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arxiv: 2604.13577 · v1 · submitted 2026-04-15 · 💻 cs.DS

Lower Bounds for Testing Directed Acyclicity in the Unidirectional Bounded-Degree Model

Yuichi Yoshida This is my paper

Pith reviewed 2026-05-10 12:26 UTC · model grok-4.3

classification 💻 cs.DS
keywords property testingdirected graphsacyclicityquery complexitylower boundsbounded-degree modelunidirectional oracle
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The pith

Testing whether a directed graph is acyclic requires tilde-Omega of n to the two-thirds queries for one-sided testers when out-degree is a large constant.

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

The paper proves new lower bounds on query complexity for testing directed acyclicity in the model where each vertex query returns only its outgoing neighbors. For any constant out-degree d above some absolute threshold, one-sided epsilon-testers must use nearly n to the two-thirds queries in the worst case. Two-sided testers require at least square-root n queries under the same degree restriction, while tolerant testing needs linear queries. These results tighten earlier bounds and show that the unidirectional information flow makes acyclicity hard to verify efficiently even when degrees stay bounded.

Core claim

There exist absolute constants d0 in natural numbers and epsilon greater than zero such that for every constant d at least d0, any one-sided epsilon-tester for acyclicity on n-vertex digraphs of maximum out-degree at most d requires tilde-Omega of n to the two-thirds queries. Under the same degree assumption any two-sided epsilon-tester requires Omega of square-root n queries, and tolerant testing requires Omega of n queries for some absolute constant out-degree bound d via reduction from bounded-degree 3-colorability.

What carries the argument

The unidirectional bounded-degree oracle model, where each query to a vertex returns only its outgoing neighbors up to a constant maximum out-degree.

If this is right

  • One-sided testers for acyclicity cannot succeed with fewer than tilde-Omega of n to the two-thirds queries once the constant out-degree exceeds d0.
  • Two-sided testers cannot succeed with fewer than Omega of square-root n queries for the same degree regime.
  • Tolerant testers cannot succeed with fewer than Omega of n queries for some fixed constant out-degree.
  • The lower bounds continue to hold for any fixed positive epsilon when d is large enough.

Where Pith is reading between the lines

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

  • The one-way nature of the oracle forces testers to explore reachability indirectly, which may make other directed properties such as strong connectivity similarly hard.
  • The linear lower bound obtained by reduction from 3-colorability indicates that acyclicity testing inherits the full hardness of coloring problems once tolerance is allowed.

Load-bearing premise

The maximum out-degree must be bounded by a sufficiently large constant and each query must reveal only outgoing neighbors.

What would settle it

An explicit one-sided epsilon-tester that uses o of n to the two-thirds queries on all n-vertex digraphs with out-degree at most some large constant d would falsify the main claim.

Figures

Figures reproduced from arXiv: 2604.13577 by Yuichi Yoshida.

Figure 1
Figure 1. Figure 1: Schematic of the hard NO distribution Dperm. The blue core B is the union of dB random permutations. Each blue vertex also sends dR edges into R≤L/2 . The red part is layered: red edges only go to later layers, and vertices in R>L/2 have outdegree 0. Since there are no edges from red to blue, every directed cycle lies inside G[B]. 3.2 Graphs from Dperm are far from acyclic Lemma 3.1 (Far from acyclic). The… view at source ↗
Figure 2
Figure 2. Figure 2: A witness configuration for the proof of Lemma 4.8. The left panel shows the blocks [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The three ingredients of the reduction R(H). Panel (a) is the selection edge for color i at vertex v. Panel (b) is the edge-constraint gadget for an undirected edge e = {u, v} and color i; if both endpoints keep color i, the gadget creates a directed cycle. Panel (c) is the at-most-one-color gadget at a single vertex v; if two colors remain active, it creates a directed cycle. Completeness (3-colorable ⇒ c… view at source ↗
read the original abstract

