Every undirected graph $G$ has a (weighted) cut-equivalent tree $T$, commonly named after Gomory and Hu who discovered it in 1961. Both $T$ and $G$ have the same node set, and for every node pair $s,t$, the minimum $(s,t)$-cut in $T$ is also an exact minimum $(s,t)$-cut in $G$.
We give the first subcubic-time algorithm that constructs such a tree for a simple graph $G$ (unweighted with no parallel edges). Its time complexity is $\tilde{O}(n^{2.5})$, for $n=|V(G)|$; previously, only $\tilde{O}(n^3)$ was known, except for restricted cases like sparse graphs. Consequently, we obtain the first algorithm for All-Pairs Max-Flow in simple graphs that breaks the cubic-time barrier.
Gomory and Hu compute this tree using $n-1$ queries to (single-pair) Max-Flow; the new algorithm can be viewed as a fine-grained reduction to $\tilde{O}(\sqrt{n})$ Max-Flow computations on $n$-node graphs.
We design an $n^{2+o(1)}$-time algorithm that constructs a cut-equivalent (Gomory-Hu) tree of a simple graph on $n$ nodes. This bound is almost-optimal in terms of $n$, and it improves on the recent $\tilde{O}(n^{2.5})$ bound by the authors (STOC 2021), which was the first to break the cubic barrier. Consequently, the All-Pairs Maximum-Flow (APMF) problem has time complexity $n^{2+o(1)}$, and for the first time in history, this problem can be solved faster than All-Pairs Shortest Paths (APSP). We further observe that an almost-linear time algorithm (in terms of the number of edges $m$) is not …
The Gomory-Hu tree or cut tree (Gomory and Hu, 1961) is a classic data structure for reporting $(s,t)$ mincuts (and by duality, the values of $(s,t)$ maxflows) for all pairs of vertices $s$ and $t$ in an undirected graph. Gomory and Hu showed that it can be computed using $n-1$ exact maxflow computations. Surprisingly, this remains the best algorithm for Gomory-Hu trees more than 50 years later, *even for approximate mincuts*. In this paper, we break this longstanding barrier and give an algorithm for computing a $(1+ε)$-approximate Gomory-Hu tree using $\text{polylog}(n)$ maxflow computations…
In 1961, Gomory and Hu showed that the All-Pairs Max-Flow problem of computing the max-flow between all $n\choose 2$ pairs of vertices in an undirected graph can be solved using only $n-1$ calls to any (single-pair) max-flow algorithm. Even assuming a linear-time max-flow algorithm, this yields a running time of $O(mn)$, which is $O(n^3)$ when $m = Θ(n^2)$. While subsequent work has improved this bound for various special graph classes, no subcubic-time algorithm has been obtained in the last 60 years for general graphs. We break this longstanding barrier by giving an $\tilde{O}(n^{2})$-time a…
We devise new cut sparsifiers that are related to the classical sparsification of Nagamochi and Ibaraki [Algorithmica, 1992], which is an algorithm that, given an unweighted graph $G$ on $n$ nodes and a parameter $k$, computes a subgraph with $O(nk)$ edges that preserves all cuts of value up to $k$. We put forward the notion of a friendly cut sparsifier, which is a minor of $G$ that preserves all friendly cuts of value up to $k$, where a cut in $G$ is called friendly if every node has more edges connecting it to its own side of the cut than to the other side. We present an algorithm that, give…
In this paper we carefully combine Fredman's trick [SICOMP'76] and Matoušek's approach for dominance product [IPL'91] to obtain powerful results in fine-grained complexity: - Under the hypothesis that APSP for undirected graphs with edge weights in $\{1, 2, \ldots, n\}$ requires $n^{3-o(1)}$ time (when $ω=2$), we show a variety of conditional lower bounds, including an $n^{7/3-o(1)}$ lower bound for unweighted directed APSP and an $n^{2.2-o(1)}$ lower bound for computing the Minimum Witness Product between two $n \times n$ Boolean matrices, even if $ω=2$, improving upon their trivial $n^2$ low…
In recent years it has become popular to study dynamic problems in a sensitivity setting: Instead of allowing for an arbitrary sequence of updates, the sensitivity model only allows to apply batch updates of small size to the original input data. The sensitivity model is particularly appealing since recent strong conditional lower bounds ruled out fast algorithms for many dynamic problems, such as shortest paths, reachability, or subgraph connectivity. In this paper we prove conditional lower bounds for sensitivity problems. For example, we show that under the Boolean Matrix Multiplication (BM…
