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arxiv: 2604.13025 · v1 · submitted 2026-04-14 · 💻 cs.DS · cs.DM· math.CO

Asymptotically faster algorithms for recognizing (k,ell)-sparse graphs

Bence De\'ak , P\'eter Madarasi This is my paper

Pith reviewed 2026-05-10 13:59 UTC · model grok-4.3

classification 💻 cs.DS cs.DMmath.CO
keywords (k,ℓ)-sparse graphsgraph recognition algorithmsrigidity theorycombinatorial optimizationarc-connectivitycentroid decompositionaugmenting paths
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The pith

New algorithms recognize (k,ℓ)-sparse graphs in subquadratic and near-linear time.

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

The paper introduces improved algorithms to determine whether a given graph satisfies the (k,ℓ)-sparsity condition, a property central to rigidity theory and combinatorial optimization. Earlier methods required quadratic or cubic time across most parameter values, but the new techniques achieve subquadratic running times by combining bounded-indegree orientations with reductions to arc-connectivity problems. Centroid decomposition helps divide the computation efficiently. The algorithms also identify explicit sets of vertices that violate the sparsity condition when the graph fails the test. This covers the classical range 0 ≤ ℓ < 2k and extends to 2k ≤ ℓ < 3k with improved bounds.

Core claim

We present new recognition algorithms for (k,ℓ)-sparse graphs in the ranges 0 ≤ ℓ ≤ k, k < ℓ < 2k, and 2k ≤ ℓ < 3k by combining bounded-indegree orientations, reductions to rooted arc-connectivity, augmenting-path techniques, and centroid decomposition. This produces the first subquadratic and near-linear-time algorithms for the classical range, with combinatorial implementations running in O(n√n) or O(n√(n log n)) time, and returns violating sets.

What carries the argument

Bounded-indegree orientations that reduce (k,ℓ)-sparsity verification to problems of rooted arc-connectivity, solved using augmenting paths and centroid decomposition.

If this is right

  • For 0 ≤ ℓ ≤ k the recognition runs in O(n √ n) time under combinatorial implementations.
  • For k < ℓ < 2k the time is O(n √(n log n)).
  • For 2k ≤ ℓ < 3k the time is O(n^{2}) when ℓ ≤ 2k+1 and O(n^{2} log n) otherwise.
  • The algorithms return an explicit vertex set certifying non-sparsity.
  • These are the first subquadratic algorithms throughout the classical range 0 ≤ ℓ < 2k.

Where Pith is reading between the lines

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

  • The speedups could make sparsity checks feasible for much larger graphs in applications like structural rigidity analysis.
  • Similar orientation and decomposition methods might accelerate other matroid intersection or sparsity-related problems.
  • Instantiating with faster connectivity subroutines could push times even closer to linear.
  • If the reductions are correct, they open the door to parallel or distributed implementations of sparsity testing.

Load-bearing premise

The reduction from (k,ℓ)-sparsity to bounded-indegree orientations and rooted arc-connectivity problems, solved via augmenting paths and centroid decomposition, correctly identifies sparse graphs and violating sets in the given ranges.

What would settle it

A concrete graph instance in one of the parameter ranges where the algorithm either misclassifies sparsity or fails to output a correct violating set, or a timing experiment on large graphs showing the runtime exceeds the stated asymptotic bounds.

Figures

Figures reproduced from arXiv: 2604.13025 by Bence De\'ak, P\'eter Madarasi.

