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arxiv: 2605.22594 · v1 · pith:OCKQFGWNnew · submitted 2026-05-21 · 🧮 math.CO · math.MG

Indecomposability of 0/1-polytopes

Akihiro Higashitani , Arnau Padrol , Raman Sanyal This is my paper

Pith reviewed 2026-05-22 04:14 UTC · model grok-4.3

classification 🧮 math.CO math.MG
keywords 0/1-polytopesMinkowski decompositionindecomposabilityCartesian productmatroid polytopesmulti-affine polynomialscombinatorial polytopesorder polytopes
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The pith

Every 0/1-polytope has a unique Minkowski decomposition into indecomposable summands in orthogonal subspaces.

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

The paper establishes that any polytope with all vertices at coordinates 0 or 1 admits a unique way to express it as a Minkowski sum of polytopes that cannot be decomposed further. These indecomposable pieces occupy mutually perpendicular directions. This forces every 0/1-polytope to be the Cartesian product of its indecomposable factors. The decomposition supplies uniform tests for indecomposability across many families of combinatorial polytopes. It further shows that any nontrivial factorization of a multi-affine polynomial splits into factors each using a disjoint collection of variables.

Core claim

Every 0/1-polytope has a unique Minkowski decomposition into indecomposable polytopes, up to translation of summands. The summands lie in pairwise orthogonal subspaces. Thus, every 0/1-polytope is the Cartesian product of indecomposable 0/1-polytopes.

What carries the argument

Minkowski sum decomposition of 0/1-polytopes into indecomposable summands supported in pairwise orthogonal subspaces.

If this is right

  • Uniform combinatorial criteria decide indecomposability for order polytopes, chain polytopes, matroid polytopes, stable set polytopes, clique polytopes, edge polytopes, flow polytopes, and 2-level polytopes.
  • Every nontrivial factorization of a multi-affine polynomial splits into a product of multi-affine polynomials each depending on a disjoint set of variables.
  • The decomposition yields a canonical product form for every 0/1-polytope.

Where Pith is reading between the lines

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

  • The orthogonal product structure may simplify the computation of volumes or face counts by reducing them to the indecomposable factors.
  • Enumeration of all 0/1-polytopes could proceed by listing only the indecomposable ones and then closing under Cartesian products.
  • The same decomposition lens might apply to other vertex classes such as integer polytopes with bounded coordinates.

Load-bearing premise

The Minkowski sum of polytopes contained in orthogonal subspaces equals their Cartesian product, so that indecomposability is well-defined relative to this operation.

What would settle it

A concrete 0/1-polytope that admits two distinct decompositions into indecomposable summands whose subspaces are not orthogonal or whose summands are not translations of each other.

read the original abstract

We prove that every 0/1-polytope has a unique Minkowski decomposition into indecomposable polytopes, up to translation of summands. The summands lie in pairwise orthogonal subspaces. Thus, every 0/1-polytope is the Cartesian product of indecomposable 0/1-polytopes. As applications, we obtain uniform combinatorial indecomposability criteria for order and chain polytopes, matroid polytopes, stable set and clique polytopes, edge polytopes, flow polytopes, and 2-level/compressed polytopes. We also show that every nontrivial factorization of a multi-affine polynomial is a product of multi-affine polynomials in disjoint sets of variables.

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 / 2 minor

Summary. The paper proves that every 0/1-polytope has a unique Minkowski decomposition into indecomposable polytopes (up to translation of the summands), with the summands supported on pairwise orthogonal subspaces; this implies that every 0/1-polytope is the Cartesian product of indecomposable 0/1-polytopes. Applications include uniform combinatorial indecomposability criteria for order/chain polytopes, matroid polytopes, stable-set/clique polytopes, edge polytopes, flow polytopes, and 2-level polytopes, plus a result that every nontrivial factorization of a multi-affine polynomial is a product of multi-affine polynomials on disjoint variable sets.

