Translating between the representations of an acyclic convex geometry of bounded degree

TL;DR

This work studies the translation between irreducible closed sets and implicational bases in finite closure systems, focusing on acyclic convex geometries and degree-based parameters. It develops top-down, incremental-enumeration algorithms and proves incremental-polynomial time results for $\mathrm{ICS\cdot Enum}$ and $\mathrm{MIB\cdot Gen}$ when premise- or conclusion-degree are bounded, while also showing that polynomial-delay achievable via flashlight search is unlikely in general. By exploiting the unique attachment of irreducible closed sets to elements, a topological ordering, solution-graph traversal, and saturation, the paper derives a polynomial-time method to generate the aggregated critical base, which minimizes several degree measures and underpins the MIB${\cdot}$Gen problem. The discussion highlights hardness barriers (e.g., $\mathrm{ICS\cdot Ext}$ is NP-hard) and outlines open questions on extending tractability to broader closure systems and achieving polynomial delay.

Abstract

We consider the problem of translating between irreducible closed sets and implicational bases in closure systems. To date, the complexity status of this problem is widely open, and it is further known to generalize the notorious hypergraph dualization problem, even in the context of acyclic convex geometries, i.e., closure systems admitting an acyclic implicational base. This paper studies this later class with a focus on the degree, which corresponds to the maximal number of implications in which an element occurs. We show that the problem is tractable for bounded values of this parameter, even when relaxed to the notions of premise- and conclusion-degree. Our algorithms rely on structural properties of acyclic convex geometries and involve various techniques from algorithmic enumeration such as solution graph traversal, saturation techniques, and a sequential approach leveraging from acyclicity. They are shown to perform in incremental-polynomial time. Finally, we complete these results by showing that our running times cannot be improved to polynomial delay using the standard framework of flashlight search.

Figures (8)

• Figure 1: A closure system over $X = \{1, 2, 3, 4, 5\}$ represented via the Hasse diagram of its closure lattice. Black dots are irreducible closed sets; they are the ones with a unique successor. We have, for instance, $\phi(25) = 235$ and $\phi(4) = 14$, $\mathrm{irr}(2) = \{35, 134\}$ and $\mathrm{mingen}(3) = \{25, 15, 24, 45\}$. A spanning set of $1235$ is $135$. The extreme points of $135$ are $1$ and $5$.
• Figure 2: The implicational bases of Example \ref{['ex:exponential']} with $\Sigma_1$ on the left and $\Sigma_2$ on the right. For clarity we only give some of the implications, drawn in grey. In both cases, an irreducible closed set attached to $x$ is highlighted in blue.
• Figure 3: The implication graph $G(\Sigma)$ of the IB $(X, \Sigma)$ of Example \ref{['ex:prelim-IB']}, associated to the closure system in Figure \ref{['fig:prelim-example']}. This directed graph is acyclic, which makes the closure system an acyclic convex geometry. The ancestors of $2$, highlighted in green, are $1, 4, 5$, i.e., $\mathrm{anc}(2) = \{1, 4, 5 \}$. Maximal elements are $4$, $5$ and the unique minimal element is $3$. A topological order, as we will use later, would be $4, 1, 5, 2, 3$ in this example.
• Figure 4: An illustration of the reduction from MIS${\cdot}$Enum to ICS${\cdot}$Enum for Theorem \ref{['thm:ics-mib-height2-hardness']}. The initial hypergraph is $\mathcal{H} = (\{1, 2, 3, 4, 5\}, \{123, 234, 45\})$ and the corresponding IB $(X, \Sigma)$ has implications $123 \to z, 234 \to z, 45 \to z$, drawn in grey. The set $125$, highlighted in blue, is a maximal independent set of $\mathcal{H}$ and hence an irreducible closed set attached to $z$.
• Figure 5: The reduction from MIS${\cdot}$Enum to ACS${\cdot}$Enum with conlusion-degree at most $2$, applied to the hypergraph of Figure \ref{['fig:mis-to-ics']}. The implications are now $123 \to z_1, 234 \to z_2, 45 \to z_3, z_1 \to z_2, z_2 \to z_3$. With $z_1 \to z_2$, $z_2 \to z_3$, none of $z_1, z_2$, highlighted in red, cannot belong to any irreducible closed set attached to $z_3$. As compared to the reduction of Theorem \ref{['thm:ics-mib-height2-hardness']}, irreducible closed sets of $z_3$ (formerly $z$) are unchanged.

Theorems & Definitions (56)

• Theorem 1
• Theorem 2
• Proposition 3: see e.g., edelman1985theory
• Proposition 5
• proof
• Proposition 6
• proof
• Example 7
• Example 8
• Example 9