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A Branch-and-Bound Approach for Maximum Low-Diameter Dense Subgraph Problems

Yi Zhou, Chunyu Luo, Zhengren Wang, Zhang-Hua Fu

TL;DR

The paper defines the maximum low-diameter $f(\cdot)$-dense subgraph problem (M$f$DS), a connectivity-respecting generalization of dense-subgraph models. It proposes a practical decomposition-based framework that partitions the input graph into small subproblems, combined with a degeneracy or two-hop degeneracy ordering and a branch-and-bound solver augmented by a sorting-based upper bound (SortBound). A mixed integer programming model (MIP-$f$D) is provided to formalize the problem, and extensive experiments on 139 real-world graphs show that the decomposition-enhanced approach outperforms standalone MIP or pure BnB, solving roughly twice as many instances within one hour. The work demonstrates that exact solutions for large-scale M$f$DS problems are feasible in practice and highlights the potential for parallelization and broader applicability to other diameter-constrained dense-subgraph tasks.

Abstract

A graph with $n$ vertices is an $f(\cdot)$-dense graph if it has at least $f(n)$ edges, $f(\cdot)$ being a well-defined function. The notion $f(\cdot)$-dense graph encompasses various clique models like $γ$-quasi cliques, $k$-defective cliques, and dense cliques, arising in cohesive subgraph extraction applications. However, the $f(\cdot)$-dense graph may be disconnected or weakly connected. To conquer this, we study the problem of finding the largest $f(\cdot)$-dense subgraph with a diameter of at most two in the paper. Specifically, we present a decomposition-based branch-and-bound algorithm to optimally solve this problem. The key feature of the algorithm is a decomposition framework that breaks the graph into $n$ smaller subgraphs, allowing independent searches in each subgraph. We also introduce decomposition strategies including degeneracy and two-hop degeneracy orderings, alongside a branch-and-bound algorithm with a novel sorting-based upper bound to solve each subproblem. Worst-case complexity for each component is provided. Empirical results on 139 real-world graphs under two $f(\cdot)$ functions show our algorithm outperforms the MIP solver and pure branch-and-bound, solving nearly twice as many instances optimally within one hour.

A Branch-and-Bound Approach for Maximum Low-Diameter Dense Subgraph Problems

TL;DR

The paper defines the maximum low-diameter -dense subgraph problem (MDS), a connectivity-respecting generalization of dense-subgraph models. It proposes a practical decomposition-based framework that partitions the input graph into small subproblems, combined with a degeneracy or two-hop degeneracy ordering and a branch-and-bound solver augmented by a sorting-based upper bound (SortBound). A mixed integer programming model (MIP-D) is provided to formalize the problem, and extensive experiments on 139 real-world graphs show that the decomposition-enhanced approach outperforms standalone MIP or pure BnB, solving roughly twice as many instances within one hour. The work demonstrates that exact solutions for large-scale MDS problems are feasible in practice and highlights the potential for parallelization and broader applicability to other diameter-constrained dense-subgraph tasks.

Abstract

A graph with vertices is an -dense graph if it has at least edges, being a well-defined function. The notion -dense graph encompasses various clique models like -quasi cliques, -defective cliques, and dense cliques, arising in cohesive subgraph extraction applications. However, the -dense graph may be disconnected or weakly connected. To conquer this, we study the problem of finding the largest -dense subgraph with a diameter of at most two in the paper. Specifically, we present a decomposition-based branch-and-bound algorithm to optimally solve this problem. The key feature of the algorithm is a decomposition framework that breaks the graph into smaller subgraphs, allowing independent searches in each subgraph. We also introduce decomposition strategies including degeneracy and two-hop degeneracy orderings, alongside a branch-and-bound algorithm with a novel sorting-based upper bound to solve each subproblem. Worst-case complexity for each component is provided. Empirical results on 139 real-world graphs under two functions show our algorithm outperforms the MIP solver and pure branch-and-bound, solving nearly twice as many instances optimally within one hour.

Paper Structure

This paper contains 39 sections, 9 theorems, 7 equations, 7 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Our Contribution
  4. Preliminary
  5. Basic Notation
  6. The Maximum $f(\cdot)$-dense Subgraph Problem
  7. $f(\cdot)$-Dense Graph
  8. Hereditary Property of $f(\cdot)$-Dense Graph
  9. The Maximum Low-Diameter $f(\cdot)$-Dense Subgraph Problem
  10. A Mixed Integer Programm of M$f$DS
  11. A Combinatorial Algorithm of Solving MfDS
  12. The Decomposition-based Framework
  13. A Efficient Heuristic Algorithm
  14. The Ordering of Vertices
  15. Degeneracy Ordering
  16. ...and 24 more sections

Key Result

Lemma 1

Given a density function $f:\mathbb{Z}_{\ge0}\rightarrow \mathbb{R}$, define $g_f(i)=\binom{i}{2}-f(i)$, for all $i\in \mathbb{Z}_{\ge0}$. If $g_f(\cdot)$ is a monotonically non-increasing, i.e., $g_f(i)\ge g_f(i+1)$, for all $i\in \mathbb{Z}_{\ge 0}$, then $f(\cdot)$ is a hereditary-induced functio

Figures (7)

  • Figure 1: $G_1[\{v_1,v_2,\dots,v_{200}\}]$ is a clique and $v_{200},v_{201},\dots,v_{210}$ forms a path between $v_{200}$ and $v_{210}$. $G_2[\{v_1,v_2,\dots,v_{200}\}]$ is a clique and $v_{201}, v_{202},\dots,v_{210}$ are all isolated vertices.
  • Figure 2: An example of computing sorting bound.
  • Figure 3: An example of computing sorting bound by Alg \ref{['alg:Sort-UB']}
  • Figure 4: Overall performance evaluation of M$f$DS algorithms for $f(i)=\gamma\binom{i}{2}$ with $\gamma=\{0.99,\ 0.95,\ 0.90,\ 0.85\}$.
  • Figure 5: Overall performance evaluation of M$f$DS algorithms for $f_2(i)=\binom{i}{2}-s$ with $s=\{1,3\}$.
  • ...and 2 more figures

Theorems & Definitions (23)

  • Definition 1: density function
  • Definition 2: $f(\cdot)$-dense graph
  • Definition 3: hereditary-induced density function
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Definition 4: degeneracy ordering
  • Theorem 1
  • Definition 5: two-hop degeneracy ordering
  • Theorem 2
  • ...and 13 more