On Finding Bipartite Graphs With a Small Number of Short Cycles and Large Girth

Abstract

The problem of finding bipartite (Tanner) graphs with given degree sequences that have large girth and few short cycles is of great interest in many applications including construction of good low-density parity-check (LDPC) codes. In this paper, we prove that for given integers $\alpha $ , $\beta $ , and $\gamma $ , and degree sequences $\pi $ and $\pi '$ , the problem of determining whether there exists a simple bipartite graph with degree sequences $(\pi, \pi ')$ that has at most $\alpha $ ( $\beta $ and $\gamma $ ) cycles of length four (six and eight, respectively) is NP-complete. This is proved by a two-step polynomial-time reduction from the 3-Partition Problem. On the other hand, using connections to linear hypergraphs, we prove that given the degree sequence $\pi $ , a polynomial time algorithm can be devised to determine whether there exists a bipartite graph whose degree sequence on one side of the bipartition is $\pi $ and has a girth of at least six. In addition to these complexity results, we devise a quasi-polynomial time algorithm that can construct a bipartite graph with given (irregular) degree sequences and a girth that increases logarithmically with the size of the graph. Compared to the well-known progressive-edge-growth (PEG) algorithm, the proposed method, in some cases, results in Tanner graphs with larger girth, particularly for scenarios where the degree sequences of the graph are strictly enforced.