We introduce the notion of {\em fair cuts} as an approach to leverage approximate $(s,t)$-mincut (equivalently $(s,t)$-maxflow) algorithms in undirected graphs to obtain near-linear time approximation algorithms for several cut problems. Informally, for any $α\geq 1$, an $α$-fair $(s,t)$-cut is an $(s,t)$-cut such that there exists an $(s,t)$-flow that uses $1/α$ fraction of the capacity of \emph{every} edge in the cut. (So, any $α$-fair cut is also an $α$-approximate mincut, but not vice-versa.) We give an algorithm for $(1+ε)$-fair $(s,t)$-cut in $\tilde{O}(m)$-time, thereby matching the bes…
Tree edit distance is a well-studied measure of dissimilarity between rooted trees with node labels. It can be computed in $O(n^3)$ time [Demaine, Mozes, Rossman, and Weimann, ICALP 2007], and fine-grained hardness results suggest that the weighted version of this problem cannot be solved in truly subcubic time unless the APSP conjecture is false [Bringmann, Gawrychowski, Mozes, and Weimann, SODA 2018]. We consider the unweighted version of tree edit distance, where every insertion, deletion, or relabeling operation has unit cost. Given a parameter $k$ as an upper bound on the distance, the pr…
Given a directed weighted graph $G=(V,E)$ undergoing vertex insertions \emph{and} deletions, the All-Pairs Shortest Paths (APSP) problem asks to maintain a data structure that processes updates efficiently and returns after each update the distance matrix to the current version of $G$. In two breakthrough results, Italiano and Demetrescu [STOC '03] presented an algorithm that requires $\tilde{O}(n^2)$ \emph{amortized} update time, and Thorup showed in [STOC '05] that \emph{worst-case} update time $\tilde{O}(n^{2+3/4})$ can be achieved. In this article, we make substantial progress on the probl…
In the Single Source Replacement Paths (SSRP) problem we are given a graph $G = (V, E)$, and a shortest paths tree $\widehat{K}$ rooted at a node $s$, and the goal is to output for every node $t \in V$ and for every edge $e$ in $\widehat{K}$ the length of the shortest path from $s$ to $t$ avoiding $e$. We present an $\tilde{O}(m\sqrt{n} + n^2)$ time randomized combinatorial algorithm for unweighted directed graphs. Previously such a bound was known in the directed case only for the seemingly easier problem of replacement path where both the source and the target nodes are fixed. Our new upper …
One of the most basic graph problems, All-Pairs Shortest Paths (APSP) is known to be solvable in $n^{3-o(1)}$ time, and it is widely open whether it has an $O(n^{3-ε})$ time algorithm for $ε> 0$. To better understand APSP, one often strives to obtain subcubic time algorithms for structured instances of APSP and problems equivalent to it, such as the Min-Plus matrix product. A natural structured version of Min-Plus product is Monotone Min-Plus product which has been studied in the context of the Batch Range Mode [SODA'20] and Dynamic Range Mode [ICALP'20] problems. This paper improves the known…
This paper considers a natural fault-tolerant shortest paths problem: for some constant integer $f$, given a directed weighted graph with no negative cycles and two fixed vertices $s$ and $t$, compute (either explicitly or implicitly) for every tuple of $f$ edges, the distance from $s$ to $t$ if these edges fail. We call this problem $f$-Fault Replacement Paths ($f$FRP). We first present an $\tilde{O}(n^3)$ time algorithm for $2$FRP in $n$-vertex directed graphs with arbitrary edge weights and no negative cycles. As $2$FRP is a generalization of the well-studied Replacement Paths problem (RP) …
We study dynamic $(1+ε)$-approximation algorithms for the all-pairs shortest paths problem in unweighted undirected $n$-node $m$-edge graphs under edge deletions. The fastest algorithm for this problem is a randomized algorithm with a total update time of $\tilde O(mn/ε)$ and constant query time by Roditty and Zwick [FOCS 2004]. The fastest deterministic algorithm is from a 1981 paper by Even and Shiloach [JACM 1981]; it has a total update time of $O(mn^2)$ and constant query time. We improve these results as follows: (1) We present an algorithm with a total update time of $\tilde O(n^{5/2}/ε)…
