On the Computational Complexity of Finding Bipartite Graphs with a Small Number of Short Cycles and Large Girth

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 a given set of 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.