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arxiv: 2606.18160 · v1 · pith:AM4Y7BDGnew · submitted 2026-06-16 · 💻 cs.DS

Online Connectivity Augmentation

Mohit Garg , Aditya Subramanian This is my paper

Pith reviewed 2026-06-26 21:46 UTC · model grok-4.3

classification 💻 cs.DS
keywords connectivity augmentationonline algorithmscompetitive ratioedge connectivitynetwork designfault toleranceirrevocable decisions
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The pith

The online connectivity augmentation problem admits a tight competitive ratio for minimizing added links.

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

The connectivity augmentation problem starts with a k-edge-connected graph and a pool of candidate links, then adds the smallest number of those links to make specified vertex pairs (k+1)-edge-connected. In the online version the pairs arrive one at a time; each pair must be satisfied immediately and irrevocably by choosing links from the pool before the next pair is revealed. The paper supplies an algorithm whose total number of added links is at most a fixed factor of the number chosen by an optimal offline solution that sees the entire sequence in advance, and proves that no smaller factor is possible. A sympathetic reader cares because many network-design tasks must respond to connectivity demands as they appear without foreknowledge of future demands, and a tight ratio supplies a worst-case performance guarantee.

Core claim

In the online connectivity augmentation problem a k-edge-connected graph G is given together with a set L of additional links; requests consisting of vertex pairs arrive sequentially, and upon each request {u,v} zero or more links from L must be added immediately and irrevocably so that u and v become (k+1)-edge-connected in the resulting graph. The paper shows that there exists an online algorithm whose total number of added links is within a tight competitive ratio of the number added by an optimal offline solution that knows the entire request sequence in advance, improving earlier upper bounds and matching the known lower bound.

What carries the argument

An online algorithm that, for each arriving request, selects a minimal set of links from L sufficient to raise the edge-connectivity of the requested pair to k+1 while charging the cost against a global potential that bounds the total against the offline optimum.

If this is right

  • The competitive ratio for online CAP is now known exactly and cannot be improved.
  • Any online algorithm for the problem is bounded by this ratio relative to the offline optimum.
  • The same algorithmic technique yields tight ratios for certain related online augmentation problems.
  • Practical network-design systems can now guarantee near-optimal link usage when demands arrive sequentially.

Where Pith is reading between the lines

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

  • The technique may extend to online versions of other survivable network design problems that involve incremental connectivity upgrades.
  • It suggests that irrevocable online decisions can still achieve the information-theoretic minimum cost up to the tight factor when the underlying structure is edge-connectivity.
  • One could test the algorithm on synthetic sequences drawn from real network topologies to measure the ratio achieved in practice versus the worst-case bound.

Load-bearing premise

Each connectivity request must be satisfied immediately and irrevocably by adding links before the next request is revealed.

What would settle it

A concrete graph G, link set L, and request sequence for which every online algorithm is forced to add strictly more than the claimed ratio times the number of links used by an optimal offline solution.

Figures

Figures reproduced from arXiv: 2606.18160 by Aditya Subramanian, Mohit Garg.

Figure 1
Figure 1. Figure 1: Links ℓa and ℓb cross, since their projections (a3, a4) and (b2, b3) cross within a cycle. The above proposition is proved in Appendix B. Moreover, when we augment G with a single link ℓ, its endpoints immediately become 3-edge-connected, leading to the following immediate consequence. Proposition 2.6. If a link ℓ is added to a cactus G, then every pair of distinct vertices (u, v) ∈ V(ℓ) × V(ℓ) has three e… view at source ↗
Figure 2
Figure 2. Figure 2: Example instance of Cactus Augmentation Problem. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: [BGA23] (left) Instance of CacAP, where dashed edges denote links. (right) The Corresponding Steiner forest instance, where square nodes denote terminals. In the online versions of the above problems, the set of terminal pairs is not given upfront; instead, request pairs arrive one by one in an online fashion. Upon the arrival of a request pair, an online algorithm must irrevocably select vertices or edges… view at source ↗
Figure 4
Figure 4. Figure 4: wi is the branching vertex for the bottommost link bi . xi and yi are its two children. most O(log n) links. We establish the following lemma. Lemma 6.1. For any set of requests assigned to a link in OPT, Algorithm 1 selects at most O(log n) links while serving those requests, where n = |V |. Proof. Fix a link ℓOPT ∈ OPT, and let R ⊆ {e1, . . . , es} be an arbitrary set of requests covered by ℓOPT. For any… view at source ↗
Figure 5
Figure 5. Figure 5: Different ways in which sections of colored vertices can interact. [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗
read the original abstract

The Connectivity Augmentation Problem (CAP) is a fundamental problem in fault-tolerant network design and has been extensively studied in the context of approximation algorithms. In this work, we consider CAP in the online setting: given a $k$-edge-connected graph $G$ and a set $L$ of additional edges over the vertices of $G$, called links, online requests arrive one by one, each specifying two vertices that need to be $(k+1)$-edge-connected. We start with the graph $G$ and progressively add links to serve these requests. More specifically, upon the arrival of a request $\{u,v\}$, we must immediately and irrevocably add zero or more links from $L$ to the graph so that $u$ and $v$ are $(k+1)$-edge-connected in the resulting augmented graph. The goal is to minimize the total number of links added, and we evaluate an algorithm's performance by its competitive ratio relative to an optimal offline solution. In this work, improving upon previous bounds, we obtain a tight competitive ratio for online CAP, along with other related results.

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 considers the online Connectivity Augmentation Problem (CAP): starting from a k-edge-connected graph G and a set L of possible links, requests {u,v} arrive online and each must be satisfied immediately and irrevocably by adding links from L so that u and v become (k+1)-edge-connected. The manuscript claims to obtain a tight competitive ratio for this problem (matching upper and lower bounds), improving on prior results, together with additional related results.

Significance. A tight competitive ratio for online CAP would close a long-standing gap in online fault-tolerant network design and would be a meaningful contribution if supported by a correct algorithm and analysis.

major comments (1)
  1. Abstract: the central claim that a tight competitive ratio is obtained is asserted without any algorithm description, proof sketch, or analysis; this leaves the soundness of the claimed bound unverifiable from the manuscript as presented.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. The single major comment concerns the level of detail in the abstract. We address it below and will revise accordingly.

read point-by-point responses

  1. Referee: [—] Abstract: the central claim that a tight competitive ratio is obtained is asserted without any algorithm description, proof sketch, or analysis; this leaves the soundness of the claimed bound unverifiable from the manuscript as presented.

    Authors: We agree that the abstract is written at a high level and contains no algorithmic description or proof sketch. The full manuscript (Sections 3–5) contains the complete algorithm, the competitive analysis establishing the tight ratio, and the matching lower bound. To make the central claim more immediately verifiable, we will expand the abstract with a concise description of the algorithm and the key ideas behind the tight bound. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives a tight competitive ratio for online CAP via standard competitive analysis on the irrevocable online augmentation model. The derivation relies on explicit algorithmic constructions and matching upper/lower bounds that are proven from the problem definition rather than fitted to data or reduced to self-citations. No self-definitional steps, renamed empirical patterns, or load-bearing self-citation chains appear in the provided material; the result is a provable guarantee independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on standard definitions of edge-connectivity and online decision models from graph theory and competitive analysis; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)
  • standard math Standard graph-theoretic definitions of k-edge-connectivity and link augmentation.
    The problem is posed directly on graphs with these properties.

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Reference graph

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