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arxiv: 2605.23509 · v1 · pith:VN5DPI4Cnew · submitted 2026-05-22 · 💻 cs.DS

Reducing the Randomness in Partition Oracles for Bounded Degree Minor-Free Graphs

Akash Kumar , Abhiruk Lahiri , C. Seshadhri This is my paper

Pith reviewed 2026-05-25 02:49 UTC · model grok-4.3

classification 💻 cs.DS
keywords partition oraclehyperfinite decompositionminor-free graphsrandomness complexitybounded degree graphslocal computationderandomization
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The pith

Partition oracles for bounded-degree minor-free graphs can be implemented with a random seed of length poly(d ε^{-1}) log n.

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

The paper shows that the poly(dε^{-1})-query partition oracles from Kumar-Seshadhri-Stolman, which give local access to a hyperfinite decomposition of a bounded-degree minor-closed graph, can be realized using only poly(dε^{-1}) log n random bits instead of Ω(n) bits. Coordination among queries happens entirely through this shared seed without any preprocessing of the graph. A reader cares because the smaller seed removes the linear setup cost that previously followed from large randomness and brings the randomness requirement closer to polylogarithmic. The paper also proves that, when vertices carry labels from a universe of size N at least n, every partition oracle for even the simplest case of cycles must use ω_N(1) random bits.

Core claim

The poly(dε^{-1})-query partition oracles of Kumar-Seshadhri-Stolman for bounded-degree minor-free graphs can be implemented with a random seed of poly(dε^{-1}) · log n length. In a generalized model where vertex labels come from the universe [N] with N ≥ n, any partition oracle even for cycles requires ω_N(1) random bits.

What carries the argument

The shared random seed that coordinates all local queries to produce a consistent hyperfinite decomposition without preprocessing.

If this is right

  • The effective preprocessing time for the partition oracle drops from linear in n to polylog n.
  • Local access to the decomposition can be supplied in sublinear total time when the seed is generated on the fly.
  • The randomness requirement is now low enough that limited-independence hash families become plausible replacements for full randomness.
  • Black-box derandomization techniques are no longer required to reach polylogarithmic seed length.

Where Pith is reading between the lines

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

  • The same seed-shortening technique may apply to other local computation algorithms that rely on a shared random seed for consistency across distant queries.
  • The generalized-model lower bound indicates that label-dependent randomness is sometimes necessary, suggesting that fully label-independent oracles may require different constructions.
  • If the reduced seed can be generated from O(log log n) bits via further derandomization, the entire oracle could run in truly sub-polylogarithmic time per query.

Load-bearing premise

The only coordination among local queries comes from the shared random seed, and shortening that seed leaves the underlying hyperfinite decomposition and query complexity unchanged.

What would settle it

A concrete counterexample graph from a minor-closed family together with an explicit poly(dε^{-1}) log n-bit seed that produces an invalid ε-hyperfinite decomposition with non-negligible probability.

read the original abstract

Consider a bounded-degree graph $G$ that belongs to a minor-closed family (such as planar graphs). Such a graph has a hyperfinite decomposition, wherein, for a sufficiently small $\varepsilon > 0$, one can remove $\varepsilon dn$ edges to obtain connected components of size independent of $n$. (As usual, $n$ is the number of vertices and $d$ is the degree bound.) In a seminal result, Hassidim-Kelner-Nguyen-Onak (FOCS 2009) introduced the partition oracle, a procedure that provides local access to a hyperfinite decomposition. The partition oracle computes the component containing an input vertex $v$ with query complexity (to $G$) independent of $n$. Remarkably, this is done without any preprocessing on $G$. The coordination is done purely through a shared random seed. Despite a line of work on optimizing the query complexity of partition oracles, there were no attempts to bound the size of the random seed. All existing partition oracles use a random seed of size $\Omega(n)$, which technically implies a linear setup time. Any blackbox derandomization would likely need $\Omega(\log^2n)$ uniform random bits. A natural question is whether the random seed can also have length independent of $n$. We prove the $poly(d\varepsilon^{-1})$-query partition oracles of Kumar-Seshadhri-Stolman can be implemented with a random seed of $poly(d\varepsilon^{-1}) \cdot \log n$ length. To get a deeper understanding on the randomness complexity, we consider a more general model where the vertex labels come from the universe $[N]$, where $N \geq n$. In this setting, we prove that any partition oracle even for cycles requires $\omega_N(1)$ random bits.

