Almost-Optimal Approximation Algorithms for Global Minimum Cut in Directed Graphs
Toyota Technological Institute at Chicago（芝加哥丰田技术学院）
Ron Mosenzon
备注：
40 pages. Full version of a paper in STOC 2026. v2: added discussion of new independent work; v3: updated section 3.1 to match proceedings version, fixed typos, and updated the bibliography
发表信息：
Proceedings of the 58th Annual ACM Symposium on Theory of Computing (STOC 2026), pp. 106-117
We develop new $(1+ε)$-approximation algorithms for finding the global minimum edge-cut in a directed edge-weighted graph, and for finding the global minimum vertex-cut in a directed vertex-weighted graph. Our algorithms are randomized, and have a running time of $O\left(m^{1+o(1)}/ε\right)$ on any $m$-edge $n$-vertex input graph, assuming all edge/vertex weights are polynomially-bounded. In particular, for any constant $ε>0$, our algorithms have an almost-optimal running time of $O\left(m^{1+o(1)}\right)$. The fastest previously-known running time for this setting, due to (Cen et al., FOCS 2021), is $\tilde{O}\left(\min\left\{n^2/ε^2,m^{1+o(1)}\sqrt{n}\right\}\right)$ for Minimum Edge-Cut, and $\tilde{O}\left(n^2/ε^2\right)$ for Minimum Vertex-Cut. Our results further extend to the rooted variants of the Minimum Edge-Cut and Minimum Vertex-Cut problems, where the algorithm is additionally given a root vertex $r$, and the goal is to find a minimum-weight cut separating any vertex from the root $r$. In terms of techniques, we build upon and extend a framework that was recently introduced by (Chuzhoy et al., SODA 2026) for solving the Minimum Vertex-Cut problem in unweighted directed graphs. Additionally, in order to obtain our result for the Global Minimum Vertex-Cut problem, we develop a novel black-box reduction from this problem to its rooted variant. Prior to our work, such reductions were only known for more restricted settings, such as when all vertex-weights are unit.
We give O(log^2 n)-approximation algorithm based on the cut-matching framework of [10, 13, 14] for computing the sparsest cut on directed graphs. Our algorithm uses only O(log^2 n) single commodity max-flow computations and thus breaks the multicommodity-flow barrier for computing the sparsest cut on directed graphs
We give an $O(n \log \log n)$ time algorithm for computing the minimum cut (or equivalently, the shortest cycle) of a weighted directed planar graph. This improves the previous fastest $O(n\log^3 n)$ solution. Interestingly, while in undirected planar graphs both min-cut and min $st$-cut have $O(n \log \log n)$ solutions, in directed planar graphs our result makes min-cut faster than min $st$-cut, which currently requires $O(n \log n)$.
In this paper, we consider the problem of designing cut sparsifiers and sketches for directed graphs. To bypass known lower bounds, we allow the sparsifier/sketch to depend on the balance of the input graph, which smoothly interpolates between undirected and directed graphs. We give nearly matching upper and lower bounds for both for-all (cf. Benczúr and Karger, STOC 1996) and for-each (Andoni et al., ITCS 2016) cut sparsifiers/sketches as a function of cut balance, defined the maximum ratio of the cut value in the two directions of a directed graph (Ene et al., STOC 2016). We also show an int…
We consider approximations for computing minimum weighted cuts in directed graphs. We consider both rooted and global minimum cuts, and both edge-cuts and vertex-cuts. For these problems we give randomized Monte Carlo algorithms that compute a $(1+ε)$-approximate minimum cut in $\tilde{O}(n^2 / ε^2)$ time. These results extend and build on recent work [4] that obtained exact algorithms with similar running times in directed graphs with small integer capacities.
Let $G=(V,E)$ be a strongly biconnected directed graph. In this paper we consider the problem of computing an edge subset $H \subseteq E$ of minimum size such that the directed subgraph $(V,H)$ is strongly biconnected.
We give an algorithm to find a mincut in an $n$-vertex, $m$-edge weighted directed graph using $\tilde O(\sqrt{n})$ calls to any maxflow subroutine. Using state of the art maxflow algorithms, this yields a directed mincut algorithm that runs in $\tilde O(m\sqrt{n} + n^2)$ time. This improves on the 30 year old bound of $\tilde O(mn)$ obtained by Hao and Orlin for this problem.