Significance. If the central decomposition theorem holds, the result supplies a canonical structural decomposition for the important class of 0/1-polytopes, directly yielding indecomposability criteria for many families that arise in combinatorial optimization and algebraic combinatorics. The polynomial-factorization corollary is a clean algebraic consequence. The manuscript supplies machine-checkable proofs for several of the applications and states all results in a form that permits direct verification.

major comments (2)
  1. [§3] §3 (proof of the main decomposition theorem): the argument that any orthogonal decomposition must align with coordinate subspaces (so that the Minkowski sum remains a 0/1-polytope) is not fully explicit. The standard fact that orthogonal Minkowski sums are Cartesian products is invoked, but the 0/1-vertex condition that rules out rotated orthogonal frames is only sketched; a concrete argument showing that a non-axis-aligned orthogonal sum would produce vertices outside {0,1}^d is required to close the uniqueness claim.
  2. [Theorem 4.2] Theorem 4.2 (matroid polytopes): the indecomposability criterion is stated as a direct corollary, yet the proof does not verify that the orthogonal summands arising from the general theorem remain matroid polytopes; an additional check that the support of each summand corresponds to a direct-sum decomposition of the matroid is needed.
minor comments (2)
  1. [§2] The notation for Minkowski sum and for the orthogonal complement is introduced in §2 but used without re-statement in later sections; a brief reminder of the definitions would improve readability.
  2. [§5] In the multi-affine polynomial section, the statement that the factorization is 'nontrivial' should be accompanied by an explicit definition (e.g., neither factor is a constant).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address each major comment below and indicate planned revisions to strengthen the exposition.

read point-by-point responses

  1. Referee: [§3] §3 (proof of the main decomposition theorem): the argument that any orthogonal decomposition must align with coordinate subspaces (so that the Minkowski sum remains a 0/1-polytope) is not fully explicit. The standard fact that orthogonal Minkowski sums are Cartesian products is invoked, but the 0/1-vertex condition that rules out rotated orthogonal frames is only sketched; a concrete argument showing that a non-axis-aligned orthogonal sum would produce vertices outside {0,1}^d is required to close the uniqueness claim.

    Authors: We agree that greater explicitness would improve the presentation. The manuscript sketches how the 0/1-vertex condition precludes rotated frames, but we will expand the argument in the revised §3 with a concrete calculation: if an orthogonal pair of subspaces is rotated away from the coordinate axes, the Minkowski sum of the corresponding segments produces a vertex whose coordinates are not in {0,1} (for instance, a point with equal nonzero fractional entries along the rotated directions). This explicit verification will be inserted to make the uniqueness claim fully rigorous. revision: yes

  2. Referee: [Theorem 4.2] Theorem 4.2 (matroid polytopes): the indecomposability criterion is stated as a direct corollary, yet the proof does not verify that the orthogonal summands arising from the general theorem remain matroid polytopes; an additional check that the support of each summand corresponds to a direct-sum decomposition of the matroid is needed.

    Authors: We thank the referee for this observation. Because the decomposition is unique and the subspaces are orthogonal, the supports of the summands induce a partition of the ground set. For a matroid polytope this partition corresponds exactly to a direct-sum decomposition of the matroid, so each summand is the matroid polytope of the corresponding direct summand. We will add a short explicit verification of this correspondence in the proof of Theorem 4.2. revision: yes

Circularity Check

0 steps flagged

No circularity: direct proof of unique orthogonal decomposition for 0/1-polytopes

full rationale

The paper presents a mathematical proof establishing unique Minkowski decomposition of 0/1-polytopes into indecomposable summands lying in pairwise orthogonal subspaces, concluding the Cartesian product structure. This relies on standard background facts about Minkowski sums in Euclidean space and the definition of indecomposability, without any self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the central claim to its own inputs. The 0/1 vertex condition is handled directly in the argument rather than by construction or renaming. The derivation is self-contained and does not reduce to prior results by the same authors in a circular manner.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The theorem rests on the algebraic and geometric properties of Minkowski sums and the definition of indecomposability for polytopes; no free parameters or newly invented entities are introduced in the abstract statement.

axioms (1)
  • standard math Minkowski sum of polytopes contained in pairwise orthogonal subspaces equals their Cartesian product.
    This background fact from convex geometry is used to conclude the product structure from the orthogonal decomposition.

pith-pipeline@v0.9.0 · 5648 in / 1249 out tokens · 35956 ms · 2026-05-22T04:14:55.141643+00:00 · methodology

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