On the Complexity of Community-aware Network Sparsification

TL;DR

This work studies community-aware network sparsification through the $\Pi$-Network Sparsification framework, focusing on $\Pi$ = connectivity (Connectivity NWS) and $\Pi$ = spanning-star containment (Stars NWS). It establishes tight fine-grained and ETH-based lower bounds, showing $2^{\Omega(n^2+c)}$ time is needed for Unweighted Connectivity NWS and Unweighted Stars NWS even on clique inputs with small communities. It also provides an XP-algorithm for Stars NWS parameterized by the solution’s feedback edge number $t$, answering long-standing questions about tractability when the solution is tree-like, while proving NP-hardness for $t=1$ in Connectivity NWS. Additionally, the paper gives a dichotomy for Stars NWS parameterized by the number of communities $c$, proving FPT results for unweighted variants but $\mathrm{W}[1]$-hardness in the weighted setting under the same parameters. Together, these results delineate when efficient, structure-exploiting algorithms exist for preserving community-wise connectivity properties during sparsification, and when the problems remain intractable under standard complexity assumptions.

Abstract

Network sparsification is the task of reducing the number of edges of a given graph while preserving some crucial graph property. In community-aware network sparsification, the preserved property concerns the subgraphs that are induced by the communities of the graph which are given as vertex subsets. This is formalized in the $Π$-Network Sparsification problem: given an edge-weighted graph $G$, a collection $Z$ of $c$ subsets of $V(G)$ (communities), and two numbers $\ell, b$, the question is whether there exists a spanning subgraph $G'$ of $G$ with at most $\ell$ edges of total weight at most $b$ such that $G'[C]$ fulfills $Π$ for each community $C$. Here, we consider two graph properties $Π$: the connectivity property (Connectivity NWS) and the property of having a spanning star (Stars NWS). Since both problems are NP-hard, we study their parameterized and fine-grained complexity. We provide a tight $2^{Ω(n^2+c)} poly(n+|Z|)$-time running time lower bound based on the ETH for both problems, where $n$ is the number of vertices in $G$. The lower bound holds even in the restricted case when all communities have size at most 4, $G$ is a clique, and every edge has unit weight. For the connectivity property, the unit weight case with $G$ being a clique is the well-studied problem of computing a hypergraph support with a minimum number of edges. We then study the complexity of both problems parameterized by the feedback edge number $t$ of the solution graph $G'$. For Stars NWS, we present an XP-algorithm for $t$. This answers an open question by Korach and Stern [Disc. Appl. Math. '08] who asked for the existence of polynomial-time algorithms for $t=0$. In contrast, we show for Connectivity NWS that known polynomial-time algorithms for $t=0$ [Korach and Stern, Math. Program. '03; Klemz et al., SWAT '14] cannot be extended by showing that Connectivity NWS is NP-hard for $t=1$.

Figures (4)

• Figure 1: $a)$ The communities (depicted in blue) and the input graph of an instance of $\Pi$-NWS with unit weights. $b)$ and $c)$ Optimal solutions (in red) for Unweighted Connectivity NWS, and Unweighted Stars NWS, respectively.
• Figure 3: Examples for solutions with and without local cycles. Red edges indicate the edges of the solution. Part a) shows an example, where both communities induce a local cycle. Part b) shows an example, where the two communities do not induce local cycle. Finally, part c) shows an example, where the solution contains a cycle but no two communities induce a local cycle.
• Figure 4: Examples for parts of the partition $\mathfrak{C}$. Only the local edges are shown. Note that $A$ and $C$ are both contained in $\mathfrak{C}(B)$, since $A$ and $C$ share at least three vertices with $B$ and no vertex of $A\cup B$ or $C\cup B$ is locally universal for $A\cap B$ or $C\cap B$, respectively. Hence, after exhaustive application of \ref{['op equivclasses']}, $\mathrm{fit}_{E^*}(A) =\mathrm{fit}_{E^*}(B) =\mathrm{fit}_{E^*}(C) = \emptyset$, since $A$ and $C$ share no vertices. Furthermore, $Y\in \mathfrak{C}(Z)$, since no vertex of $Y\cup Z$ is locally universal for $Y \cap Z$. Note that $X\notin\mathfrak{C}(Z)$, since the black vertex of $X$ is locally universal for $X\cap Y$ and $X\cap Z$. Observe that an exhaustive application of \ref{['op equivclasses']} yields $\mathrm{fit}_{E^*}(Z) \subseteq Y \cap Z$ and an exhaustive application of \ref{['op local coms 3']} yields $\mathrm{fit}_{E^*}(Z) \cap (X\cap Z) = \mathrm{fit}_{E^*}(Z) \cap (Y\cap Z) = \emptyset$, since $X$ contains at least one local edge and $X\cap Z$ contains no local edge. Hence, for both shown hypergraphs, there is no fitting solution for the given set of local edges.
• Figure 5: Examples of applications of \ref{['op endpoints of local', 'op equivclasses', 'op nonlocal coms', 'op local coms 2', 'op local coms 3']}. The black edges represent the local edges, the solid (for $C$) or dashed (for $D$) red edges show the non-local edges resulting from choosing the respective center for community $C$ or $D$. For example in 2), $z$ is the center of community $C$ and $v$ is the center of community $D$, and the edges $\{z,y\}$ and $\{v,y\}$ are non-local edges in the solution. For each operation, the violation of the property of being a fitting solution is shown, if a vertex $a$ is selected as a center of a community $A$ where the application of the corresponding operation would remove $a$ from $\mathrm{fit}_{E^*}(A)$. In $1)$, $2)$, $3)$, and $4.2)$, the vertex selected as center for community $C$ is removed from $\mathrm{fit}_{E^*}(C)$ by the respective operation. For example, in $4.2)$, (assuming $\mathrm{fit}_{E^*}(C)\cap \{x,y\} = \{x\}$) \ref{['op local coms 2']} removes $v$ from $\mathrm{fit}_{E^*}(C)$, as otherwise selecting $v$ as center of $C$ results in the depicted non-fitting solution. In $4.1)$ and $5)$, the vertex selected as center for community $C$ is removed from $\mathrm{fit}_{E^*}(C)$ by the respective operation. For example in $4.1)$, (assuming $\mathrm{fit}_{E^*}(C)\cap \{x,y\} = \emptyset$) \ref{['op local coms 2']} removes $y$ from $\mathrm{fit}_{E^*}(D)$, as otherwise selecting $y$ as center of $D$ results in the depicted non-fitting solution.

Theorems & Definitions (26)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Theorem 6
• Corollary 8
• Corollary 9
• Theorem 10
• Theorem 11