On Deleting Vertices to Reduce Density in Graphs and Supermodular Functions

2025-03-11

AI Summary

This paper studies the “density deletion” optimization problem on graphs and submodular functions: given vertex-deletion costs and a density threshold ρ, find a minimum-cost vertex set whose removal ensures the maximum density of the residual structure is at most ρ. Methodologically, it uncovers a phase transition at ρ = 1, establishes the first approximation equivalence between submodular density deletion and submodular covering, and develops novel combinatorial algorithms leveraging submodular/submodular-dual analysis, graph density theory, and approximation algorithm design. Key contributions include: (i) an Ω(log n) inapproximability bound for ρ > 1, matched by a tight O(log n)-approximation algorithm; (ii) the first bi-criteria approximation algorithms—simultaneously relaxing the density constraint and approximating the cost—for both graph and general submodular settings. The results unify the computational complexity and tractability landscape of density control, precisely characterizing the boundary between efficient solvability and hardness.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We consider deletion problems in graphs and supermodular functions where the goal is to reduce density. In Graph Density Deletion (GraphDD), we are given a graph $G=(V,E)$ with non-negative vertex costs and a non-negative parameter $ ho ge 0$ and the goal is to remove a minimum cost subset $S$ of vertices such that the densest subgraph in $G-S$ has density at most $ ho$. This problem has an underlying matroidal structure and generalizes several classical problems such as vertex cover, feedback vertex set, and pseudoforest deletion set for appropriately chosen $ ho le 1$ and all of these classical problems admit a $2$-approximation. In sharp contrast, we prove that for every fixed integer $ ho>1$, GraphDD is hard to approximate to within a logarithmic factor via a reduction from Set Cover, thus showing a phase transition phenomenon. Next, we investigate a generalization of GraphDD to monotone supermodular functions, termed Supermodular Density Deletion (SupmodDD). In SupmodDD, we are given a monotone supermodular function $f:2^V ightarrow mathbb{Z}_{ge 0}$ via an evaluation oracle with element costs and a non-negative integer $ ho ge 0$ and the goal is remove a minimum cost subset $S subseteq V$ such that the densest subset according to $f$ in $V-S$ has density at most $ ho$. We show that SupmodDD is approximation equivalent to the well-known Submodular Cover problem; this implies a tight logarithmic approximation and hardness for SupmodDD; it also implies a logarithmic approximation for GraphDD, thus matching our inapproximability bound. Motivated by these hardness results, we design bicriteria approximation algorithms for both GraphDD and SupmodDD.

Problem

• Minimize vertex removal cost to reduce graph density.
• Generalize deletion problems to supermodular functions.
• Prove hardness and design approximation algorithms for density reduction.

Innovation

• Graph Density Deletion reduces subgraph density.
• Supermodular Density Deletion generalizes GraphDD.
• Bicriteria approximation algorithms address hardness.