Maximal Biclique Enumeration with Improved Worst-Case Time Complexity Guarantee: A Partition-Oriented Strategy

TL;DR

This paper introduces a new branch-and-bound (BB) algorithm \texttt{IPS), and proposes an improved pivot selection strategy, which well aligns with the new termination condition to achieve better theoretical and practical performance.

Abstract

The maximal biclique enumeration problem in bipartite graphs is fundamental and has numerous applications in E-commerce and transaction networks. Most existing studies adopt a branch-and-bound framework, which recursively expands a partial biclique with a vertex until no further vertices can be added. Equipped with a basic pivot selection strategy, all state-of-the-art methods have a worst-case time complexity no better than $O(m\cdot (\sqrt{2})^n)$}, where $m$ and $n$ are the number of edges and vertices in the graph, respectively. In this paper, we introduce a new branch-and-bound (BB) algorithm \texttt{IPS}. In \texttt{IPS}, we relax the strict stopping criterion of existing methods by allowing termination when all maximal bicliques within the current branch can be outputted in the time proportional to the number of maximal bicliques inside, reducing the total number of branches required. Second, to fully unleash the power of the new termination condition, we propose an improved pivot selection strategy, which well aligns with the new termination condition to achieve better theoretical and practical performance. Formally, \texttt{IPS} improves the worst-case time complexity to $O(m\cdot α^n + n\cdot β)$, where $α(\approx 1.3954)$ is the largest positive root of $x^4-2x-1=0$ and $β$ represents the number of maximal bicliques in the graph, respectively. This result surpasses that of all existing algorithms given that $α$ is strictly smaller than $\sqrt{2}$ and $β$ is at most $(\sqrt{2})^n-2$ theoretically. Furthermore, we apply an inclusion-exclusion-based framework to boost the performance of \texttt{IPS}, improving the worst-case time complexity to $O(n\cdot γ^2\cdotα^γ+ γ\cdot β)$ for large sparse graphs ($γ$ is a parameter satisfying $γ\ll n$ for sparse graphs).

Figures (5)

• Figure 1: An example graph with $n$ vertices ($n$ is even), in which each $u_i$ is disconnected only from $v_i$ for $i=1,2,\ldots,n/2$, as indicated by the dashed edges. In total, the graph contains exactly $(\sqrt{2})^n - 2$ maximal bicliques. To see this, observe that any non-empty subset of vertices chosen from $U$ uniquely determines a maximal biclique by taking its common neighbors in $V$. Since $|U|=|V|=n/2$, the number of such choices is $2^{n/2}-2$ (equiv. $(\sqrt{2})^n - 2$), excluding the empty set and the full set so that both sides of the biclique are non-empty.
• Figure 2: An illustration of the reason why the conventional pivoting strategy with the new stopping criterion would still produce much more branches than necessary.
• Figure 3: Results of delay testing.
• Figure 4: Comparison among baselines when enumerating size-constrained maximal bicliques (varying $\tau=\tau_L=\tau_R$).
• Figure 5: Results of scalability testing on TV dataset.

Theorems & Definitions (6)

• Definition 2.1: Biclique bondy1976graph
• lemma 1
• proof
• theorem 1
• proof : Proof
• theorem 2