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arxiv: 2606.31560 · v1 · pith:W3DDNLRGnew · submitted 2026-06-30 · 💻 cs.DS

Directed Low Diameter Decomposition for Structured Digraphs

Shinwoo An , Arnold Filtser This is my paper

Pith reviewed 2026-07-01 03:06 UTC · model grok-4.3

classification 💻 cs.DS
keywords directed graphslow diameter decompositionpathwidthtreewidthsparsest cutintegrality gapquasipartition
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The pith

Directed graphs with pathwidth pw admit low-diameter decompositions with diameter linear in pw.

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

The paper establishes that any directed graph of pathwidth pw has an (O(pw), Δ)-LDD. This improves an earlier construction whose diameter depended exponentially on pw. The same refined quasipartition analysis also shows that the integrality gap of the directed non-bipartite sparsest-cut LP is O(tw log n) on any n-vertex graph of treewidth tw. These bounds matter because LDDs and sparsest-cut approximations are used as subroutines in many graph algorithms; linear or near-linear dependence on the width parameter therefore yields faster procedures precisely on the graphs that admit small width. The results are obtained by tightening the analysis of an existing quasipartition method rather than by inventing an entirely new construction.

Core claim

Any directed graph with pathwidth pw admits an (O(pw), Δ)-LDD. The integrality gap of the Directed Non-Bipartite Sparsest-Cut LP relaxation on an n-vertex graph with treewidth tw is O(tw log n). Both statements follow from a refined analysis of the quasipartition construction that works on bounded-treewidth and bounded-pathwidth digraphs.

What carries the argument

The refined quasipartition construction that produces both the linear-in-pw LDD and the O(tw log n) integrality gap.

If this is right

  • Every directed graph of pathwidth pw has an (O(pw), Δ)-LDD.
  • The directed non-bipartite sparsest-cut LP has integrality gap O(tw log n) on all treewidth-tw graphs.
  • The previous exponential dependence on pw for LDDs is replaced by linear dependence.
  • The previous O(tw log² n) integrality-gap bound is tightened by one logarithmic factor.

Where Pith is reading between the lines

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

  • The same refinement technique may produce improved LDDs or cut approximations for other width measures such as branchwidth.
  • Faster approximation algorithms for directed cut and clustering problems become possible on the subclass of low-pathwidth or low-treewidth digraphs.
  • The linear LDD bound may combine with dynamic-programming methods that already run in 2^O(pw) n time to give end-to-end polynomial dependence on pw for some problems.

Load-bearing premise

The existing quasipartition construction admits a tighter analysis that removes the extra logarithmic factor from the gap and yields linear dependence on pathwidth.

What would settle it

An explicit directed graph of pathwidth pw on which every LDD requires diameter ω(pw), or an n-vertex treewidth-tw digraph whose directed non-bipartite sparsest-cut LP gap is ω(tw log n).

Figures

Figures reproduced from arXiv: 2606.31560 by Arnold Filtser, Shinwoo An.

Figure 1
Figure 1. Figure 1: Counter-example of Lemma 1 for directed graphs. Consider a directed graph [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Due to the definition of δ ∗ (·), every SCC of H \δ ∗ (B1) is either contained in B1 or disjoint from B1. The same holds symmetrically for B2. Therefore, every SCC of H \ (δ ∗ (B1) ∪ δ ∗ ′ (B2)) is fully contained in one of the regions of the Venn diagram. This proves both items of Lemma 3 for the special case of C = G. See Section 4 for the general case. C, denoted as P[C], as the u-v subpath of P. Note t… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Illustration of the first stage. The gray region denotes two balls B + G (u, R) and B − G (v, R) where R is sampled from [ 1 2 τ, 2 3 τ ] with τ = dG(u, v). For an SCC C ′ (yellow region) contained in B1 = B + G (u, R) and x, y ∈ P ∩ C ′ , we have that dG(x, y) ≤ dG(u, y) ≤ 2 3 τ , since P is a shortest path in G. (b) Illustration of the second stage. The blue region denotes B − G (u, R), and P ′ repre… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Illustration of the graph partition from the first two stages. In the first stage, an out-ball B + G (w, ∆) (light gray region) containing u, v is computed. In the second stage, an out-ball B + G (v, 8∆) (dark gray region) containing u is computed. (b) At the start of the third stage, an SCC (blue region) might have a large diameter, as no upper bounds for dG(x, u) and dG(y, u) are assumed. In the thir… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the critical path and critical sequence of [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
read the original abstract

