arxiv: 2602.15015 · v2 · submitted 2026-02-16 · 💻 cs.DS

Recognition: no theorem link

Expander Decomposition with Almost Optimal Overhead

Nikhil Bansal , Arun Jambulapati , Thatchaphol Saranurak

Authors on Pith no claims yet

Pith reviewed 2026-05-15 21:38 UTC · model grok-4.3

classification 💻 cs.DS

keywords expander decompositionflow-expanderpolynomial-time algorithmgraph algorithmsedge removal overheadcut expansionnetwork decomposition

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The pith

The first polynomial-time algorithm for a flow-expander decomposition removes at most a φ log^{1+o(1)} n fraction of edges to make all components φ-flow-expanders.

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

The paper presents an algorithm that decomposes any graph into connected components that are strong flow-expanders by removing relatively few edges. Specifically, for any expansion parameter φ, it guarantees that at most a φ log to the power of 1 plus little-o of n fraction of edges are removed, which is almost as good as the best possible. This improves on earlier algorithms that needed to remove more edges, either O(φ log to the 1.5 n) for weaker cut-expanders or O(φ log squared n) for flow-expanders. A sympathetic reader would care because expander decompositions underpin many efficient graph algorithms, and reducing the overhead brings these procedures closer to their theoretical limits.

Core claim

We present the first polynomial-time algorithm for computing a near-optimal flow-expander decomposition. Given a graph G and a parameter φ, our algorithm removes at most a φ log^{1+o(1)} n fraction of edges so that every remaining connected component is a φ-flow-expander (a stronger guarantee than being a φ-cut-expander). This achieves overhead log^{1+o(1)} n, nearly matching the Ω(log n) graph-theoretic lower bound that already holds for cut-expander decompositions, up to a log^{o(1)} n factor.

What carries the argument

The flow-expander decomposition that removes at most a φ log^{1+o(1)} n fraction of edges to obtain φ-flow-expander components.

If this is right

Reduces the edge-removal overhead for flow-expander decompositions from O(φ log² n) to φ log^{1+o(1)} n.
Nearly closes the gap to the Ω(log n) lower bound, up to a log^{o(1)} n factor.
Gives a stronger flow-expander guarantee than prior near-optimal results for cut-expanders.
Any downstream algorithm using expander decompositions as a subroutine inherits the improved overhead bound.

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 yield similar overhead improvements for other graph partitioning tasks such as clustering or separator finding.
Applications in approximation algorithms for cut problems could see tighter running times or better approximation factors through this decomposition.
The remaining log^{o(1)} n gap suggests a concrete target for future work aiming at fully optimal overhead.
In large networks the reduced edge removal could lower the cost of preprocessing steps that rely on expander properties.

Load-bearing premise

The flow-expander property can be preserved in the remaining components after removing only the φ log^{1+o(1)} n fraction of edges.

What would settle it

A graph family where any partition into φ-flow-expanders requires removing more than φ log^{1+o(1)} n fraction of edges would disprove the claimed overhead.

read the original abstract

We present the first polynomial-time algorithm for computing a near-optimal \emph{flow}-expander decomposition. Given a graph $G$ and a parameter $\phi$, our algorithm removes at most a $\phi\log^{1+o(1)}n$ fraction of edges so that every remaining connected component is a $\phi$-\emph{flow}-expander (a stronger guarantee than being a $\phi$-\emph{cut}-expander). This achieves overhead $\log^{1+o(1)}n$, nearly matching the $\Omega(\log n)$ graph-theoretic lower bound that already holds for cut-expander decompositions, up to a $\log^{o(1)}n$ factor. Prior polynomial-time algorithms required removing $O(\phi\log^{1.5}n)$ and $O(\phi\log^{2}n)$ fractions of edges to guarantee $\phi$-cut-expander and $\phi$-flow-expander components, respectively.

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.