The multi-commodity flow-cut gap is a fundamental parameter that affects the performance of several divide \& conquer algorithms, and has been extensively studied for various classes of undirected graphs. It has been shown by Linial, London and Rabinovich and by Aumann and Rabani that for general $n$-vertex graphs it is bounded by $O(\log n)$ and the Gupta-Newman-Rabinovich-Sinclair conjecture asserts that it is $O(1)$ for any family of graphs that excludes some fixed minor. We show that the multicommodity flow-cut gap on \emph{directed} planar graphs is $O(\log^3 n)$. This is the first \emph{…
We give an algorithm to find a minimum cut in an edge-weighted directed graph with $n$ vertices and $m$ edges in $\tilde O(n\cdot \max(m^{2/3}, n))$ time. This improves on the 30 year old bound of $\tilde O(nm)$ obtained by Hao and Orlin for this problem. Our main technique is to reduce the directed mincut problem to $\tilde O(\min(n/m^{1/3}, \sqrt{n}))$ calls of {\em any} maxflow subroutine. Using state-of-the-art maxflow algorithms, this yields the above running time. Our techniques also yield fast {\em approximation} algorithms for finding minimum cuts in directed graphs. For both edge and …
The multi-commodity flow-cut gap is a fundamental parameter that affects the performance of several divide \& conquer algorithms, and has been extensively studied for various classes of undirected graphs. It has been shown by Linial, London and Rabinovich \cite{linial1994geometry} and by Aumann and Rabani \cite{aumann1998log} that for general $n$-vertex graphs it is bounded by $O(\log n)$ and the Gupta-Newman-Rabinovich-Sinclair conjecture \cite{gupta2004cuts} asserts that it is $O(1)$ for any family of graphs that excludes some fixed minor. The flow-cut gap is poorly understood for the case o…
Wu and Grumbach introduced the concept of strongly biconnected directed graphs. A directed graph $G=(V,E)$ is called strongly biconnected if the directed graph $G$ is strongly connected and the underlying undirected graph of $G$ is biconnected. A strongly biconnected directed graph $G=(V,E)$ is said to be $2$- edge strongly biconnected if it has at least three vertices and the directed subgraph $(V,E\setminus\left\lbrace e\right\rbrace )$ is strongly biconnected for all $e \in E$. Let $G=(V,E)$ be a $2$-edge-strongly biconnected directed graph. In this paper we study the problem of computing a…
We investigate the odd multiway node (edge) cut problem where the input is a graph with a specified collection of terminal nodes and the goal is to find a smallest subset of nonterminal nodes (edges) to delete so that the terminal nodes do not have an odd length path between them. In an earlier work, Lokshtanov and Ramanujan showed that both odd multiway node cut and odd multiway edge cut are fixed-parameter tractable (FPT) when parameterized by the size of the solution in undirected graphs. In this work, we focus on directed acyclic graphs (DAGs) and design a fixed-parameter algorithm. Our ma…
A directed graph $G=(V,E)$ is strongly biconnected if $G$ is strongly connected and its underlying graph is biconnected. A strongly biconnected directed graph $G=(V,E)$ is called $2$-vertex-strongly biconnected if $|V|\geq 3$ and the induced subgraph on $V\setminus\left\lbrace w\right\rbrace $ is strongly biconnected for every vertex $w\in V$. In this paper we study the following problem. Given a $2$-vertex-strongly biconnected directed graph $G=(V,E)$, compute an edge subset $E^{2sb} \subseteq E$ of minimum size such that the subgraph $(V,E^{2sb})$ is $2$-vertex-strongly biconnected.
Given a directed graph $G$, a set of $k$ terminals and an integer $p$, the \textsc{Directed Vertex Multiway Cut} problem asks if there is a set $S$ of at most $p$ (nonterminal) vertices whose removal disconnects each terminal from all other terminals. \textsc{Directed Edge Multiway Cut} is the analogous problem where $S$ is a set of at most $p$ edges. These two problems indeed are known to be equivalent. A natural generalization of the multiway cut is the \emph{multicut} problem, in which we want to disconnect only a set of $k$ given pairs instead of all pairs. Marx (Theor. Comp. Sci. 2006) sh…
We present a practically efficient algorithm that finds all global minimum cuts in huge undirected graphs. Our algorithm uses a multitude of kernelization rules to reduce the graph to a small equivalent instance and then finds all minimum cuts using an optimized version of the algorithm of Nagamochi, Nakao and Ibaraki. In shared memory we are able to find all minimum cuts of graphs with up to billions of edges and millions of minimum cuts in a few minutes. We also give a new linear time algorithm to find the most balanced minimum cuts given as input the representation of all minimum cuts.
