相关论文: Flow-augmentation II: Undirected graphs

We present an $\tilde{O}\left(m^{\frac{10}{7}}U^{\frac{1}{7}}\right)$-time algorithm for the maximum $s$-$t$ flow problem and the minimum $s$-$t$ cut problem in directed graphs with $m$ arcs and largest integer capacity $U$. This matches…

Consider the following 2-respecting min-cut problem. Given a weighted graph $G$ and its spanning tree $T$, find the minimum cut among the cuts that contain at most two edges in $T$. This problem is an important subroutine in Karger's…

We present a combinatorial algorithm for computing exact maximum flows in directed graphs with $n$ vertices and edge capacities from $\{1,\dots,U\}$ in $n^{2+o(1)}\log U$ time, which is almost optimal in dense graphs. Our algorithm is a…

We study the problem of finding a small sparse cut in an undirected graph. Given an undirected graph G=(V,E) and a parameter k <= |E|, the small sparsest cut problem is to find a subset of vertices S with minimum conductance among all sets…

In the $k$-Steiner Orientation problem, we are given a mixed graph, that is, with both directed and undirected edges, and a set of $k$ terminal pairs. The goal is to find an orientation of the undirected edges that maximizes the number of…

We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, {\sc $k$-Leaf…

We study two "above guarantee" versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, the task is to…

In this paper we describe a randomized algorithm which returns a maximal spanning forest of an unknown {\em weighted} undirected graph making $O(n)$ $\mathsf{CUT}$ queries in expectation. For weighted graphs, this is optimal due to a result…

A min-cut that seperates vertices s and t in a network is an edge set of minimum weight whose removal will disconnect s and t. This problem is the dual of the well known s-t max-flow problem. Several algorithms for the min-cut problem are…

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…

We show the existence of length-constrained expander decomposition in directed graphs and undirected vertex-capacitated graphs. Previously, its existence was shown only in undirected edge-capacitated graphs [Haeupler-R\"acke-Ghaffari, STOC…

Let G be a directed graph with n vertices and non-negative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe a randomized algorithm to preprocess the graph in O(gn log n) time…

We consider the Hypergraph-$k$-cut problem. The input consists of a hypergraph $G=(V,E)$ with non-negative hyperedge-costs $c: E\rightarrow R_+$ and a positive integer $k$. The objective is to find a least-cost subset $F\subseteq E$ such…

We prove an approximate max-multiflow min-multicut theorem for bounded treewidth graphs. In particular, we show the following: Given a treewidth-$r$ graph, there exists a (fractional) multicommodity flow of value $f$, and a multicut of…

Given an undirected graph $G=(V,E)$ with positive edge lengths and two vertices $s$ and $t$, the next-to-shortest path problem is to find an $st$-path which length is minimum amongst all $st$-paths strictly longer than the shortest path…

We give an algorithm for augmenting the edge connectivity of an undirected graph by using the isolating cuts framework (Li and Panigrahi, FOCS '20). Our algorithm uses poly-logarithmic calls to any max-flow algorithm, which yields a running…

We propose an $O(\log n)$-approximation algorithm for the bipartiteness ratio of undirected graphs introduced by Trevisan (SIAM Journal on Computing, vol. 41, no. 6, 2012), where $n$ is the number of vertices. Our approach extends the…

We present faster algorithms for approximate maximum flow in undirected graphs with good separator structures, such as bounded genus, minor free, and geometric graphs. Given such a graph with $n$ vertices, $m$ edges along with a recursive…

We give the first almost-linear total time algorithm for deciding if a flow of cost at most $F$ still exists in a directed graph, with edge costs and capacities, undergoing decremental updates, i.e., edge deletions, capacity decreases, 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…