Hedgegraph Polymatroids

TL;DR

This work develops a polymatroid-based framework to study connectivity in hedgegraphs, a generalization of hypergraphs that groups hyperedges into hedges to model interdependencies. By defining the hedgegraph polymatroid $f_G(A)=|V|- \\#Comps(V,A)$ and partition-based connectivity measures, the authors obtain a suite of structural and algorithmic results, including a polynomial-time computation of partition connectivity and a min-max decomposition theorem via hedgegraph matroids. They show that hedgegraphs are k-partition connected iff they contain k hedge-disjoint 1-partition connected sub-hedgegraphs, with trimming-based spanning-tree characterizations and orientation consequences. Additional contributions include the notions of weak partition connectivity, sampling guarantees that preserve connectivity, and partition sparsifiers derived from polymatroid quotient sparsification, all generalizing classical graph/hypergraph results to hedges. The collective framework provides principled, tractable avenues for understanding connectivity in hedged dependencies and offers potential practical impacts for reliability, decomposition, and sparsification in complex network models.

Abstract

Graphs and hypergraphs combine expressive modeling power with algorithmic efficiency for a wide range of applications. Hedgegraphs generalize hypergraphs further by grouping hyperedges under a color/hedge. This allows hedgegraphs to model dependencies between hyperedges and leads to several applications. However, it poses algorithmic challenges. In particular, the cut function is not submodular, which has been a barrier to algorithms for connectivity. In this work, we introduce two alternative partition-based measures of connectivity in hedgegraphs and study their structural and algorithmic aspects. Instead of the cut function, we investigate a polymatroid associated with hedgegraphs. The polymatroidal lens leads to new tractability results as well as insightful generalizations of classical results on graphs and hypergraphs.

Figures (6)

• Figure 1: Leftmost figure shows a hedgegraph with $V = \{A,B,C,D,E,F\}$ and three hedges, where the black hedge $e_1:=\{\{A,B,C\}, \{D,E,F\}\}$, the red hedge $e_2:=\{\{B,D\},\{E,F\}\}$, and the blue hedge $e_3:=\{\{C,E\}, \{B,D\}\}$. It can also be viewed as a coloring of hyperedges of the hypergraph shown in the middle. Rightmost figure shows an example of a graph.
• Figure 2: An example of a hedgegraph $G=(V, E)$ whose cut function is not submodular. Here, $V=\{A,B,C,D\}$ and $E:=\{e_1, e_2\}$ where $e_1:=\{\{A,B\},\{C,D\}\}$ and $e_2=\{\{A,C\},\{B,D\}\}$. We note that $d(\{A,B\})=d(\{A,C\})=1$ and $d(\{A\})=d(\{A,B,C\})=2$ which implies that $d(\{A,B\})+d(\{A,C\})<d(\{A\})+g(\{A,B,C\})$.
• Figure 3: An example of an acyclic-trimming. The left side shows a hedgegraph $G=(V,E)$ with five hedges, where $e_1:=\{\{A,E\},\{B,C\}\}$, $e_2:=\{\{A,B\}\}$, $e_3:=\{\{C,E\}\}$, $e_4:=\{\{C,D,E\}\}$, and $e_5:=\{\{B,E\}\}$. The right side shows an acyclic-trimming of $\{e_1, e_2, e_3, e_4\}$, where $e_1$ is trimmed into $\{A,E\}$, $e_2$ is trimmed into $\{A,B\}$, $e_3$ is trimmed into $\{C,E\}$, and $e_4$ is trimmed into $\{D,E\}$. We may verify that $\{e_1, e_2, e_3, e_5\}$ does not have an acyclic-trimming.
• Figure 4: An example of hedgegraph orientation. The left side shows a hedgegraph $G=(V,E)$ with three hedges, where $e_1:=\{\{A,C,D\}\}$, $e_2:=\{\{A,B\},\{C,D\}\}$, and $e_3:=\{\{B,D\}\}$. The right side shows an orientation of $G$, where hyperedge $\{A,C,D\}$ is picked in $e_1$ with $A$ being its head, hyperedge $\{A,B\}$ is picked in $e_2$ with $A$ being its head, and hyperedge $\{B,D\}$ is picked in $e_3$ with $B$ being its head. If we let $r:=A$ being the root, then it is a rooted $1$-out-hyperarc-connected hypergraph. We note that this is not rooted $2$-out-hyperarc-connected since $d^{out}_{\overrightarrow{G}}(\{A,B,D\})=d^{out}_{\overrightarrow{G}}(\{A,C,D\})=1$.
• Figure 5: An example of vertex partition $\mathcal{P}$ and hedgegraph $\mathcal{P}(e)$. The left side shows a partition $\mathcal{P}:=\{\{B\}, \{C\},\{D\},\{E\},\{A,F\}\}$ of vertex set $V$ and a hedge $e:=\{\{A,B,C\},\{E,F\}\}$. The right side shows the hedgegraph obtained from the hedgegraph $(V,\{e\})$ by contracting every part in $\mathcal{P}$ into a single vertex. We note that $|\mathcal{P}|=5$ and $\#\text{Comps}(\mathcal{P}(e))=2$.

Theorems & Definitions (50)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Corollary 1
• Lemma 1
• Theorem 5
• Theorem 6
• Theorem 7
• Definition 1