We study property testing of directed acyclicity in the unidirectional bounded-degree oracle model, where a query to a vertex reveals its outgoing neighbors. We prove that there exist absolute constants $d_0\in\mathbb{N}$ and $\varepsilon>0$ such that for every constant $d\ge d_0$, any one-sided $\varepsilon$-tester for acyclicity on $n$-vertex digraphs of maximum outdegree at most $d$ requires $\widetilde{\Omega}(n^{2/3})$ queries. This improves the previous $\widetilde{\Omega}(n^{5/9})$ lower bound for one-sided testing of acyclicity in the same model. We also prove that, under the same degree assumption, any two-sided $\varepsilon$-tester requires $\Omega(\sqrt n)$ queries, improving the previous $\Omega(n^{1/3})$ lower bound. We further prove an $\Omega(n)$ lower bound for tolerant testing for some absolute constant outdegree bound $d$ by reduction from bounded-degree $3$-colorability.

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 / 3 minor

Summary. The paper studies property testing of directed acyclicity in the unidirectional bounded-degree oracle model (queries reveal outgoing neighbors only). It proves that for absolute constants d0 and ε>0, and all constant d≥d0, any one-sided ε-tester requires Õ(n^{2/3}) queries (improving the prior Õ(n^{5/9}) bound); any two-sided ε-tester requires Ω(√n) queries (improving Ω(n^{1/3})); and tolerant testing requires Ω(n) queries for some constant d, via reduction from bounded-degree 3-colorability.

Significance. If the reductions and query-complexity arguments hold, the results tighten the known landscape for directed-graph property testing: they establish a stronger separation between one-sided and two-sided testers and give the first linear lower bound for tolerant acyclicity testing. The explicit parametrization by a sufficiently large constant out-degree d0 and the use of standard edge-edit distance make the claims directly comparable to prior work in the area.

minor comments (3)
  1. [§1] §1 (Introduction): the statement of the one-sided lower bound uses Õ notation without defining the hidden polylog factors; a brief sentence clarifying the precise form of the bound would aid readability.
  2. [§4] §4 (Reduction for tolerant testing): the distance measure in the constructed instance is normalized by n, but the reduction from 3-colorability should explicitly verify that the edit-distance parameter ε maps to a constant fraction of the original instance size.
  3. [Preliminaries] Figure 1 (or the corresponding diagram in the preliminaries): the illustration of the unidirectional oracle does not label the out-degree bound d, which is central to the statement of all theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review and the recommendation of minor revision. We appreciate the assessment that the results tighten the known landscape for directed-graph property testing by establishing a stronger separation between one-sided and two-sided testers and providing the first linear lower bound for tolerant acyclicity testing.

Circularity Check

0 steps flagged

No significant circularity; lower bounds derived via external reductions

full rationale

The paper establishes query lower bounds for testing directed acyclicity via explicit reductions from independent problems (e.g., bounded-degree 3-colorability for the tolerant-testing case, and presumably other communication or query-complexity problems for the one-sided and two-sided cases). The abstract and described contributions contain no self-definitional steps, no fitted parameters renamed as predictions, and no load-bearing self-citations whose validity depends on the current paper. The parametrization by a sufficiently large constant out-degree d0 is an explicit assumption stated up-front rather than derived circularly. All claimed improvements (from n^{5/9} to n^{2/3} and from n^{1/3} to sqrt(n)) are comparisons to prior external results, not internal re-derivations of the same quantity. The derivation chain is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on standard graph-theoretic assumptions and reductions from known NP-hard problems; no free parameters, ad-hoc constants, or new entities are introduced in the abstract.

axioms (1)
  • standard math Standard assumptions of graph theory and property testing, including existence of hard instances for reductions from 3-colorability.
    The Omega(n) tolerant-testing bound is obtained by reduction from bounded-degree 3-colorability, which is treated as an established hard problem.

pith-pipeline@v0.9.0 · 5477 in / 1331 out tokens · 43614 ms · 2026-05-10T12:26:09.444082+00:00 · methodology

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Reference graph

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