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arxiv: 2605.20897 · v1 · pith:SU2FK7SMnew · submitted 2026-05-20 · 💻 cs.DS

Creating Robust and Fair Graph Structures for Connectivity and Clustering

Kushagra Chatterjee This is my paper

Pith reviewed 2026-05-21 01:55 UTC · model grok-4.3

classification 💻 cs.DS
keywords fault-tolerant reachability preserversfair clusteringconsensus clusteringstreaming algorithmsdirected graphsapproximation algorithmsgraph connectivitymulti-group fairness
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The pith

Dual fault-tolerant reachability preservers achieve O(n^{4/3} |P|^{1/3}) size in directed graphs while fair clustering gains log-memory streaming.

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

This work focuses on two challenges for graph algorithms in applications such as navigation and social networks: resilience to failures and fairness across protected groups. For the first challenge the paper gives the first non-trivial constructions of dual fault-tolerant pairwise reachability preservers that survive two edge or vertex failures while staying sparse. For the second it introduces closest fair clustering, supplies approximation algorithms and hardness results for multi-group cases, and produces the first streaming algorithm for fair consensus clustering that uses only logarithmic memory.

Core claim

We present the first non-trivial constructions of dual fault-tolerant pairwise reachability preservers that remain resilient to two edge or vertex failures, achieving a sparse construction of size O(n^{4/3}|P|^{1/3}). We develop approximation algorithms for fair consensus clustering and introduce the framework of closest fair clustering, establishing hardness results and efficient algorithms for multi-group settings, plus the first streaming algorithm for fair consensus clustering using only logarithmic memory.

What carries the argument

Dual fault-tolerant pairwise reachability preserver (a subgraph preserving reachability between pairs after two failures) together with the closest fair clustering framework (which enforces balanced representation of protected groups).

If this is right

  • Connectivity queries in directed graphs become reliable even after two failures.
  • Fair consensus clustering obtains provable approximation guarantees for multiple protected groups.
  • Large-scale datasets can be clustered fairly using only logarithmic memory.
  • Hardness results delimit what fairness levels are computationally feasible.
  • Graph systems can be designed with both failure resilience and group balance.

Where Pith is reading between the lines

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

  • The preserver constructions may extend to three or more failures if the same ratio-symmetric techniques apply.
  • Closest fair clustering could be merged with the fault-tolerant preservers to produce graphs that are simultaneously robust and equitable.
  • The logarithmic-memory streaming method may transfer to other fairness-constrained graph problems such as correlation clustering.
  • Practical tests on real navigation or social-network graphs would reveal whether the asymptotic size bound translates to measurable savings.

Load-bearing premise

The directed-graph and edge-or-vertex failure models are assumed to match the real-world instances in which these algorithms would be deployed.

What would settle it

A concrete directed graph and pair set P for which every dual fault-tolerant preserver requires more than O(n^{4/3} |P|^{1/3}) edges, or a multi-group dataset on which the streaming fair-consensus algorithm uses super-logarithmic memory while preserving the claimed fairness guarantee.

Figures

Figures reproduced from arXiv: 2605.20897 by Kushagra Chatterjee.

Figure 1.1
Figure 1.1. Figure 1.1: H(s,t)=2-FTRS for a single pair (s, t). Two black paths are the outer strands and the purple paths are the coupling paths between them. Consider any two failure edges f1, f2. W.l.o.g. assume, they do not form an s − t cut-set; otherwise, after the failure there won’t be any s − t path. Now if both f1, f2 lie on one of the two outer strands (i.e., either on P 1 (s,t) or P 2 (s,t) ), then since these two s… view at source ↗
Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p026_1.png] view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: Visualization of the scenario when only merge clusters remain after executing Algo [PITH_FULL_IMAGE:figures/full_fig_p027_1_2.png] view at source ↗
Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p027_1.png] view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: Visualization of the scenario when only cut clusters remain after executing AlgoGeneral. [PITH_FULL_IMAGE:figures/full_fig_p028_1_3.png] view at source ↗
Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p030_1.png] view at source ↗
Figure 1.4
Figure 1.4. Figure 1.4: Visualization of the fairpower-of-two algorithm with four colors {c1, c2, c3, c4} where nci denotes that the number vertices of color ci is n. In the 1st iteration the color blocks are B1 1 = {c1, c2} and B1 2 = {c3, c4}. So, in N 1 we make number of vertices of color c1 equal to number of vertices of color c2 and number of vertices of color c3 equal to number of vertices of color c4. In the 2nd iteratio… view at source ↗
Figure 1.5
Figure 1.5. Figure 1.5: Visualization of the make-pdc-fair algorithm for three colors {c1, c2, c3}, with 20, 12, and 8 vertices respectively, corresponding to a global ratio of 5: 3: 2. In the first iteration, the color blocks are B1 1 = {c1, c2} and B1 2 = {c3}. Hence, in the intermediate clustering F 1 , each cluster maintains the local ratio c1 : c2 = 5: 3. The scaling factor of block B1 1 in cluster F 1 1 is x = 2, while th… view at source ↗
Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p033_1.png] view at source ↗
read the original abstract

