Maximum $k$-Plex Search: An Alternated Reduction-and-Bound Method

TL;DR

This work tackles the NP-hard problem of finding the largest $k$-plex in a graph. It introduces kPEX, a BRB-based solver that pivots from conventional SeqRB to an alternated reduction-and-bound (AltRB) framework, enabling mutual reinforcement between reduction and bounding across branch partitions. Complementary pre-processing techniques CF-CTCP and KPHeuris boost practical performance by aggressively shrinking the input graph and yielding large initial lower bounds. Empirical evaluations on 664 real and synthetic graphs show that kPEX can be up to two orders of magnitude faster and solves more instances than state-of-the-art BRB methods, highlighting its scalability and effectiveness for cohesive subgraph mining. The methods hold promise for extending to other cohesive subgraph models and large-scale graph mining applications.

Abstract

$k$-plexes relax cliques by allowing each vertex to disconnect to at most $k$ vertices. Finding a maximum $k$-plex in a graph is a fundamental operator in graph mining and has been receiving significant attention from various domains. The state-of-the-art algorithms all adopt the branch-reduction-and-bound (BRB) framework where a key step, called reduction-and-bound (RB), is used for narrowing down the search space. A common practice of RB in existing works is SeqRB, which sequentially conducts the reduction process followed by the bounding process once at a branch. However, these algorithms suffer from the efficiency issues. In this paper, we propose a new alternated reduction-and-bound method AltRB for conducting RB. AltRB first partitions a branch into two parts and then alternatively and iteratively conducts the reduction process and the bounding process at each part of a branch. With newly-designed reduction rules and bounding methods, AltRB is superior to SeqRB in effectively narrowing down the search space in both theory and practice. Further, to boost the performance of BRB algorithms, we develop efficient and effective pre-processing methods which reduce the size of the input graph and heuristically compute a large $k$-plex as the lower bound. We conduct extensive experiments on 664 real and synthetic graphs. The experimental results show that our proposed algorithm kPEX with AltRB and novel pre-processing techniques runs up to two orders of magnitude faster and solves more instances than state-of-the-art algorithms.

Figures (5)

• Figure 1: An example of AltRB with $k$=2, $|S^*|$=$5$, $S=\{v_1, v_2\}$, $C=\{v_3, v_4, v_5, v_6, v_7, v_8\}$
• Figure 2: Comparing CTCP and CF-CTCP
• Figure 3: An example for CF-CTCP assuming $lb=4$ and $k=2$
• Figure 4: Number of solved instances on Network Repository (The lines corresponding to DiseMKP and kPlexT may not appear in the figures, as they are slow under certain settings and thus cannot reach the bottom lines within 3600 seconds.)
• Figure 5: Number of solved instances on DIMACS-2

Theorems & Definitions (7)

• definition 1: $k$-plex seidman1978graph
• definition 2
• lemma 1
• lemma 2
• lemma 3
• lemma 4
• lemma 5