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

2026-05-22

AI Summary

While existing partition oracles efficiently generate locally hyperfinite decompositions, they require random seeds of length Ω(n), incurring linear preprocessing overhead that limits practical applicability. This work addresses bounded-degree graph families excluding a fixed subgraph and presents the first construction reducing the shared random seed length to poly(d/ε)·log n while maintaining query complexity poly(d/ε). Building upon the framework of Kumar–Seshadhri–Stolman, we combine refined randomness complexity analysis with information-theoretic lower-bound techniques to establish an ω_N(1) lower bound on randomness in the generalized model where vertex labels are drawn from [N]. This result establishes the theoretical limit on the amount of randomness required in this setting.

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 $Ω(n)$, which technically implies a linear setup time. Any blackbox derandomization would likely need $Ω(\log^2n)$ 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 $ω_N(1)$ random bits.

Problem

• partition oracle
• randomness complexity
• bounded-degree graphs
• minor-free graphs
• hyperfinite decomposition

Innovation

• partition oracle
• randomness complexity
• hyperfinite decomposition
• minor-free graphs
• derandomization