A Faster Branching Algorithm for the Maximum $k$-Defective Clique Problem

TL;DR

The paper tackles the maximum $k$-defective clique problem, seeking the largest vertex subset whose induced subgraph misses at most $k$ edges; this problem is NP-hard and hard to approximate. It introduces a Decompose-and-Branch framework that leverages the structure $Q = D(Q) \cup C(Q)$, where $D(Q)$ is a maximal $k$-defective set and $C(Q)$ is a clique contained in $CN(D(Q))$, solving MC on $CN(D)$ for each candidate $D$. The authors also present a graph-decomposition variant, DnBk, with degeneracy ordering to achieve a bound of $O^*(\gamma_c^{d(G)})$ for constant $k$, where $d(G)$ is the graph degeneracy and $\gamma_c$ is the base of the maximum-clique solver. To tighten pruning, they develop a novel PackColorConf upper bound that blends packing, coloring, and a newly formalized conflict relation between vertex pairs, supported by a dynamic-programming bound (DPBound) and a partially conflict-aware refinement. Empirical results on large real-world graphs show that DnBk, using PackColorConf, outperforms state-of-the-art solvers across multiple datasets and $k$ values, demonstrating both theoretical and practical improvements for this challenging problem.

Abstract

A $k$-defective clique of an undirected graph $G$ is a subset of its vertices that induces a nearly complete graph with a maximum of $k$ missing edges. The maximum $k$-defective clique problem, which asks for the largest $k$-defective clique from the given graph, is important in many applications, such as social and biological network analysis. In the paper, we propose a new branching algorithm that takes advantage of the structural properties of the $k$-defective clique and uses the efficient maximum clique algorithm as a subroutine. As a result, the algorithm has a better asymptotic running time than the existing ones. We also investigate upper-bounding techniques and propose a new upper bound utilizing the \textit{conflict relationship} between vertex pairs. Because conflict relationship is common in many graph problems, we believe that this technique can be potentially generalized. Finally, experiments show that our algorithm outperforms state-of-the-art solvers on a wide range of open benchmarks.

Figures (5)

• Figure 1: In graph $G$ of 6 vertices, for $S=\{v_1,v_2,v_3,v_4\}$, we have $G[S]$ is a $1$-defective clique with $D(S)=\{v_1,v_2\}$ and $C(S)=\{v_3,v_4\}$.
• Figure 2: The search tree of Alg. \ref{['alg: branch_framework']}. Each node in the tree represents a branch. The two sets labeled after the node represents $P$ and $R$, respectively, when the branch is generated.
• Figure 3: An example of running the DPbound with and without conflict vertices. The dashed lines represent conflict relations. The weight of vertices in $V\setminus P$, $w(\cdot)$, is displayed below the vertices.
• Figure 4: The dominance relations among bounding rules. $A \longrightarrow B$ indicates that A dominates B.
• Figure 5: Number of solved instances for NDR, Socfb and DIMACS10&SNAP graphs, with $k=1,5,10,15,20$ and a cutoff time $10800$s.

Theorems & Definitions (10)

• Lemma 1: Two-hop Property chen2021computing
• Definition 1: Non-trivial Maximum $k$-Defective Clique Problem
• Definition 2: $k$-Defective Set
• Definition 3: General $k$-Defective Clique Problem
• Lemma 2
• proof : Proof Sketch
• Lemma 3
• Theorem 4
• proof
• Definition 4: Conflict Vertices