Finding cohesive subgraphs from a directed graph is a fundamental approach to analyze directed graph data. We consider a new model called directed$(k,\ell )$-plex for a cohesive directed subgraph, which is generalized from the concept of$k$-plex that is only applicable to undirected graphs. Directed$(k,\ell )$-plex (or DPlex) has the connection requirements on both inbound and outbound directions of each vertex inside, i.e., each vertex disconnects at most$k$vertices and is meanwhile not pointed to by at most$\ell$vertices. In this paper, we study the maximum DPlex search problem which finds a DPlex with the most vertices. We formally prove the NP-hardness of the problem. We then design a heuristic algorithm calledDPHeuris, which finds a DPlex with the size close to the maximum one and runs practically fast in polynomial time. Furthermore, we propose a branch-and-bound algorithm calledDPBBto find the exact maximum DPlex and develop effective graph reduction strategies for boosting the empirical performance. We also consider the problem of querying personalized maximum DPlex, and design a new method calledDPBBQfor the problem. Finally, we conduct extensive experiments on real directed graphs. The experimental results show that (1) our heuristic method can quickly find a near-optimal solution and (2) our branch-and-bound method runs up to six orders of magnitude faster than other baselines.

Finding cohesive subgraphs from a directed graph is a fundamental approach to analyze directed graph data. We consider a new model called directed$(k,\ell)$-plex for a cohesive directed subgraph, which is generalized from the concept of$k$-plex that is only applicable to undirected graphs. Directed$(k,\ell)$-plex has the connection requirements on both inbound and outbound directions of each vertex inside, i.e., each vertex disconnects at most$K$vertices and is meanwhile not pointed to by at most$\ell$vertices. In this paper, we study the maximum directed$(k, \ell)$-plex search problem which finds a directed$(k, \ell)$-plex with the most vertices. We formally prove the NP-hardness of the problem. We then design a heuristic algorithm called DPHeuris, which finds a directed$(k, \ell)$-plex with the size close to the maximum one and runs practically fast in polynomial time. Furthermore, we propose a branch-and-bound algorithm called DPBB to find the exact maximum directed$(k, \ell)$-plex and develop effective graph reduction strategies for boosting the empirical performance. Finally, we conduct extensive experiments on real directed graphs. The experimental results show that (1) our heuristic method can quickly find a near-optimal solution and (2) our branch-and-bound method runs up to six orders of magnitude faster than other baselines.

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 preprocessing techniques runs up to two orders of magnitude faster and solves more instances than state-of-the-art algorithms.