The problem of realizing a given degree sequence by a multigraph can be thought of as a relaxation of the classical degree realization problem (where the realizing graph is simple). This paper concerns the case where the realizing multigraph is required to be bipartite. The problem of characterizing sequences that can be realized by a bipartite graph has two variants. In the simpler one, termed BDR$^P$, the partition of the sequence into two sides is given as part of the input. A complete characterization for realizability in this variant was given by Gale and Ryser over sixty years ago. However, the variant where the partition is not given, termed BDR, is still open. For bipartite multigraph realizations, there are also two variants. For BDR$^P$, where the partition is given as part of the input, a characterization was known for determining whether there is a multigraph realization whose underlying graph is bipartite, such that the maximum number of copies of an edge is at most $r$. We present a characterization for determining if there is a bipartite multigraph realization such that the total number of excess edges is at most $t$. We show that optimizing these two measures may lead to different realizations, and that optimizing by one measure may increase the other substantially. As for the variant BDR, where the partition is not given, we show that determining whether a given (single) sequence admits a bipartite multigraph realization is NP-hard. Moreover, we show that this hardness result extends to any graph family which is a sub-family of bipartite graphs and a super-family of paths. On the positive side, we provide an algorithm that computes optimal realizations for the case where the number of balanced partitions is polynomial, and present sufficient conditions for the existence of bipartite multigraph realizations that depend only on the largest degree of the sequence.

翻译：通过多重图实现给定度序列的问题可视为经典度序列实现问题（要求实现图为简单图）的一种松弛形式。本文研究要求实现多重图为二部图的情形。刻画可由二部图实现的序列问题存在两种变体：在较简单的变体BDR$^P$中，序列划分为两侧的分区作为输入的一部分给出。Gale和Ryser在六十余年前已给出该变体可实现性的完整刻画。然而，对于分区未给定的变体BDR，其刻画问题至今尚未解决。对于二部多重图实现问题，同样存在两种变体。在分区作为输入给定的BDR$^P$变体中，已有刻画方法可用于判断是否存在底层图为二部、且单边最大重数不超过$r$的多重图实现。本文提出一种刻画方法，用于判断是否存在总超额边数不超过$t$的二部多重图实现。我们证明优化这两个度量可能导致不同的实现方案，且优化某一度量可能显著增加另一度量。对于分区未给定的BDR变体，我们证明判断给定（单一）序列是否允许二部多重图实现是NP难问题。此外，该硬度结果可推广至任何属于二部图子族且包含路径图超族的图族。在积极方面，我们提出一种算法，可在平衡分区数量为多项式时计算最优实现，并给出仅依赖于序列最大度的二部多重图实现存在性充分条件。