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 α, β, and -y, and degree sequences π and π', the problem of determining whether there exists a simple bipartite graph with degree sequences (π, π') that has at most α (β and -y) 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 π, 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 π 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.

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关于寻找少量短环和大周长的二分图


寻找具有大周长和很少短周期的给定度数序列的二分（Tanner）图的问题在许多应用中引起了极大的兴趣，包括构建良好的低密度奇偶校验（LDPC）码。在本文中，我们证明了对于给定的整数 α、β 和 -y 以及度数序列 π 和 π'，确定是否存在度数序列 (π, π') 至多有 的简单二部图的问题长度为 4（分别为 6 和 8）的 α（β 和 -y）循环是 NP 完全的。这通过三划分问题的两步多项式时间简化得到了证明。另一方面，使用与线性超图的连接，我们证明给定度数序列 π，可以设计一个多项式时间算法来确定是否存在二分图，其二分一侧的度数序列为 π 并且周长为至少有六个。除了这些复杂性结果之外，我们还设计了一种拟多项式时间算法，该算法可以构造具有给定（不规则）度序列和周长的二分图，周长随图的大小以对数方式增加。与众所周知的渐进边缘增长（PEG）算法相比，所提出的方法在某些情况下会产生具有更大周长的 Tanner 图，特别是对于严格执行图的度数序列的情况。