This paper introduces a special family of randomized algorithms for Max DICUT that we call oblivious algorithms. Let the bias of a vertex be the ratio between the total weight of its outgoing edges and the total weight of all its edges. An oblivious algorithm selects at random in which side of the cut to place a vertex v, with probability that only depends on the bias of v, independently of other vertices. The reader may observe that the algorithm that ignores the bias and chooses each side with probability 1/2 has an approximation ratio of 1/4, whereas no oblivious algorithm can have an appro…
Let $G$ be an $n$-node simple directed planar graph with nonnegative edge weights. We study the fundamental problems of computing (1) a global cut of $G$ with minimum weight and (2) a~cycle of $G$ with minimum weight. The best previously known algorithm for the former problem, running in $O(n\log^3 n)$ time, can be obtained from the algorithm of \Lacki, Nussbaum, Sankowski, and Wulff-Nilsen for single-source all-sinks maximum flows. The best previously known result for the latter problem is the $O(n\log^3 n)$-time algorithm of Wulff-Nilsen. By exploiting duality between the two problems in pla…
We show that every $α$-approximate minimum cut in a connected graph is the unique minimum $(S,T)$-terminal cut for some subsets $S$ and $T$ of vertices each of size at most $\lfloor2α\rfloor+1$. This leads to an alternative proof that the number of $α$-approximate minimum cuts in a $n$-vertex connected graph is $n^{O(α)}$ and they can all be enumerated in deterministic polynomial time for constant $α$.
We consider the Minimum Steiner Cut problem on undirected planar graphs with non-negative edge weights. This problem involves finding the minimum cut of the graph that separates a specified subset $X$ of vertices (terminals) into two parts. This problem is of theoretical interest because it generalizes two classical optimization problems, Minimum $s$-$t$ Cut and Minimum Cut, and of practical importance because of its application to computing a lower bound for Steiner (Subset) TSP. Our algorithm has running time $O(n\log{n}\log{k})$ where $k$ is the number of terminals.
In this paper, we present two approximation algorithms for the directed multi-multiway cut and directed multicut problems. The so called region growing paradigm \cite{1} is modified and used for these two cut problems on directed graphs. By using this paradigm, we give for each problem an approximation algorithm such that both algorithms have the approximate factor $O(k)$ the same as the previous works done on these problems. However, the previous works need to solve $k$ linear programming, whereas our algorithms require only one linear programming. Therefore, our algorithms improve the runnin…
We study multi-budgeted variants of the classic minimum cut problem and graph separation problems that turned out to be important in parameterized complexity: Skew Multicut and Directed Feedback Arc Set. In our generalization, we assign colors $1,2,...,\ell$ to some edges and give separate budgets $k_{1},k_{2},...,k_{\ell}$. Let $E_{i}$ be the set of edges of color $i$. The solution $C$ for the multi-budgeted variant of a graph separation problem not only needs to satisfy the usual separation requirements, but also needs to satisfy that $|C\cap E_{i}|\leq k_{i}$ for every $i\in \{1,...,\ell\}$…
We consider the minimum cut problem in undirected, weighted graphs. We give a simple algorithm to find a minimum cut that $2$-respects (cuts two edges of) a spanning tree $T$ of a graph $G$. This procedure can be used in place of the complicated subroutine given in Karger's near-linear time minimum cut algorithm (J. ACM, 2000). We give a self-contained version of Karger's algorithm with the new procedure, which is easy to state and relatively simple to implement. It produces a minimum cut on an $m$-edge, $n$-vertex graph in $O(m \log^3 n)$ time with high probability, matching the complexity of…
We give query-efficient algorithms for the global min-cut and the s-t cut problem in unweighted, undirected graphs. Our oracle model is inspired by the submodular function minimization problem: on query $S \subset V$, the oracle returns the size of the cut between $S$ and $V \setminus S$. We provide algorithms computing an exact minimum $s$-$t$ cut in $G$ with $\tilde{O}(n^{5/3})$ queries, and computing an exact global minimum cut of $G$ with only $\tilde{O}(n)$ queries (while learning the graph requires $\tildeΘ(n^2)$ queries).
