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arxiv: 2606.24845 · v1 · pith:ZEVYBYXMnew · submitted 2026-06-23 · 💻 cs.DS · cs.CC

A Near-Optimal Parallel Algorithm for Finding Matroid Bases

Sanjeev Khanna , Aaron Putterman , Junkai Song This is my paper

Pith reviewed 2026-06-25 22:22 UTC · model grok-4.3

classification 💻 cs.DS cs.CC
keywords parallel algorithmsmatroid basisround complexityindependence oraclecombinatorial optimizationparallel complexity
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The pith

Parallel algorithm computes matroid basis in O(n^{1/3} log^{1/3} n) rounds.

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

The paper gives a parallel algorithm that finds a basis of any matroid in O(n^{1/3} log^{1/3} n) rounds. This bound nearly matches the Omega(n^{1/3}) lower bound shown by Karp, Upfal and Wigderson in 1985. The result resolves the open question on the parallel complexity of matroid basis computation. A reader would care because matroids model many optimization tasks and parallel round bounds directly affect scalability on large instances.

Core claim

We settle the classic question of the parallel complexity of computing a matroid basis by giving an algorithm that runs in O(n^{1/3} log^{1/3} n) rounds, matching the KUW lower bound up to a log^{2/3}(n) factor.

What carries the argument

The parallel algorithm that finds a maximum independent set using an independence oracle in the stated round bound.

If this is right

  • Matroid basis computation is now possible in near-optimal parallel rounds.
  • The logarithmic gap to the lower bound is the only remaining separation.
  • Many matroid-representable problems inherit improved parallel complexity.
  • The algorithm works for any matroid given by an independence oracle.

Where Pith is reading between the lines

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

  • The same round bound may extend to matroid intersection when oracles are available.
  • Graphic matroids (spanning-tree problems) would immediately gain the same parallel speedup.
  • Closing the remaining log^{2/3} n factor would require a different technique.

Load-bearing premise

The cost of parallel independence-oracle calls is fully absorbed into the round count without extra logarithmic overhead from the model.

What would settle it

An explicit matroid instance on which the given algorithm requires more than O(n^{1/3} log^{1/3} n) rounds to output a basis.

read the original abstract

We settle the classic question of the parallel complexity of computing a matroid basis, as first posed in the seminal work of Karp, Upfal, and Wigderson (FOCS 1985, JCSS 1988). Our algorithm runs in $O(n^{1/3}\log^{1/3}n)$ rounds, matching the lower bound of KUW up to a $\log^{2/3}(n)$ factor.

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

1 major / 0 minor

Summary. The manuscript claims to settle the classic question on the parallel complexity of computing a matroid basis (posed by Karp, Upfal, and Wigderson in 1985) by giving an explicit algorithm that runs in O(n^{1/3} log^{1/3} n) rounds and matches the KUW lower bound up to a log^{2/3} n factor.

Significance. If correct, the result would be a significant contribution to parallel algorithms and matroid theory by nearly closing a long-standing gap between upper and lower bounds on round complexity.

major comments (1)
  1. [Abstract] Abstract: the stated O(n^{1/3} log^{1/3} n) round bound is presented with no derivation, proof sketch, model definition (PRAM/BSP/etc.), or protocol for batched independence-oracle access. This is load-bearing for the central near-optimality claim, because any unaccounted per-batch synchronization or communication latency would increase the bound by an extra log n factor and invalidate the 'up to log^{2/3} n' comparison to KUW.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful comments. Below we address the concern raised about the abstract.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the stated O(n^{1/3} log^{1/3} n) round bound is presented with no derivation, proof sketch, model definition (PRAM/BSP/etc.), or protocol for batched independence-oracle access. This is load-bearing for the central near-optimality claim, because any unaccounted per-batch synchronization or communication latency would increase the bound by an extra log n factor and invalidate the 'up to log^{2/3} n' comparison to KUW.

    Authors: We respectfully disagree that the abstract requires a derivation or proof sketch, as abstracts are summaries by nature. The model is the standard one from KUW: the parallel random access machine (PRAM) model with access to a batched independence oracle, where each round allows an arbitrary number of parallel independence queries. The protocol for batched access is described in the algorithm pseudocode and analysis in Sections 3 and 4, where we prove that the total number of rounds is O(n^{1/3} log^{1/3} n) with the logarithmic factors arising only from the algorithm's structure, not from additional synchronization. The comparison to the KUW lower bound is valid because it is in the identical model. If the editor prefers, we can append a parenthetical note on the model to the abstract in the revised version. revision: partial

Circularity Check

0 steps flagged

No significant circularity; explicit algorithmic construction

full rationale

The paper presents a direct algorithmic construction achieving O(n^{1/3} log^{1/3} n) rounds for matroid basis computation, with the bound positioned as matching an external KUW lower bound up to a logarithmic factor. No equations, fitted parameters, self-citations, or ansatzes appear in the abstract or description that reduce the claimed result to its own inputs by construction. The derivation relies on an independence oracle model whose costs are stated to be absorbed, but this is an explicit modeling choice rather than a self-referential reduction. The work is self-contained against external benchmarks with no load-bearing self-citation chains or renamings of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; ledger left empty due to lack of technical content.

pith-pipeline@v0.9.1-grok · 5595 in / 1081 out tokens · 32269 ms · 2026-06-25T22:22:16.280746+00:00 · methodology

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