Counting Short Cycles in Bipartite Graphs: A Fast Technique/Algorithm and a Hardness Result

Abstract

In this paper, we propose a new technique, based on the so-called breadth-first search algorithm, to count the short cycles of a bipartite graph. For a general bipartite graph with $|V|$ nodes and girth $g$ , our technique has a time complexity of $\mathcal {O}(|V|^{2}\Delta )$ to count $g$ -cycles and $(g+2)$ -cycles, and a time complexity of $\mathcal {O}(|V|^{2}\Delta ^{2})$ to count $(g+4)$ -cycles, where $\Delta $ is the maximum node degree in the graph. Moreover, for bi-regular bipartite graphs, the latter complexity is further reduced to $\mathcal {O}(|V|^{2}\Delta )$ . Compared to the fastest known algorithm, which has a complexity $\mathcal {O}(g |V|^{2} \Delta ^{2})$ , the proposed method always has a lower complexity for counting $g$ -cycles and $(g+2)$ -cycles. It also has a lower complexity for counting $(g+4)$ -cycles in bi-regular graphs and in scenarios where $g$ is increased with the size of the graph. Related to the problem of counting short cycles, we also demonstrate, using a long-standing conjecture, that there is no algorithm with time complexity less than $\mathcal {O}\left({|V|^{2-\frac {2}{1+i}}}\right)$ that can determine whether a given sparse bipartite graph has a cycle of length $4i$ . An important application of the results presented here is to count the short cycles of Tanner graphs of low-density parity-check (LDPC) codes.