We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a ``semi-duality'' between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized algorithm that finds a minimum cut in an m-edge, n-vertex graph with high probability in O(m log^3 n) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O(n^2 log n) time. This variant has an optimal RNC parallelization. Both variants improve on the previous best time bound of…
We give a randomized algorithm that finds a minimum cut in an undirected weighted $m$-edge $n$-vertex graph $G$ with high probability in $O(m \log^2 n)$ time. This is the first improvement to Karger's celebrated $O(m \log^3 n)$ time algorithm from 1996. Our main technical contribution is a deterministic $O(m \log n)$ time algorithm that, given a spanning tree $T$ of $G$, finds a minimum cut of $G$ that 2-respects (cuts two edges of) $T$.
Given an edge-weighted graph, how many minimum $k$-cuts can it have? This is a fundamental question in the intersection of algorithms, extremal combinatorics, and graph theory. It is particularly interesting in that the best known bounds are algorithmic: they stem from algorithms that compute the minimum $k$-cut. In 1994, Karger and Stein obtained a randomized contraction algorithm that finds a minimum $k$-cut in $O(n^{(2-o(1))k})$ time. It can also enumerate all such $k$-cuts in the same running time, establishing a corresponding extremal bound of $O(n^{(2-o(1))k})$. Since then, the algorithm…
In an undirected graph, a $k$-cut is a set of edges whose removal breaks the graph into at least $k$ connected components. The minimum weight $k$-cut can be computed in $O(n^{O(k)})$ time, but when $k$ is treated as part of the input, computing the minimum weight $k$-cut is NP-Hard [Holdschmidt and Hochbaum 1994]. For $\operatorname{poly}(m,n,k)$-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi 2017]. Saran and Vazirani [1995] showed that a $(2 - 2/k)$-approximately minimum weight $k$-cut can be computed by $O(k)$ min…
Given a directed graph $G$ on $n$ vertices with a special vertex $s$, the directed minimum degree spanning tree problem requires computing a incoming spanning tree rooted at $s$ whose maximum tree in-degree is the smallest among all such trees. The problem is known to be NP-hard, since it generalizes the Hamiltonian path problem. The best LP-based polynomial time algorithm can achieve an approximation of $Δ^*+2$ [Bansal et al, 2009], where $Δ^*$ denotes the optimal maximum tree in-degree. As for purely combinatorial algorithms (algorithms that do not use LP), the best approximation is $O(Δ^*+\…
We present a practically efficient algorithm for maintaining a global minimum cut in large dynamic graphs under both edge insertions and deletions. While there has been theoretical work on this problem, our algorithm is the first implementation of a fully-dynamic algorithm. The algorithm uses the theoretical foundation and combines it with efficient and finely-tuned implementations to give an algorithm that can maintain the global minimum cut of a graph with rapid update times. We show that our algorithm gives up to multiple orders of magnitude speedup compared to static approaches both on edg…
In this paper, we study a generalization of the classical minimum cut prob- lem, called Connectivity Preserving Minimum Cut (CPMC) problem, which seeks a minimum cut to separate a pair (or pairs) of source and destination nodes and meanwhile ensure the connectivity between the source and its partner node(s). The CPMC problem is a rather powerful formulation for a set of problems and finds applications in many other areas, such as network security, image processing, data mining, pattern recognition, and machine learning. For this important problem, we consider two variants, connectiv- ity prese…
In this work, we resolve the query complexity of global minimum cut problem for a graph by designing a randomized algorithm for approximating the size of minimum cut in a graph, where the graph can be accessed through local queries like {\sc Degree}, {\sc Neighbor}, and {\sc Adjacency} queries. Given $ε\in (0,1)$, the algorithm with high probability outputs an estimate $\hat{t}$ satisfying the following $(1-ε) t \leq \hat{t} \leq (1+ε) t$, where $m$ is the number of edges in the graph and $t$ is the size of minimum cut in the graph. The expected number of local queries used by our algorithm is…