Figure 1
Figure 1. Figure 1: Application of Lemma 2.2.2. Let U0 be an independent set of G, and suppose we already have a k-indegree-bounded U0-source orientation. By Lemma 2.2.2, the question of whether there exists a strict superset of U0 that violates the (k, ℓ)-sparsity of G reduces to a rooted arc-connectivity problem. This yields the following algorithm: Algorithm 1 Check Superset Sparsity Input: A k-indegree-bounded digraph D0 … view at source ↗
Figure 2
Figure 2. Figure 2: Two steps of the centroid decomposition. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
read the original abstract

The family of $(k,\ell)$-sparse graphs, introduced by Lorea, plays a central role in combinatorial optimization and has a wide range of applications, particularly in rigidity theory. A key algorithmic problem is to decide whether a given graph is $(k,\ell)$-sparse and, if not, to produce a vertex set certifying the failure of sparsity. While pebble game algorithms have long yielded $O(n^2)$-time recognition throughout the classical range $0 \leq \ell < 2k$, and $O(n^3)$-time algorithms in the extended range $2k \leq \ell < 3k$, substantially faster bounds were previously known only in a few special cases. We present new recognition algorithms for the parameter ranges $0 \le \ell \le k$, $k < \ell < 2k$, and $2k \leq \ell < 3k$. Our approach combines bounded-indegree orientations, reductions to rooted arc-connectivity, augmenting-path techniques, and a divide-and-conquer method based on centroid decomposition. This yields the first subquadratic, and in fact near-linear-time, recognition algorithms throughout the classical range when instantiated with the fastest currently available subroutines. Under purely combinatorial implementations, the running times become $O(n\sqrt n)$ for $0 \leq \ell \leq k$ and $O(n\sqrt{n\log n})$ for $k< \ell <2k$. For $2k \leq \ell < 3k$, we obtain an $O(n^2)$-time algorithm when $\ell \leq 2k+1$ and an $O(n^2\log n)$-time algorithm otherwise. In each case, the algorithm can also return an explicit violating set certifying that the input graph is not $(k,\ell)$-sparse.

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 paper claims new recognition algorithms for (k,ℓ)-sparse graphs (and explicit violating sets) in the ranges 0 ≤ ℓ ≤ k, k < ℓ < 2k, and 2k ≤ ℓ < 3k. The approach reduces the problem to finding bounded-indegree orientations, then solving rooted arc-connectivity queries via augmenting paths and centroid decomposition. This yields near-linear time with fast subroutines, O(n√n) combinatorially for 0 ≤ ℓ ≤ k, O(n√(n log n)) for k < ℓ < 2k, and O(n²) or O(n² log n) for 2k ≤ ℓ < 3k (depending on whether ℓ ≤ 2k+1).

Significance. If the reductions and implementations are correct, the work supplies the first subquadratic (and near-linear) algorithms for the classical range 0 ≤ ℓ < 2k, improving on the long-standing O(n²) pebble-game bound and the O(n³) bound for the extended range. The constructive character, use of standard primitives (orientations, augmenting paths, centroid decomposition), and ability to return violating sets are strengths that make the result directly usable in rigidity theory and matroid applications.

minor comments (2)
  1. The abstract and introduction would benefit from an explicit table or bullet list summarizing the new running times versus prior bounds for each interval of ℓ/k; this would make the improvement immediately visible to readers.
  2. Notation for the parameter ranges is repeated in several places with slight variations (e.g., “0 ≤ ℓ ≤ k” vs. “0 ≤ ℓ < 3k”); a single consolidated statement of the three cases would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript, accurate summary of the results, and recommendation for minor revision. We are pleased that the significance for rigidity theory and matroid applications is recognized.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper presents a constructive algorithmic approach that reduces (k,ℓ)-sparsity testing to bounded-indegree orientations followed by rooted arc-connectivity queries solved via augmenting paths and centroid decomposition. These are standard, externally studied primitives whose combination produces the stated running-time bounds through ordinary analysis of the described subroutines; no equation or claim reduces to a self-definition, a fitted input renamed as a prediction, or a load-bearing self-citation chain. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on established graph-theoretic definitions and standard algorithmic primitives without introducing new free parameters, invented entities, or non-standard axioms.

axioms (1)
  • domain assumption Standard definition of (k,ℓ)-sparsity and properties of graph orientations and connectivity.
    The algorithms build upon the established definition of (k,ℓ)-sparse graphs and common combinatorial tools.

pith-pipeline@v0.9.0 · 5643 in / 1430 out tokens · 45932 ms · 2026-05-10T13:59:53.611515+00:00 · methodology

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