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Measuring Depth of Matroids

Jakub Balabán, Petr Hliněný, Jan Jedelský, Kristýna Pekárková

Abstract

Motivated by recently discovered connections between matroid depth measures and block-structured integer programming [ICALP 2020, 2022], we undertake a systematic study of recursive depth parameters for matrices and matroids, aiming to unify recently introduced and scattered concepts. We propose a general framework that naturally yields eight different depth measures for matroids, prove their fundamental properties and relationships, and relate them to two established notions in the field: matroid branch-depth and a newly introduced natural depth counterpart of matroid tree-width. In particular, we show that six of our eight measures are mutually functionally inequivalent, and among these, one is functionally equivalent to matroid branch-depth and another to matroid tree-depth. Importantly, we also prove that these depth measures coincide on matroids and on matrices over any field, which is (somehow surprisingly) not a trivial task. Finally, we provide a comparison between the matroid parameters and classical depth measures of graphs.

Measuring Depth of Matroids

Abstract

Motivated by recently discovered connections between matroid depth measures and block-structured integer programming [ICALP 2020, 2022], we undertake a systematic study of recursive depth parameters for matrices and matroids, aiming to unify recently introduced and scattered concepts. We propose a general framework that naturally yields eight different depth measures for matroids, prove their fundamental properties and relationships, and relate them to two established notions in the field: matroid branch-depth and a newly introduced natural depth counterpart of matroid tree-width. In particular, we show that six of our eight measures are mutually functionally inequivalent, and among these, one is functionally equivalent to matroid branch-depth and another to matroid tree-depth. Importantly, we also prove that these depth measures coincide on matroids and on matrices over any field, which is (somehow surprisingly) not a trivial task. Finally, we provide a comparison between the matroid parameters and classical depth measures of graphs.

Paper Structure

This paper contains 22 sections, 51 theorems, 4 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Basic terminology and notation
  3. Graphs
  4. Matrices
  5. Matroids
  6. Technical overview
  7. Matroid depth and width parameters
  8. Branch-width and branch-depth
  9. Tree-width and tree-depth
  10. The family of "recursive" depth measures
  11. Connection to Integer Programming
  12. Basic parameter properties
  13. Understanding c$^*$-depth and contraction$^*$-depth
  14. Comparing the parameters
  15. Relation to matroid tree-depth
  16. ...and 7 more sections

Key Result

Theorem 2

For every matrix ${\hbox{\boldmath$\bf A$}}$ it holds that

Figures (3)

  • Figure 1: The Fano matroid $F_7$ -- the projective plane over the binary field (its $7$ lines are the six line segments and the central cycle), which is not representable over fields whose characteristic is different from $2$.
  • Figure 2: A comparison of the considered matroid depth parameters in \ref{['thm:depthcompar']}. See \ref{['sec:terminology']} for the definitions of functional comparison $p \mathop{\mathrm{\sim_{\rm f}}}\nolimits p'$ ("equal") and $p \mathop{\mathrm{\le_{\rm f}}}\nolimits p'$ ("at most") between measures. An arrow connection $p\to_{\rm f} q$ in the picture means functional "strictly less", i.e., $q \mathop{\mathrm{\le_{\rm f}}}\nolimits p$ but $p \mathop{\mathrm{\nleq_{\rm f}}}\nolimits q$. Moreover, c-depth and d-depth are dual to each other, but they are functionally incomparable both ways, as well as c$^*$d-depth and cd$^*$-depth.
  • Figure 3: Left: The "fat cycle" $C_{6, 5}$. Right: The graph $D_{5, 5}$. Crucially, by contracting a single edge, we obtain $C_{5, 5}$.

Theorems & Definitions (68)

  • Definition 1: \ref{['def:eight-parameters']}
  • Theorem 2: ChaCKKP22 and BriKKPS24
  • Definition 3
  • Theorem 4: \ref{['cor:csdmatrixeq']} and \ref{['cor:csxxmatrixeq']}
  • Theorem 5: \ref{['cor:xd_circuits']}, \ref{['lem:starredineq']}, \ref{['thm:bd-csdsd']}, \ref{['pro:allstrict']}, \ref{['pro:allvalid']}
  • Theorem 6: \ref{['thm:csclosure']}
  • Definition 7
  • Definition 8: DeVKO20
  • Definition 9: matroid-tree-width
  • Definition 10
  • ...and 58 more