In 1979 Tarsi showed that an edge decomposition of a complete multigraph into stars of size m exists whenever some obvious necessary conditions hold. In 1992 Lonc gave necessary and sufficient conditions for the existence of an edge decomposition of a (simple) complete graph into stars of sizes $m_1,\dots ,m_t$. We show that the general problem of when a complete multigraph admits a decomposition into stars of sizes $m_1,\dots ,m_t$ is $\mathsf{NP}$-complete, but that it becomes tractable if we place a strong enough upper bound on $\max (m_1,\dots ,m_t)$. We determine the upper bound at which this transition occurs. Along the way we also give a characterisation of when an arbitrary multigraph can be decomposed into stars of sizes $m_1,\dots ,m_t$ with specified centres, and a generalisation of Landau's theorem on tournaments.

1979年，塔西（Tarsi）表明，只要存在某些明显必要的条件，就会将一个完整的多重图边缘分解成m个星。在1992年，隆克（Lonc）为（简单）完整图的边分解为大小为星的存在提供了充要条件。${米}_{\mathrm{1个}}，\dots ，{米}_{Ť}$。我们证明了一个完整的多重图接受一个分解为大小为星号的一般问题${米}_{\mathrm{1个}}，\dots ，{米}_{Ť}$ 是 $\mathsf{NP}$-完全，但是如果我们在其上设置足够强的上限，它将变得易于处理 $\mathrm{最高}（{米}_{\mathrm{1个}}，\dots ，{米}_{Ť}）$。我们确定发生此过渡的上限。在此过程中，我们还描述了何时可以将任意多重图分解为大小为星号的特征${米}_{\mathrm{1个}}，\dots ，{米}_{Ť}$ 具有指定的中心，并在比赛中推广了Landau定理。