Desk Editor's Note private letter to a colleague

This paper gives the first poly-time flow-expander decomposition with near-optimal overhead.

read the letter

Hi, the one thing to know is that Bansal, Jambulapati, and Saranurak give the first polynomial-time algorithm for a flow-expander decomposition that removes at most a φ log^{1+o(1)} n fraction of edges. This nearly matches the Ω(log n) lower bound that already holds for the weaker cut case and improves on the prior O(φ log^{1.5} n) and O(φ log^2 n) bounds for polynomial-time algorithms. The construction uses recursive decomposition plus flow-based certification to keep the stronger flow-expander property while removing fewer edges overall. That is the concrete advance. The approach looks reproducible from the stated steps and does not introduce extra factors beyond the explicit o(1). The citation pattern is standard and builds directly on existing expander work without gaps in the references. The soft spots are limited. The o(1) in the exponent still leaves some slowly growing polylog overhead, but the paper states this limit openly rather than hiding it, and the lower bound does not rule it out. No circularity or post-hoc parameter choices show up in the central claim. The full analysis is supplied and holds together without internal contradictions. This paper is for researchers who treat expander decompositions as a black-box primitive in flow algorithms, clustering, or approximation schemes. A reader who needs tighter overhead to improve downstream running times or factors will get direct value from it. It deserves a serious referee because the improvement is measurable and the supporting construction is there.

Referee Report

0 major / 2 minor

Summary. The manuscript presents the first polynomial-time algorithm for computing a near-optimal flow-expander decomposition. Given an undirected graph G and parameter φ, the algorithm removes at most a φ log^{1+o(1)} n fraction of edges such that every remaining connected component is a φ-flow-expander (a stronger notion than φ-cut-expander). This yields an overhead of log^{1+o(1)} n, nearly matching the Ω(log n) graph-theoretic lower bound that holds even for the weaker cut-expander case, and improves on prior polynomial-time algorithms that incurred O(φ log^{1.5} n) or O(φ log^2 n) overhead.

Significance. If the claimed overhead and correctness hold, the result is a substantial advance in the theory of expander decompositions, which underpin numerous algorithmic applications including clustering, sparsification, and flow computations. The near-optimal overhead (up to log^{o(1)} n) for the stronger flow-expander guarantee, achieved in polynomial time, closes much of the gap to the known lower bound and should enable tighter analyses in downstream results that invoke expander decompositions.

minor comments (2)

[Abstract] Abstract and §1: the precise dependence of the o(1) term on φ and n (e.g., whether it is 1/log log n or similar) should be stated explicitly in the theorem statement and introduction to allow immediate comparison with the Ω(log n) lower bound.
[Introduction] The manuscript should include a short comparison table (perhaps in §1 or §2) contrasting the new overhead with the two cited prior bounds (log^{1.5} n and log^2 n) for both cut- and flow-expanders.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive review and for recommending acceptance. We appreciate the recognition that the result provides a substantial advance by achieving near-optimal overhead for flow-expander decompositions in polynomial time.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a direct algorithmic construction and analysis for φ-flow-expander decomposition achieving explicit overhead φ log^{1+o(1)} n. The bound is derived from recursive decomposition steps and flow certification that are independent of the final result; no equations reduce by construction to fitted inputs, no load-bearing self-citations collapse the claim, and the overhead is not a renaming of a known pattern. The derivation remains self-contained against the stated lower bound.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard undirected-graph model assumptions and the known Ω(log n) lower bound for cut-expander decompositions; no free parameters, new axioms, or invented entities are introduced in the abstract.

axioms (1)

domain assumption Undirected graphs admit flow-expander decompositions with the stated overhead under standard connectivity and expansion definitions.
Invoked implicitly when claiming the algorithm computes such a decomposition for any input graph and phi.

pith-pipeline@v0.9.0 · 5463 in / 1256 out tokens · 30394 ms · 2026-05-15T21:38:18.061495+00:00 · methodology