therefore practical for large changing networks where recomputing from scratch would be too slow.

Core claim

The paper establishes that there exists a dynamic algorithm maintaining a maximal independent set of G at any time t of the stream S with amortized update time O(log³ n), where the vertex set V is fixed with n = |V| and updates consist only of edge insertions and deletions in the standard dynamic graph stream model.

What carries the argument

The dynamic algorithm that maintains a maximal independent set while processing edge updates in amortized O(log³ n) time.

If this is right

  • The algorithm maintains a valid maximal independent set after every single update.
  • Both insertions and deletions are supported with the same amortized bound.
  • The bound holds in the standard dynamic graph stream model with fixed vertices.
  • The time is amortized over the entire sequence of updates.

Where Pith is reading between the lines

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

  • Similar techniques might extend to maintaining other maximal structures such as matchings under the same update model.
  • The fixed-vertex assumption could be tested by allowing vertex additions in a follow-up construction.
  • The polylogarithmic bound suggests the method may combine with other dynamic primitives that tolerate logarithmic overhead.

Load-bearing premise

The vertex set stays fixed throughout the stream and only edges are inserted or deleted.

What would settle it

A sequence of edge updates on a graph with n vertices where the algorithm requires more than O(log³ n) amortized time per update on some update.

read the original abstract

Given a stream $\mathcal{S}$ of insertions and deletions of edges of an underlying graph $G$ (with fixed vertex set $V$ where $n=|V|$ is the number of vertices of $G$), we propose a dynamic algorithm that maintains a maximal independent set (MIS) of $G$ (at any time $t$ of the stream $\mathcal{S}$) with amortized update time $O(\log^3 n)$.

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 paper claims to give a dynamic algorithm maintaining a maximal independent set of a graph G (fixed vertex set V, n = |V|) under a stream of edge insertions and deletions, achieving amortized O(log^3 n) update time per update.

Significance. A correct algorithm with this bound would be a notable contribution to dynamic graph algorithms, as it supplies an efficient polylogarithmic-time solution for maintaining an MIS in the standard edge-update model.

major comments (1)
  1. Abstract: the central claim is an existence result for an algorithm and its analysis, yet the provided text supplies neither a high-level description of the data structure nor a proof sketch, so the derivation cannot be verified from the given material.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their feedback. Below we respond to the single major comment.

read point-by-point responses

  1. Referee: Abstract: the central claim is an existence result for an algorithm and its analysis, yet the provided text supplies neither a high-level description of the data structure nor a proof sketch, so the derivation cannot be verified from the given material.

    Authors: We agree that the abstract, as currently written, is limited to a statement of the result without a high-level description of the data structure or a proof sketch. This is a valid observation. We will revise the abstract to incorporate a concise high-level overview of the algorithmic approach and the key ideas in the analysis. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper asserts existence of a dynamic algorithm maintaining a maximal independent set under edge insertions and deletions in the standard model with fixed vertex set, achieving amortized O(log³ n) update time. No equations, fitted parameters, predictions, or self-citations appear in the abstract or described structure. The central claim is an algorithmic construction and analysis result rather than a reduction of any output quantity to its own inputs by definition or fitting. The derivation chain consists of standard algorithmic techniques whose correctness is independent of the final time bound.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard dynamic graph model with fixed vertices; no free parameters, invented entities, or non-standard axioms are visible in the abstract.

axioms (1)
  • domain assumption Vertex set V is fixed with size n; updates are edge insertions and deletions only.
    Explicitly stated in the opening sentence of the abstract.

pith-pipeline@v0.9.0 · 5576 in / 995 out tokens · 21448 ms · 2026-05-25T17:56:44.722632+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · 2 internal anchors

  1. [1]

    Fully dynamic maximal inde- pendent set with sublinear update time

    Sepehr Assadi, Krzysztof Onak, Baruch Schieber, and Sha y Solomon. Fully dynamic maximal inde- pendent set with sublinear update time. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 815–826, 2018

    work page 2018

  2. [2]

    Fully dynamic maximal indepen- dent set with sublinear in n update time

    Sepehr Assadi, Krzysztof Onak, Baruch Schieber, and Sha y Solomon. Fully dynamic maximal indepen- dent set with sublinear in n update time. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1919–1936, 2019

    work page 2019

  3. [3]

    Chernoff

    H. Chernoff. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observa- tions. Annals of Mathematical Statistics , 23(4):493 – 507, 1952

    work page 1952

  4. [4]

    Improved Algorithms for Fully Dynamic Maximal Independent Set

    Y uhao Du and Hengjie Zhang. Improved algorithms for full y dynamic maximal independent set. CoRR, abs/1804.08908, 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  5. [5]

    Simple dynamic algorithms for Maximal Independent Set and other problems

    Manoj Gupta and Shahbaz Khan. Simple dynamic algorithms for maximal independent set and other problems. CoRR, abs/1804.01823, 2018. 12

    work page internal anchor Pith review Pith/arXiv arXiv 2018