[Paper Review] Representability of Matroids by c-Arrangements is Undecidable

Lukas Kühne, Geva Yashfe|arXiv (Cornell University)|Dec 12, 2019

graph theory and CDMA systems4 citations

TL;DR

This paper proves that determining whether a given matroid can be represented by a c-arrangement—where subspaces of dimension c satisfy a dimension-multiplicity condition—is algorithmically undecidable for any field. The authors reduce the uniform word problem for finite groups to c-arrangement representability using generalized Dowling geometries, establishing that multilinear matroid representability is undecidable, with implications for network coding and secret sharing.

ABSTRACT

For a natural number $c$, a $c$-arrangement is an arrangement of dimension $c$ subspaces satisfying the following condition: the sum of any subset of the subspaces has dimension a multiple of $c$. Matroids arising as normalized rank functions of $c$-arrangements are also known as multilinear matroids. We prove that it is algorithmically undecidable whether there exists a $c$ such that a given matroid has a $c$-arrangement representation, or equivalently whether the matroid is multilinear. It follows that certain network coding problems are also undecidable. In the proof, we introduce a generalized Dowling geometry to encode an instance of the uniform word problem for finite groups in matroids of rank three. The $c$-arrangement condition gives rise to some difficulties and their resolution is the main part of the paper.

Motivation & Objective

• To resolve the long-open question of whether multilinear matroid representability is decidable.
• To establish a connection between c-arrangement representability and the uniform word problem for finite groups (UWPFG).
• To demonstrate that the existence of a c-arrangement representation for a matroid is undecidable, even over algebraically closed fields.
• To extend Dowling geometry constructions to encode group presentations in matroids of rank three.
• To show that the undecidability of UWPFG implies the undecidability of c-arrangement representability via a computable reduction.

Proposed method

• Construct a generalized Dowling geometry from a finite group presentation ⟨S∣R⟩ to encode group elements and relations into a rank-3 matroid.
• Define a triangle matroid N_{S,R} with a distinguished basis and elements corresponding to group words, using a construction from 4.3.
• Introduce a polymatroid g derived from N_{S,R} via Theorem 7.6, which captures the structure of the group presentation.
• Construct a double arrangement W from a well-separated extension of g, using a c-basis to ensure separation of elements.
• Use Malcev’s theorem to pass from infinite matrix groups to finite quotients, preserving non-equality of representations.
• Apply a reduction from the UWPFG: a matroid has a weak c-representation with A_x ≠ A_y if and only if the group word is non-trivial, linking to c-arrangement representability.

Experimental results

Research questions

• RQ1 Is there an algorithm to decide whether a given matroid admits a c-arrangement representation over any field?
• RQ2 Can the uniform word problem for finite groups be reduced to the c-arrangement representability problem for matroids?
• RQ3 Does the existence of a weak c-representation with distinct subspaces for two elements imply the existence of a c-arrangement for an expanded matroid?
• RQ4 Is multilinear matroid representability strictly more expressive than 1-arrangement representability, and is it decidable?
• RQ5 What is the computational complexity of determining whether a network coding instance admits a linear solution, given its associated matroid?

Key findings

• The multilinear representability problem for matroids is undecidable over any field, including algebraically closed fields.
• The undecidability arises from a computable reduction from the uniform word problem for finite groups (UWPFG), which is known to be undecidable by Slobodskoi’s theorem.
• A matroid N_{S,R} constructed from a group presentation ⟨S∣R⟩ has a weak c-representation with A_x ≠ A_y if and only if the word w is non-trivial in the group.
• If such a weak c-representation exists, then some expansion of 2g (where g is the polymatroid from N_{S,R}) admits a c-arrangement representation that separates x and y.
• The existence of a c-arrangement representation for any of finitely many associated matroids M_i is equivalent to the UWPFG instance having a negative answer.
• The result implies that deciding whether a network coding problem has a linear vector solution is also undecidable, due to the known equivalence with multilinear representability.