Low diameter decompositions, or LDDs for short, are a fundamental primitive in the design of efficient graph algorithms. Roughly speaking, an LDD is a distribution over partitions of the vertices into bounded-diameter clusters such that nearby vertices are likely to be clustered together. Recently, there has been growing interest in lifting the notion of LDDs into \emph{directed graphs}. In particular, there are two natural directed analogues. The first is a directed LDD, where after removing a random subset of edges, every strongly connected component has a small diameter. The second is a quasipartition, which imposes the stronger requirement that whenever one vertex can still reach another after the edge removal, the two vertices must be close in the original directed metric. Every quasipartition yields an LDD, but the converse does not necessarily hold. In this work, we initiate the systematic study of LDDs in structured directed graphs. As our first main result, we show that any directed graph with pathwidth $\mathsf{pw}$ admits an $(O(\mathsf{pw}), \Delta)$-LDD. This improves upon the previous best-known $(2^{O(\mathsf{pw}^2)}, \Delta)$-LDD construction, which was implicitly derived from the quasipartition result of Salmasi, Sidiropoulos, and Sridhar [SODA'19]. As our second result, we show that the integrality gap of the Directed Non-Bipartite Sparsest-Cut LP relaxation on an $n$-vertex graph with treewidth $\mathsf{tw}$ is $O(\mathsf{tw} \log n)$. This improves upon the $O(\mathsf{tw}\log^2 n)$ bound of M\'emoli, Sidiropoulos, and Sridhar [ICALP'16, Algorithmica'18]. We obtain this result through the refined analysis of the quasipartition construction of M\'emoli et al. for bounded treewidth graphs.

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 two main results on directed low-diameter decompositions (LDDs) and related quasipartitions in structured digraphs. For any directed graph with pathwidth pw it constructs an (O(pw), Δ)-LDD, improving the prior (2^{O(pw²)}, Δ) bound implicitly obtained from Salmasi et al. (SODA'19). For the Directed Non-Bipartite Sparsest-Cut LP on n-vertex graphs of treewidth tw it proves an integrality gap of O(tw log n), improving the O(tw log² n) bound of Mémoli et al. (ICALP'16, Algorithmica'18). Both improvements are obtained by a refined analysis of the existing quasipartition construction of Mémoli, Sidiropoulos and Sridhar rather than a new construction.

Significance. If the refined analysis is correct, the results tighten the dependence on pathwidth and treewidth for two fundamental primitives, removing an exponential factor in the LDD bound and a logarithmic factor in the integrality gap. Such improvements are useful for approximation algorithms and exact algorithms on low-treewidth/pathwidth digraphs. The paper explicitly builds on and credits the prior quasipartition construction, which is a positive feature when the improvement arises from careful re-analysis.

minor comments (2)
  1. [Abstract] The abstract and introduction should explicitly state whether the O(pw) LDD bound is with respect to the directed metric or the underlying undirected metric, and whether Δ is the maximum out-degree or total degree; this notation is used without definition in the provided abstract.
  2. [Introduction / Technical sections] The claim that the new integrality gap follows from 'refined analysis' of the Mémoli et al. quasipartition should be accompanied, in §3 or the relevant technical section, by a short comparison table or paragraph highlighting exactly which step in the original analysis is tightened and by how much.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the improvements, and recommendation for minor revision. We appreciate the note that the work builds on and credits the prior quasipartition construction.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central results—an (O(pw), Δ)-LDD for pathwidth-pw digraphs and an O(tw log n) integrality gap for the directed sparsest-cut LP on treewidth-tw graphs—are obtained by refining the analysis of an existing quasipartition construction due to Mémoli, Sidiropoulos, and Sridhar (ICALP'16/Algorithmica'18). This prior construction is external to the present authors and is not derived from or fitted to the target bounds; the improvement consists of a tighter analysis that removes a log factor. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The claims rest on independent analytic refinement rather than reduction to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard definitions of pathwidth, treewidth, and quasipartitions together with the existence of prior constructions; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)
  • standard math Standard definitions and metric properties of pathwidth and treewidth for directed graphs.
    The statements are parameterized by these graph-width measures.
  • domain assumption The quasipartition construction of Mémoli et al. exists and admits further analysis.
    The paper obtains its bounds by refining that prior construction.

pith-pipeline@v0.9.1-grok · 5897 in / 1365 out tokens · 59392 ms · 2026-07-01T03:06:11.748438+00:00 · methodology

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Reference graph

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