Hive-type polytopes for quiver multiplicities and the membership problem for quiver moment cones

Let $Q$ be a bipartite quiver with vertex set $Q_0$ such that the number of arrows between any source vertex and any sink vertex is constant. Let $\beta=(\beta(x))_{x \in Q_0}$ be a dimension vector of $Q$ with positive integer coordinates. Let $rep(Q, \beta)$ be the representation space of $\beta$-dimensional representations of $Q$ and $GL(\beta)$ the base change group acting on $rep(Q, \beta)$ be simultaneous conjugation. Let $K^{\beta}_{\underline{\lambda}}$ be the multiplicity of the irreducible representation of $GL(\beta)$ of highest weight $\underline{\lambda}$ in the ring of polynomial functions on $rep(Q, \beta)$. We show that $K^{\beta}_{\underline{\lambda}}$ can be expressed as the number of lattice points of a polytope obtained by gluing together two Knutson-Tao hive polytopes. Furthermore, this polytopal description together with Derksen-Weyman's Saturation Theorem for quiver semi-invariants allows us to use Tardos' algorithm to solve the membership problem for the moment cone associated to $(Q,\beta)$ in strongly polynomial time.