Graph algorithms are central to large-scale applications such as navigation systems, social networks, and data analysis platforms. This thesis studies two important challenges in such systems: robustness to failures and fairness in clustering outcomes. In the first part, we investigate fault-tolerant reachability preservers in directed graphs. We present the first non-trivial constructions of dual fault-tolerant pairwise reachability preservers that remain resilient to two edge or vertex failures, achieving a sparse construction of size $O(n^{4/3}|\mathcal{P}|^{1/3})$. In the second part, we study fair clustering algorithms that ensure balanced representation of protected groups. We develop approximation algorithms for fair consensus clustering and introduce the framework of closest fair clustering, establishing hardness results and efficient algorithms for multi-group settings. Building on this framework, we obtain improved guarantees for fair correlation clustering and design the first streaming algorithm for fair consensus clustering using only logarithmic memory. Together, these results contribute toward the design of graph algorithms that are both robust and socially responsible.

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 / 2 minor

Summary. The manuscript studies two challenges in graph algorithms for large-scale applications such as navigation systems and social networks: robustness to failures and fairness in clustering. In the first part, it presents the first non-trivial constructions of dual fault-tolerant pairwise reachability preservers in directed graphs that remain resilient to two edge or vertex failures, achieving a sparse construction of size O(n^{4/3}|P|^{1/3}). In the second part, it develops approximation algorithms for fair consensus clustering, introduces the closest fair clustering framework with hardness results and efficient algorithms for multi-group settings, obtains improved guarantees for fair correlation clustering, and designs the first streaming algorithm for fair consensus clustering using only logarithmic memory.

Significance. If the constructions and proofs hold, the results advance the design of robust graph structures for connectivity under failures and socially responsible clustering methods. The sparsity bound for dual fault-tolerant preservers and the logarithmic-memory streaming algorithm represent concrete theoretical progress with potential impact on real-world systems. The work provides both algorithmic frameworks and hardness results, strengthening the case for fair and resilient graph algorithms.

minor comments (2)
  1. [Introduction] The introduction could more explicitly connect the two parts by discussing how fault-tolerance and fairness both relate to reliable graph structures in applications.
  2. [Streaming Algorithm Section] In the description of the streaming algorithm, clarify whether the logarithmic memory bound holds in the worst case or under additional assumptions on the input stream.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive review, accurate summary of our contributions on dual fault-tolerant reachability preservers and fair clustering, and recommendation for minor revision. We appreciate the recognition of the sparsity bound and the logarithmic-memory streaming algorithm.

read point-by-point responses

  1. Referee: No major comments were listed in the report.

    Authors: We will incorporate any minor suggestions or corrections identified by the referee into the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper presents explicit algorithmic constructions for dual fault-tolerant pairwise reachability preservers (size O(n^{4/3}|P|^{1/3})) and approximation/streaming algorithms for fair consensus clustering and closest fair clustering. No equations, fitted parameters, or self-referential definitions appear in the provided abstract or description. Claims rest on stated constructions, hardness results, and algorithmic frameworks rather than any reduction by construction to inputs, self-citations that are load-bearing, or renamed known results. The central results are independent of the outputs they produce and do not invoke uniqueness theorems or ansatzes from prior self-work in a circular manner.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract does not introduce or rely on any explicit free parameters, ad-hoc axioms, or invented entities; all claims appear to build on standard graph-theoretic assumptions such as directed graphs and failure models.

axioms (1)
  • domain assumption Standard assumptions of directed graphs with edge and vertex failures
    Invoked implicitly when describing fault-tolerant reachability preservers and clustering on graphs.

pith-pipeline@v0.9.0 · 5693 in / 1351 out tokens · 26373 ms · 2026-05-21T01:55:18.127669+00:00 · methodology

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Reference graph

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