AI 中文总结
本文提出首个多项式时间算法解决Győri-Lovász定理这一算法图论核心开放问题，引入流-本质分配概念，还获其有向版本多项式算法及DAG的近线性算法，及加权推广的多项式算法。
AI 中文摘要
历经半个世纪，我们为著名的Győri-Lovász定理提供了首个多项式时间算法，该定理解决了Frank于1975年提出的猜想。该定理是最易解释的存在性定理之一，指出每个k-连通图都可划分为k个互不相交、大小为任意指定正值的连通子图。这是一项具有广泛应用的基础结构性结果，例如在足够连通的云基础设施中灵活分配指定大小的连通子网。尽管Lovász于1977年利用代数拓扑给出了更强有向版本的高度非构造性证明，Győri于1976年的原始构造性证明需要指数时间；历经50余年的努力，即使对于k>4的情况，也未发现多项式时间算法。确定Győri-Lovász定理的计算复杂性——其是否存在亚指数时间算法或计算困难（尤其是PLS完全或PPAD完全）——一直是算法图论的核心开放问题之一。本文通过引入全新的“流-本质分配”概念，该概念真正结合了匹配与割结构，最终解决了这一长期存在的问题，为该存在性定理提供了全新证明，得到了Győri-Lovász定理的首个多项式时间构造性算法。事实上，我们获得了Lovász更强有向版本的多项式时间算法，其证明即使对于DAG也非构造性；对于DAG，我们进一步得到了近线性时间算法。我们还为加权推广开发了多项式时间算法，而Chen、Kleinberg、Lovász、Rajaraman、Sundaram和Vetta（JACM'07）关于汇合流的开创性工作仅建立了存在性非构造性结果。
英文摘要
We give the first polynomial-time algorithm, after half a century, for the celebrated Győri-Lovász theorem, which resolved a conjecture of Frank (1975). The theorem, one of the simplest existential theorems to explain, states that every $k$-connected graph can be partitioned into $k$ disjoint connected subgraphs of arbitrary prescribed positive sizes. This is a fundamental structural result with broad applications, such as flexible allocation of connected subnetworks of prescribed sizes in sufficiently connected cloud infrastructures. While Lovász (1977) gave a highly non-constructive proof for a stronger directed version using algebraic topology, Győri's original constructive proof (1976) requires exponential time. Despite more than 50 years of effort, no polynomial-time algorithm was known even for $k>4$. Determining the computational complexity of the Győri-Lovász theorem---whether it admits even a sub-exponential-time algorithm or is computationally hard (in particular, PLS-complete or PPAD)---has remained one of the central open problems in algorithmic graph theory. In this paper, we finally resolve this long-standing problem by a fundamentally new proof of the existential theorem via introducing the novel concept of \emph{flow-essential assignment}, which genuinely marries matching and cut structures and yields the first polynomial-time constructive algorithm for the Győri-Lovász theorem. In fact, we obtain a polynomial-time algorithm for Lovász's stronger directed version, whose proof was non-constructive even for DAGs; for DAGs, we further obtain a near-linear-time algorithm. We also develop polynomial-time algorithms for weighted generalizations where the seminal work of Chen, Kleinberg, Lovász, Rajaraman, Sundaram, and Vetta (JACM'07) on confluent flows established only existential non-constructive results.