We give a deterministic algorithm for finding the minimum (weight) cut of an undirected graph on $n$ vertices and $m$ edges using $\text{polylog}(n)$ calls to any maximum flow subroutine. Using the current best deterministic maximum flow algorithms, this yields an overall running time of $\tilde O(m \cdot \min(\sqrt{m}, n^{2/3}))$ for weighted graphs, and $m^{4/3+o(1)}$ for unweighted (multi)-graphs. This marks the first improvement for this problem since a running time bound of $\tilde O(mn)$ was established by several papers in the early 1990s. Our global minimum cut algorithm is obtained as…
Computing a directed minimum spanning tree, called arborescence, is a fundamental algorithmic problem, although not as common as its undirected counterpart. In 1967, Edmonds discussed an elegant solution. It was refined to run in $O(\min(n^2, m\log n))$ by Tarjan which is optimal for very dense and very sparse graphs. Gabow et al.~gave a version of Edmonds' algorithm that runs in $O(n\log n + m)$, thus asymptotically beating the Tarjan variant in the regime between sparse and dense. Despite the attention the problem received theoretically, there exists, to the best of our knowledge, no empiric…
In Directed Multiway Cut(Dir-MC) the input is an edge-weighted directed graph $G=(V,E)$ and a set of $k$ terminal nodes $\{s_1,s_2,\ldots,s_k\} \subseteq V$; the goal is to find a min-weight subset of edges whose removal ensures that there is no path from $s_i$ to $s_j$ for any $i \neq j$. In Node-weighted Multiway Cut(Node-MC) the input is a node-weighted undirected graph $G$ and a set of $k$ terminal nodes $\{s_1,s_2,\ldots,s_k\} \subseteq V$; the goal is to remove a min-weight subset of nodes to disconnect each pair of terminals. Dir-MC admits a $2$-approximation [Naor, Zosin '97] and Node-…
The minimum cut problem for an undirected edge-weighted graph asks us to divide its set of nodes into two blocks while minimizing the weight sum of the cut edges. In this paper, we engineer the fastest known exact algorithm for the problem. State-of-the-art algorithms like the algorithm of Padberg and Rinaldi or the algorithm of Nagamochi, Ono and Ibaraki identify edges that can be contracted to reduce the graph size such that at least one minimum cut is maintained in the contracted graph. Our algorithm achieves improvements in running time over these algorithms by a multitude of techniques. F…
We consider the (exact, minimum) $k$-cut problem: given a graph and an integer $k$, delete a minimum-weight set of edges so that the remaining graph has at least $k$ connected components. This problem is a natural generalization of the global minimum cut problem, where the goal is to break the graph into $k=2$ pieces. Our main result is a (combinatorial) $k$-cut algorithm on simple graphs that runs in $n^{(1+o(1))k}$ time for any constant $k$, improving upon the previously best $n^{(2ω/3+o(1))k}$ time algorithm of Gupta et al.~[FOCS'18] and the previously best $n^{(1.981+o(1))k}$ time combinat…
The $k$-cut problem asks, given a connected graph $G$ and a positive integer $k$, to find a minimum-weight set of edges whose removal splits $G$ into $k$ connected components. We give the first polynomial-time algorithm with approximation factor $2-ε$ (with constant $ε> 0$) for the $k$-cut problem in planar and minor-free graphs. Applying more complex techniques, we further improve our method and give a polynomial-time approximation scheme for the $k$-cut problem in both planar and minor-free graphs. Despite persistent effort, to the best of our knowledge, this is the first improvement for the…
We study the problem of finding the cycle of minimum cost-to-time ratio in a directed graph with $ n $ nodes and $ m $ edges. This problem has a long history in combinatorial optimization and has recently seen interesting applications in the context of quantitative verification. We focus on strongly polynomial algorithms to cover the use-case where the weights are relatively large compared to the size of the graph. Our main result is an algorithm with running time $ \tilde O (m^{3/4} n^{3/2}) $, which gives the first improvement over Megiddo's $ \tilde O (n^3) $ algorithm [JACM'83] for sparse …
In this paper, we present new incremental algorithms for maintaining data structures that represent all connectivity cuts of size one in directed graphs (digraphs), and the strongly connected components that result by the removal of each of those cuts. We give a conditional lower bound that provides evidence that our algorithms may be tight up to sub-polynomial factors. As an additional result, with our approach we can also maintain dynamically the $2$-vertex-connected components of a digraph during any sequence of edge insertions in a total of $O(mn)$ time. This matches the bounds for the inc…
