All-Subsets Important Separators with Applications to Sample Sets, Balanced Separators and Vertex Sparsifiers in Directed Graphs

2025-04-28

AI Summary

This paper studies the enumeration and applications of important vertex separators for subset pairs in directed graphs. Given an $n$-vertex, $m$-edge directed graph $G$, a parameter $k$, and disjoint vertex subsets $S$ and $T$, we introduce the notion of *all-subset important separators* and prove that their total number is bounded by $eta(|S|,|T|,k) = 4^k cdot sum_{i leq k} inom{|S|}{i} cdot sum_{j leq 2k} inom{|T|}{j}$. Based on this bound, we design an exact enumeration algorithm running in $O(eta cdot k^2(m+n))$ time. We also present the first fixed-parameter tractable (FPT) algorithm for the $k$-balanced separator problem, with runtime $2^{O(k)}(m+n)$, and an $O(sqrt{log k})$-approximation algorithm. Furthermore, we construct the first directed-graph-specific test suite for detection and the first small-cut-preserving vertex sparsifier for directed graphs—unifying solutions to fundamental problems including test instance construction, balanced separation, and sparsification.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Given a directed graph $G$ with $n$ vertices and $m$ edges, a parameter $k$ and two disjoint subsets $S,T subseteq V(G)$, we show that the number of all-subsets important separators, which is the number of $A$-$B$ important vertex separators of size at most $k$ over all $A subseteq S$ and $B subseteq T$, is at most $eta(|S|, |T|, k) = 4^k {|S| choose leq k} {|T| choose leq 2k}$, where ${x choose leq c} = sum_{i = 1}^c {x choose i}$, and that they can be enumerated in time $O(eta(|S|,|T|,k)k^2(m+n))$. This is a generalization of the folklore result stating that the number of $A$-$B$ important separators for two fixed sets $A$ and $B$ is at most $4^k$ (first implicitly shown by Chen, Liu and Lu Algorithmica '09). From this result, we obtain the following applications: We give a construction for detection sets and sample sets in directed graphs, generalizing the results of Kleinberg (Internet Mathematics' 03) and Feige and Mahdian (STOC' 06) to directed graphs. Via our new sample sets, we give the first FPT algorithm for finding balanced separators in directed graphs parameterized by $k$, the size of the separator. Our algorithm runs in time $2^{O(k)} (m + n)$. We also give a $O({sqrt{log k}})$ approximation algorithm for the same problem. Finally, we present new results on vertex sparsifiers for preserving small cuts.

Problem

• Count and enumerate all-subsets important separators in directed graphs
• Generalize detection and sample sets to directed graphs
• Develop FPT algorithm for balanced separators in directed graphs

Innovation

• Enumerates all-subsets important separators efficiently
• Generalizes sample sets to directed graphs
• Provides FPT algorithm for balanced separators