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The Minimum Linear Ordering Problem for submodular functions admits a tight O(sqrt(n/ln n)) approximation in the value oracle model.

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Establishes tight O(sqrt(n/ln n)) approximation and matching hardness for the minimum linear ordering problem with arbitrary submodular functions.

T0 review reviewed 2026-06-26 challenge →

load-bearing objection This paper gives the first tight O(sqrt(n/ln n)) approximation and matching information-theoretic hardness for general submodular linear ordering.

arxiv 2606.20202 v1 pith:TOLGVN5Z submitted 2026-06-18 cs.DS

Tight Algorithm and Hardness for Submodular Linear Ordering

Evan Abboud , Roy Schwartz This is my paper
classification cs.DS
keywords submodular functionslinear orderingapproximation algorithmshardness of approximationvalue oracle modelminimum linear arrangementprefix sums
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

The paper considers the problem of ordering a ground set of size n to minimize the sum of a non-negative submodular function f evaluated on all prefixes of the ordering. It supplies a polynomial-time algorithm that returns an ordering whose cost is at most O(sqrt(n/ln n)) times the optimum. It also proves that any algorithm allowed only polynomially many direct evaluations of f cannot achieve an asymptotically better ratio. This pair of results pins down the approximability of the problem exactly, improving on a prior constant-factor hardness and on an approximation known only for the symmetric special case.

Core claim

We resolve the approximability of the Minimum Linear Ordering Problem for a general submodular f by establishing matching upper and lower bounds and present: (1) a polynomial-time algorithm achieving an O(sqrt(n/ln n))-approximation; and (2) a matching information-theoretic hardness result, showing that no algorithm evaluating f a polynomial number of times can achieve an o(sqrt(n/ln n))-approximation.

What carries the argument

The value-oracle access model for the submodular function f together with a new polynomial-time ordering procedure that produces the stated approximation ratio.

Load-bearing premise

Any algorithm is restricted to learning the submodular function only through polynomially many direct evaluations.

What would settle it

An algorithm that makes only polynomially many evaluations of f and returns an ordering whose cost is o(sqrt(n/ln n)) times the optimum on every instance would falsify the hardness claim.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The approximability gap for general submodular linear ordering is closed in the value-oracle model.
  • The same O(sqrt(n/ln n)) guarantee applies to every special case previously studied, including Minimum Linear Arrangement.
  • No polynomial-query algorithm can improve the ratio by more than a constant factor.
  • The earlier 2-hardness result is strengthened to a tight super-constant bound.

Where Pith is reading between the lines

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

  • If an algorithm were allowed arbitrary non-oracle access to f, the information-theoretic barrier would no longer apply.
  • The result suggests that submodular ordering problems may require query complexity that grows with n to beat the sqrt(n) barrier.
  • Similar tight bounds may hold for other ordering objectives that reduce to prefix-sum minimization under submodularity.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The paper considers the Minimum Linear Ordering Problem for a non-negative submodular set function f on a ground set of size n. It claims to resolve the approximability gap by giving a polynomial-time O(√(n / ln n))-approximation algorithm and a matching information-theoretic hardness result: no algorithm making polynomially many queries to an oracle for f can achieve an o(√(n / ln n))-approximation. The result improves on a prior constant-factor hardness of 2 and extends an O(√(n / ln n)) guarantee that was previously known only for symmetric submodular functions.

Significance. If the stated bounds hold, the work closes a long-standing gap in submodular optimization by establishing asymptotically tight approximability for a problem that generalizes Minimum Linear Arrangement and Minimum Containing Interval Graph. The combination of a new polynomial-time algorithm with a matching query-complexity lower bound is a notable strength; the information-theoretic nature of the hardness makes the result robust within the standard value-oracle model.

minor comments (3)
  1. The abstract and introduction should explicitly state the precise query model (value oracle with polynomially many calls) in the hardness statement to avoid any ambiguity about the scope of the lower bound.
  2. Notation for the ordering π and the prefix sums should be introduced once and used consistently; a short table of notation would improve readability in the technical sections.
  3. The running-time analysis of the algorithm should include the exact number of oracle calls as a function of n, even if it is polynomial.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. The report correctly captures our main results on the tight O(√(n/ln n)) approximation and matching information-theoretic hardness for the Minimum Linear Ordering Problem under general non-negative submodular functions.

Circularity Check

0 steps flagged

No significant circularity; bounds are independently established

full rationale

The paper presents a new polynomial-time algorithm achieving the O(√(n/ln n)) approximation and a separate information-theoretic hardness proof showing that o(√(n/ln n)) is impossible under polynomially many oracle queries to f. These are matching but not circular: the algorithm is an explicit construction whose query cost matches the hardness model, while the lower bound is proven directly in the query model without reducing to the algorithm or to any self-citation chain. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the stated claims or abstract. The result is self-contained against the stated query-model benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard assumptions of theoretical computer science rather than new free parameters or invented entities.

axioms (1)
  • domain assumption Algorithms have access to the set function f only via direct evaluations, and the hardness holds in the polynomial-query information-theoretic model.
    Explicitly stated in the abstract as the model for the hardness result.

reviewed 2026-06-26 · how reviews work

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Cite this review

Pith. "Pith review of Tight Algorithm and Hardness for Submodular Linear Ordering." pith.science (2026). https://pith.science/paper/TOLGVN5Z

@misc{pith2026260620202,
  author       = {Pith},
  title        = {Pith review of: Tight Algorithm and Hardness for Submodular Linear Ordering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TOLGVN5Z}},
  note         = {Machine review of arXiv:2606.20202}
}
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abstract

We consider the Minimum Linear Ordering Problem: given a ground set $N$ of cardinality $n$ and a non-negative set function $f\colon 2^N\rightarrow \mathbb{R}_{\geq 0}$, the goal is to find an ordering $\pi$ of $N$ that minimizes the sum of the values of $f$ over all prefixes of $\pi$. This problem has been studied for various classes of set functions, and the case of a submodular $f$ is of special interest, as it captures classic problems including Minimum Linear Arrangement and Minimum Containing Interval Graph. In this work, we resolve the approximability of the Minimum Linear Ordering Problem for a general submodular $f$ by establishing matching upper and lower bounds and present: $(1)$ a polynomial-time algorithm achieving an $O(\sqrt{n/\ln n})$-approximation; and $(2)$ a matching information-theoretic hardness result, showing that no algorithm evaluating $f$ a polynomial number of times can achieve an $o(\sqrt{n/\ln n})$-approximation. Previously, the best known hardness of approximation was $2$, and an $O(\sqrt{n/\ln n})$-approximation was known only for the special case where $f$ is both submodular and symmetric.

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This paper was first reviewed by grok-4.3 on June 26, 2026.