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On Deleting Vertices to Reduce Density in Graphs and Supermodular Functions

Karthekeyan Chandrasekaran, Chandra Chekuri, Shubhang Kulkarni

TL;DR

The paper studies vertex-deletion problems to reduce graph and supermodular density, introducing $ ho$-GraphDD and $ ho$-SupmodDD. It establishes a phase transition in approximability: a 2-approximation for $ ho ext{≤}1$ and $ ilde{ heta}( ext{log }n)$-hardness for fixed integers $ ho ext{≥}2$, via a Set Cover reduction. It then ties SupmodDD to Submodular Cover, yielding tight logarithmic approximations and hardness, and designs bicriteria algorithms for both GraphDD and SupmodDD using orientation-based LPs and dense-decomposition techniques. The results illuminate the structure behind density deletion problems, linking matroidal and submodular perspectives and offering practical bicriteria tools. This work advances understanding of robustness of densest subgraphs and provides a framework for approximations under density constraints.

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 $ρ\ge 0$ 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 $ρ$. 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 $ρ\le 1$ and all of these classical problems admit a $2$-approximation. In sharp contrast, we prove that for every fixed integer $ρ> 1$, 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 $f:2^V \rightarrow \mathbb{Z}_{\ge 0}$ via an evaluation oracle with element costs and a non-negative integer $ρ\ge 0$ 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 $ρ$. 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.

On Deleting Vertices to Reduce Density in Graphs and Supermodular Functions

TL;DR

The paper studies vertex-deletion problems to reduce graph and supermodular density, introducing -GraphDD and -SupmodDD. It establishes a phase transition in approximability: a 2-approximation for and -hardness for fixed integers , via a Set Cover reduction. It then ties SupmodDD to Submodular Cover, yielding tight logarithmic approximations and hardness, and designs bicriteria algorithms for both GraphDD and SupmodDD using orientation-based LPs and dense-decomposition techniques. The results illuminate the structure behind density deletion problems, linking matroidal and submodular perspectives and offering practical bicriteria tools. This work advances understanding of robustness of densest subgraphs and provides a framework for approximations under density constraints.

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 with non-negative vertex costs and a non-negative parameter and the goal is to remove a minimum cost subset of vertices such that the densest subgraph in has density at most . 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 and all of these classical problems admit a -approximation. In sharp contrast, we prove that for every fixed integer , 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 via an evaluation oracle with element costs and a non-negative integer and the goal is remove a minimum cost subset such that the densest subset according to in has density at most . 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.

Paper Structure

This paper contains 26 sections, 21 theorems, 16 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Density Deletion Problems.
  3. Motivations and Connections.
  4. Results
  5. Connections between SubmodCover and $\textsc{SupmodDD}$
  6. Hardness of Approximation
  7. Bicriteria Approximations
  8. Organization.
  9. Preliminaries: Characterizing Density Using Orientations
  10. Approximation Hardness
  11. Warmup.
  12. Bicriteria Approximations
  13. Bicriteria for GraphDD
  14. Orientation ILP
  15. Rounding the Orientation LP
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $f: 2^V\rightarrow \Z_{\ge 0}$ be an integer-valued normalized supermodular function and $\rho$ be a rational number. Then, there exists a normalized non-decreasing submodular function $h:2^V\rightarrow\R_{\geq 0}$ such that

Figures (2)

  • Figure 1: Reductions between problems of interest to this work. Arrow from Problem A to Problem B implies that Problem A has an approximation-preserving reduction to Problem B. Figure (a) consists of the connections between problems known prior to our work. Figure (b) showcases our results.
  • Figure 2: The figure in (a) depicts the subgraph of the construction corresponding to element $e \in \calU$. Here, $f = 8$. The figure in (b) depicts the intermediate orientation $\vec{H}$ for the subgraph of $H$ corresponding to an element $e \in \calU$. The greyed-out set-vertex at the bottom represents that this vertex is in $X_F$. The figure in (c) depicts the final orientation for the subgraph from the figure in (b) after reorientation. The highlighted edges are those that have been reoriented.

Theorems & Definitions (54)

  • Definition 1.1: $\rho\text{-}\textsc{GraphDD}\xspace$
  • Definition 1.2: $\rho\text{-}\textsc{SupmodDD}\xspace$
  • Definition 1.3: MatroidFVS
  • Definition 1.4: SubmodCover
  • Theorem 1.1
  • Corollary 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • ...and 44 more