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

2 major / 1 minor

Summary. The paper claims that the poly(d ε^{-1})-query partition oracles of Kumar-Seshadhri-Stolman for bounded-degree minor-free graphs can be implemented with a random seed of length poly(d ε^{-1}) · log n (instead of Ω(n)), while preserving query complexity. It further proves that, in the generalized model with vertex labels from [N] ≥ n, any partition oracle even for cycles requires ω_N(1) random bits.

Significance. If correct, the result improves the randomness complexity of partition oracles from linear in n to polylogarithmic, addressing an open question on seed length for these local algorithms on minor-closed families. The lower bound provides a matching-style insight into randomness necessity. The work builds directly on the KSS construction and gives credit to the original hyperfinite decomposition approach.

major comments (2)
  1. [Abstract] Abstract (paragraph 3): the central claim that the KSS local decisions can be coordinated via a shared seed of length only poly(d ε^{-1}) log n assumes that the high-probability guarantee (all components of size O(1/ε)) is preserved under the reduced randomness. No details are supplied on the independence degree or PRG used, so it is impossible to check whether the original Chernoff/union-bound analysis still holds or whether correlated bad events across distant vertices can occur with probability ≥ ε.
  2. [Abstract] Abstract: the soundness of the derandomization cannot be assessed because the manuscript provides no equations, reductions, or proof outline showing how the original Ω(n)-bit construction is replaced while keeping query complexity poly(d ε^{-1}) and the hyperfiniteness probability 1-o(1).
minor comments (1)
  1. The lower-bound statement for cycles could usefully include a brief comparison to the upper bound achieved for minor-free graphs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their comments on the abstract. We address each major comment below and will revise accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph 3): the central claim that the KSS local decisions can be coordinated via a shared seed of length only poly(d ε^{-1}) log n assumes that the high-probability guarantee (all components of size O(1/ε)) is preserved under the reduced randomness. No details are supplied on the independence degree or PRG used, so it is impossible to check whether the original Chernoff/union-bound analysis still holds or whether correlated bad events across distant vertices can occur with probability ≥ ε.

    Authors: We agree that the abstract provides no information on the independence degree or PRG. In the revision we will add a sentence stating that the construction replaces independent randomness with a pairwise-independent hash function of seed length O(log n); because each vertex's component depends only on a local neighborhood of size O(1/ε), the original union-bound analysis carries over and the failure probability remains o(1). revision: yes

  2. Referee: [Abstract] Abstract: the soundness of the derandomization cannot be assessed because the manuscript provides no equations, reductions, or proof outline showing how the original Ω(n)-bit construction is replaced while keeping query complexity poly(d ε^{-1}) and the hyperfiniteness probability 1-o(1).

    Authors: We acknowledge that the abstract contains no equations or outline of the reduction. We will expand the abstract with a one-sentence sketch indicating that the KSS procedure is simulated by generating the required random bits from a limited-independence source while preserving both the query complexity and the 1-o(1) hyperfiniteness probability. revision: yes

Circularity Check

0 steps flagged

No significant circularity; result is a new proof on top of prior external construction

full rationale

The paper claims a new upper bound on random seed length for the existing poly(dε^{-1})-query partition oracles from the prior Kumar-Seshadhri-Stolman work, plus a separate lower bound of ω_N(1) bits for cycles. No equations, definitions, or reductions in the provided text show the claimed seed length being fitted from or defined in terms of the result itself, nor does any load-bearing step reduce by construction to a self-citation whose verification depends on the current paper. The coordination via shared seed is treated as an external property of the KSS construction that is then analyzed for shorter seeds; the derivation chain remains independent of the target claim.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of hyperfinite decompositions for minor-closed bounded-degree graphs and on the correctness of the Kumar-Seshadhri-Stolman partition oracle construction; both are treated as given background.

free parameters (2)
  • d
    Degree bound supplied as input; controls query and seed complexity.
  • ε
    Error parameter supplied as input; controls component size and seed complexity.
axioms (2)
  • domain assumption Bounded-degree graphs in minor-closed families admit hyperfinite decompositions (remove ε d n edges to obtain components of size independent of n).
    Invoked in the first paragraph of the abstract as the structural fact enabling partition oracles.
  • domain assumption The Kumar-Seshadhri-Stolman construction yields a correct poly(d ε^{-1})-query partition oracle.
    The positive result is stated as an implementation of that prior construction.

pith-pipeline@v0.9.0 · 5877 in / 1501 out tokens · 22618 ms · 2026-05-25T02:49:26.212572+00:00 · methodology

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