Characterizing the optimum bases of a convex geometry using quasi-closed hypergraphs

Abstract

Optimizing an implicational base of a closure system consists in turning this implicational base into an equivalent one with premises and conclusions as small as possible. This task is known to be hard in general but tractable for a number of classes of closure systems. In particular, several classes of convex geometries are known to have tractable optimization, while the problem was recently claimed to remain hard in general convex geometries. Continuing this line of research, we give a characterization of the optimum bases of a convex geometry in terms of what we call quasi-closed hypergraphs. We then use this characterization to show that when each quasi-closed hypergraph has disjoint edges, any implicational base of the convex geometry can be optimized in polynomial time with existing minimization and reduction algorithms. Finally, we prove that this property applies to double-shelling, acyclic, affine and acceptant convex geometries, thus unifying the existing results regarding the tractability of optimization for the first three classes.

Figures (5)

• Figure 1: The closure system of Example \ref{['ex:carac-valid-qc']}. It has three essential sets (yellow square nodes) being $\mathit{ab}$, $\mathit{cdef}$, $\mathit{abcdef}$. To each essential set is associated a box representing its quasi-closed sets (circle nodes) ordered by inclusion within its spanning sets (shaded zone). For instance the quasi-closed sets of $\mathit{abcdef}$ are $\mathit{abd}$, $\mathit{abe}$, $\mathit{abcd}$. Minimal quasi-closed sets are green (e.g., $\mathit{abd}$), maximal ones red (e.g., $\mathit{abcd}$), and those that are both minimal and maximal purple (e.g., $\mathit{abe}$).
• Figure 2: A convex geometry with $3$ essential sets being $\mathit{acde}$, $\mathit{bcde}$ and $\mathit{abcde}$ (yellow square nodes). Some closed sets are unlabeled for readability. Each essential set has an associated box representing its spanning sets (shaded zone) and its quasi-closed sets (circle nodes). Minimal quasi-closed sets (e.g., $ab$) are green, maximal ones (e.g., $abd$) are red and those that are both minimal and maximal (e.g., $ae$) are purple. The canonical base has implications $\mathit{ae} \rightarrow \mathit{cd}$, $\mathit{bcd} \rightarrow \mathit{e}$ and $\mathit{ab} \rightarrow \mathit{cde}$. We have $\mathcal{Q}(\mathit{abcde}) = \{de, ce\}$ where $de$ and $ce$ are not disjoint.
• Figure 3: On the left the poset $P$ of Example \ref{['ex:poset-cg']}. The set $\mathit{cdeh}$ (circled in purple, solid) is convex. The interval $[a, y]$ (circled in green, dotted) is $\mathit{afgy}$. It is also convex. The comparability graph $G_P$ is pictured in the middle and the graph $G_P(x, y)$ on the right. It has three connected component: $\mathit{abgf}$, $d$, and $\mathit{ceh}$.
• Figure 4: The 3 edges of of the quasi-closed hypergraph associated to $X$ in the double-shelling convex geometry of Example \ref{['ex:poset-cg']} (solid yellow zones). The dotted blue zones thus indicate the corresponding quasi-closed sets. The set $\mathit{ade}$ (bold nodes) form a minimum hitting set of this hypergraph.
• Figure 5: Examples of $2$-acceptant convex geometries. Closed sets with at most $2$ elements are trivial, other ones are non-trivial and all have exactly $2$ extreme points.

Theorems & Definitions (29)

• Theorem 1: restate=THMcgoptimib, label=thm:cg-optim-ib
• Theorem 2: restate=THMqcdisjoint, label=thm:qc-disjoint
• Theorem 3: restate=THMcgclasses, label=thm:cg-classes
• Theorem 4: see, e.g., ausiello1986minimummaier1983theorybichoupan2022complexitymannila1983relationship
• Theorem 5: see, e.g., guigues1986familleswild1994theory
• Lemma 1
• Proposition 1: Lemma 2 in bichoupan2023independence
• proof
• Example 1
• Example 2: continued