Let $Q$ be a bipartite quiver with vertex set $Q_0$ such that the number of arrows between any two source and sink vertices is constant. Let $\beta=(\beta(x))_{x \in Q_0}$ be a dimension vector of $Q$ with positive integer coordinates, and let $\Delta(Q, \beta)$ be the moment cone associated to $(Q, \beta)$. We show that the membership problem for $\Delta(Q, \beta)$ can be solved in strongly polynomial time. As a key step in our approach, we first solve the polytopal problem for semi-invariants of $Q$ and its flag-extensions. Specifically, let $Q_{\beta}$ be the flag-extension of $Q$ obtained by attaching a flag $\mathcal{F}(x)$ of length $\beta(x)-1$ at every vertex $x$ of $Q$, and let $\widetilde{\beta}$ be the extension of $\beta$ to $Q_{\beta}$ that takes values $1, \ldots, \beta(x)$ along the vertices of the flag $\mathcal{F}(x)$ for every vertex $x$ of $Q$. For an integral weight $\widetilde{\sigma}$ of $Q_{\beta}$, let $K_{\widetilde{\sigma}}$ be the dimension of the space of semi-invariants of weight $\widetilde{\sigma}$ on the representation space of $\widetilde{\beta}$-dimensional complex representations of $Q_{\beta}$. We show that $K_{\widetilde{\sigma}}$ can be expressed as the number of lattice points of a certain hive-type polytope. This polytopal description together with Derksen-Weyman's Saturation Theorem for quiver semi-invariants allows us to use Tardos's algorithm to solve the membership problem for $\Delta(Q,\beta)$ in strongly polynomial time.

翻译：设$Q$为顶点集$Q_0$的二部箭图，其中任意源顶点与汇顶点之间的箭条数为常数。令$\beta=(\beta(x))_{x \in Q_0}$为$Q$的维数向量（分量均为正整数），$\Delta(Q, \beta)$为$(Q, \beta)$关联的矩锥。我们证明$\Delta(Q, \beta)$的成员关系问题可在强多项式时间内求解。作为方法的关键步骤，我们首先解决了$Q$及其旗扩张的半不变量多面体问题。具体而言，设$Q_{\beta}$为在$Q$的每个顶点$x$处附加长度为$\beta(x)-1$的旗$\mathcal{F}(x)$得到的旗扩张箭图，$\widetilde{\beta}$为$\beta$在$Q_{\beta}$上的扩张：对每个顶点$x$，沿旗$\mathcal{F}(x)$的顶点依次取值$1, \ldots, \beta(x)$。对$Q_{\beta}$的整权$\widetilde{\sigma}$，令$K_{\widetilde{\sigma}}$表示$Q_{\beta}$的$\widetilde{\beta}$维复表示空间上权为$\widetilde{\sigma}$的半不变量空间的维数。我们证明$K_{\widetilde{\sigma}}$可表示为特定蜂巢型多面体的格点数。结合该多面体描述与Derksen-Weyman箭图半不变量饱和定理，我们得以利用Tardos算法在强多项式时间内求解$\Delta(Q,\beta)$的成员关系问题。