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The Polymatroid Representation of a Greedoid, and Associated Galois Connections

Robert Streit, Vijay K. Garg

Abstract

A greedoid is a generalization of a matroid allowing for more flexible analyses and modeling of combinatorial optimization problems. However, these structures decimate many matroid properties contributing to their pervasive nature. A polymatroid greedoid [KL85a] presents an interesting middle ground, so we further develop this class. First we prove every local poset greedoid for which the greedy algorithm correctly solves linear optimizations over its basic words must have a polymatroid representation. For this, we use relationships between the lattices of greedoid flats and closed sets of a polymatroid to generalize concepts in [KL85a]. Then, we show our generalization induces a Galois injection between the greedoid flats and closed sets of a representation. Finally, we apply this duality to identify a subclass of polymatroid greedoids with a maximum representation, giving a partial answer to an open problem of [KL85a]. As technical tools for our analyses, we introduce optimism and the Forking Lemma for interval greedoids. Both are pervasive in our work, and are of independent interest.

The Polymatroid Representation of a Greedoid, and Associated Galois Connections

Abstract

A greedoid is a generalization of a matroid allowing for more flexible analyses and modeling of combinatorial optimization problems. However, these structures decimate many matroid properties contributing to their pervasive nature. A polymatroid greedoid [KL85a] presents an interesting middle ground, so we further develop this class. First we prove every local poset greedoid for which the greedy algorithm correctly solves linear optimizations over its basic words must have a polymatroid representation. For this, we use relationships between the lattices of greedoid flats and closed sets of a polymatroid to generalize concepts in [KL85a]. Then, we show our generalization induces a Galois injection between the greedoid flats and closed sets of a representation. Finally, we apply this duality to identify a subclass of polymatroid greedoids with a maximum representation, giving a partial answer to an open problem of [KL85a]. As technical tools for our analyses, we introduce optimism and the Forking Lemma for interval greedoids. Both are pervasive in our work, and are of independent interest.

Paper Structure

This paper contains 20 sections, 33 theorems, 118 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Organization
  3. Related Work
  4. Preliminaries
  5. Lattices
  6. Polymatroids
  7. Greedoids
  8. Polymatroid Representations
  9. Alignment
  10. Optimism
  11. The Forking Lemma
  12. Polymatroid Greedoids and the Greedy Algorithm
  13. The Basis Rank of the Contraction Minors of Interval Greedoids
  14. An Aligned Representation of Greedily Optimizable Local Poset Greedoids
  15. Galois Connections
  16. ...and 5 more sections

Key Result

Lemma 3.2

Let $\mathscr{P}$ and $\mathscr{Q}$ be posets with a map $\varphi:\mathscr{P}\to\mathscr{Q}$. Suppose further that both $\mathscr{P}$ and $\mathscr{Q}$ are finite lattices. Then, we have that $\varphi$ is (meet/join)-preserving if and only if $\varphi$ is the (upper/lower) adjoint of some Galois con

Figures (10)

  • Figure 1: The class of polymatroid greedoids with aligned representations is shown by the gray-shaded rectangle. By Theorem \ref{['thm:lp-greedy=>pm-greedoid']}, the intersection of strong-exchange greedoids (those for which the greedy algorithm correctly solves linear optimizations goetschel1986linearkorte1984greedoids) and local poset greedoids is contained this class. This intersection includes matroids and antimatroids with the local poset property. By Theorem \ref{['thm:pm->optimism']}, polymatroid greedoids with aligned representations are contained in the intersection of optimistic and local poset greedoids. However, one can adapt an example from korte1988intersection to show this inclusion is strict (see Appendix \ref{['app:local-augmentation']}). Moreover, Appendix \ref{['app:optimism']} shows all strong-exchange greedoids are optimistic, and gives a trimmed matroid construction for the existence of an optimistic interval greedoid without the local poset property. Finally, the end of Section \ref{['subsec:lattice-embeddings']} discusses why optimistic distributive supermatroids are polymatroid greedoids, and all other shown inclusions pre-date our work (see korte2012greedoids).
  • Figure 2: A graph with root $s$ and the lattice of flats of the corresponding undirected branching greedoid. We see that $[ac]\sqcap[ad] = [a]$ because $ac \not\sim ad$, for example. Furthermore, the reader should note (or convince themselves) that the top flat of a greedoid is always given by the basic words.
  • Figure 3: Values of the representation from Example \ref{['ex:ubg-rep']} applied to the greedoid in Figure \ref{['fig:ubg']}.
  • Figure 4: The Hasse diagram of the closed sets of $\rho$ from Table \ref{['table:ubg-rep']}, ordered by containment.
  • Figure 5: Above is a sublattice of $\mathscr{L}(\Lambda)$ satisfying the premise described in the Forking Lemma. We draw the covering relation by solid lines, labelled by $x$ as it is a continuation advancing words in the lower neighbor to words in the upper neighbor, and denote a flat lying below another in a way which may or may not be covering by the dotted lines. Then, $F$ and $F'$ are such that the letter $x$ is a continuation of the meet $F\sqcap F' = [\mu]$, with $[\mu x]$ lying below $F'$ but not $F$. Then, $x$ must be a continuation of $F$ as a consequence. By semimodularity $F \sqcup [\mu x]$ covers $F$, so $F \prec \kappa^{-1}(\kappa(F) + x)$ (what we just found) and $\kappa^{-1}(\kappa(F) + x) \sqsubseteq F\sqcup [\mu x]$ imply $\kappa^{-1}(\kappa(F) + x) = F\sqcup [\mu x]$.
  • ...and 5 more figures

Theorems & Definitions (85)

  • Definition 3.1: Galois Connection
  • Lemma 3.2: davey2002introduction
  • Definition 3.3: Greedoid
  • Example 3.4: Undirected Branching Greedoid
  • Definition 3.5: Interval Property
  • Example 4.1: Undirected Branching Greedoid
  • Definition 4.2: Local Poset Property korte1985polymatroid
  • Proposition 4.2: korte1985polymatroid
  • Definition 4.3: Aligned Representation
  • Lemma 4.3
  • ...and 75 more