The \emph{maximal $k$-edge-connected subgraphs} problem is a classical graph clustering problem studied since the 70's. Surprisingly, no non-trivial technique for this problem in weighted graphs is known: a very straightforward recursive-mincut algorithm with $\Omega(mn)$ time has remained the fastest algorithm until now. All previous progress gives a speed-up only when the graph is unweighted, and $k$ is small enough (e.g.~Henzinger~et~al.~(ICALP'15), Chechik~et~al.~(SODA'17), and Forster~et~al.~(SODA'20)). We give the first algorithm that breaks through the long-standing $\tilde{O}(mn)$-time barrier in \emph{weighted undirected} graphs. More specifically, we show a maximal $k$-edge-connected subgraphs algorithm that takes only $\tilde{O}(m\cdot\min\{m^{3/4},n^{4/5}\})$ time. As an immediate application, we can $(1+\epsilon)$-approximate the \emph{strength} of all edges in undirected graphs in the same running time. Our key technique is the first local cut algorithm with \emph{exact} cut-value guarantees whose running time depends only on the output size. All previous local cut algorithms either have running time depending on the cut value of the output, which can be arbitrarily slow in weighted graphs or have approximate cut guarantees.

翻译：最大 $k$-边连通子图问题是一个自70年代以来就被研究的经典图聚类问题。令人惊讶的是，目前尚无针对加权图中该问题的非平凡技术：一个非常直接的递归最小割算法（时间复杂度为 $\Omega(mn)$）至今仍是最快的算法。以往的所有进展仅在图为非加权且 $k$ 足够小的情况下提供了加速（例如 Henzinger 等人（ICALP'15）、Chechik 等人（SODA'17）和 Forster 等人（SODA'20））。我们提出了首个突破长期存在的 $\tilde{O}(mn)$ 时间壁垒的算法，适用于\emph{加权无向}图。具体而言，我们展示了一个最大 $k$-边连通子图算法，仅需 $\tilde{O}(m\cdot\min\{m^{3/4},n^{4/5}\})$ 时间。作为一个直接应用，我们可以在相同运行时间内对无向图中所有边的\emph{强度}进行 $(1+\epsilon)$-近似。我们的关键技术是首个具有\emph{精确}割值保证的局部割算法，其运行时间仅取决于输出规模。以往所有局部割算法要么运行时间取决于输出的割值（在加权图中可能任意慢），要么仅具有近似割值保证。