arXiv subjects

Sushant Sachdeva

Publications and source records attributed to Sushant Sachdeva.

At least 19 recordsLinked to original sources

arXiv2026

A Tight Bound on Localization of Electrical Flows

We prove that for any unweighted graph on n vertices the L1 norm of a unit electric current between the endpoints of a random edge is at most 2 log n. Furthermore, we show that on any weighted graph the spectral norm of the entry-wise absolute value of the symmetric transfer-current matrix is at most 2 log n. This bound is tight up to constants and improves the O(log^2 n) bound from [Schild-Rao-Srivastava, SODA '18]. The initial proofs were generated by OpenAI's ChatGPT 5.5 Pro; the authors have verified and rewritten them to enhance readability and provide additional context.

cs.DS↗

arXiv2025

PREM: Privately Answering Statistical Queries with Relative Error

We introduce $\mathsf{PREM}$ (Private Relative Error Multiplicative weight update), a new framework for generating synthetic data that achieves a relative error guarantee for statistical queries under $(\varepsilon, \delta)$ differential privacy (DP). Namely, for a domain ${\cal X}$, a family ${\cal F}$ of queries $f : {\cal X} \to \{0, 1\}$, and $\zeta > 0$, our framework yields a mechanism that on input dataset $D \in {\cal X}^n$ outputs a synthetic dataset $\widehat{D} \in {\cal X}^n$ such that all statistical queries in ${\cal F}$ on $D$, namely $\sum_{x \in D} f(x)$ for $f \in {\cal F}$, are within a $1 \pm \zeta$ multiplicative factor of the corresponding value on $\widehat{D}$ up to an additive error that is polynomial in $\log |{\cal F}|$, $\log |{\cal X}|$, $\log n$, $\log(1/\delta)$, $1/\varepsilon$, and $1/\zeta$. In contrast, any $(\varepsilon, \delta)$-DP mechanism is known to require worst-case additive error that is polynomial in at least one of $n, |{\cal F}|$, or $|{\cal X}|$. We complement our algorithm with nearly matching lower bounds.

cs.LG↗

arXiv2024

Eulerian Graph Sparsification by Effective Resistance Decomposition

We provide an algorithm that, given an $n$-vertex $m$-edge Eulerian graph with polynomially bounded weights, computes an $\breve{O}(n\log^{2} n \cdot \varepsilon^{-2})$-edge $\varepsilon$-approximate Eulerian sparsifier with high probability in $\breve{O}(m\log^3 n)$ time (where $\breve{O}(\cdot)$ hides $\text{polyloglog}(n)$ factors). Due to a reduction from [Peng-Song, STOC '22], this yields an $\breve{O}(m\log^3 n + n\log^6 n)$-time algorithm for solving $n$-vertex $m$-edge Eulerian Laplacian systems with polynomially-bounded weights with high probability, improving upon the previous state-of-the-art runtime of $\Omega(m\log^8 n + n\log^{23} n)$. We also give a polynomial-time algorithm that computes $O(\min(n\log n \cdot \varepsilon^{-2} + n\log^{5/3} n \cdot \varepsilon^{-4/3}, n\log^{3/2} n \cdot \varepsilon^{-2}))$-edge sparsifiers, improving the best such sparsity bound of $O(n\log^2 n \cdot \varepsilon^{-2} + n\log^{8/3} n \cdot \varepsilon^{-4/3})$ [Sachdeva-Thudi-Zhao, ICALP '24]. Finally, we show that our techniques extend to yield the first $O(m\cdot\text{polylog}(n))$ time algorithm for computing $O(n\varepsilon^{-1}\cdot\text{polylog}(n))$-edge graphical spectral sketches, as well as a natural Eulerian generalization we introduce. In contrast to prior Eulerian graph sparsification algorithms which used either short cycle or expander decompositions, our algorithms use a simple efficient effective resistance decomposition scheme we introduce. Our algorithms apply a natural sampling scheme and electrical routing (to achieve degree balance) to such decompositions. Our analysis leverages new asymmetric variance bounds specialized to Eulerian Laplacians and tools from discrepancy theory.

cs.DS↗

arXiv2024

Almost-Linear Time Algorithms for Decremental Graphs: Min-Cost Flow and More via Duality

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 cost increases. This implies almost-linear time algorithms for approximating the minimum-cost flow value and $s$-$t$ distance on such decremental graphs. Our framework additionally allows us to maintain decremental strongly connected components in almost-linear time deterministically. These algorithms also improve over the current best known runtimes for statically computing minimum-cost flow, in both the randomized and deterministic settings. We obtain our algorithms by taking the dual perspective, which yields cut-based algorithms. More precisely, our algorithm computes the flow via a sequence of $m^{1+o(1)}$ dynamic min-ratio cut problems, the dual analog of the dynamic min-ratio cycle problem that underlies recent fast algorithms for minimum-cost flow. Our main technical contribution is a new data structure that returns an approximately optimal min-ratio cut in amortized $m^{o(1)}$ time by maintaining a tree-cut sparsifier. This is achieved by devising a new algorithm to maintain the dynamic expander hierarchy of [Goranci-R\"{a}cke-Saranurak-Tan, SODA 2021] that also works in capacitated graphs. All our algorithms are deterministc, though they can be sped up further using randomized techniques while still working against an adaptive adversary.

cs.DS↗

arXiv2024

Optimal Electrical Oblivious Routing on Expanders

In this paper, we investigate the question of whether the electrical flow routing is a good oblivious routing scheme on an $m$-edge graph $G = (V, E)$ that is a $\Phi$-expander, i.e. where $\lvert \partial S \rvert \geq \Phi \cdot \mathrm{vol}(S)$ for every $S \subseteq V, \mathrm{vol}(S) \leq \mathrm{vol}(V)/2$. Beyond its simplicity and structural importance, this question is well-motivated by the current state-of-the-art of fast algorithms for $\ell_{\infty}$ oblivious routings that reduce to the expander-case which is in turn solved by electrical flow routing. Our main result proves that the electrical routing is an $O(\Phi^{-1} \log m)$-competitive oblivious routing in the $\ell_1$- and $\ell_\infty$-norms. We further observe that the oblivious routing is $O(\log^2 m)$-competitive in the $\ell_2$-norm and, in fact, $O(\log m)$-competitive if $\ell_2$-localization is $O(\log m)$ which is widely believed. Using these three upper bounds, we can smoothly interpolate to obtain upper bounds for every $p \in [2, \infty]$ and $q$ given by $1/p + 1/q = 1$. Assuming $\ell_2$-localization in $O(\log m)$, we obtain that in $\ell_p$ and $\ell_q$, the electrical oblivious routing is $O(\Phi^{-(1-2/p)}\log m)$ competitive. Using the currently known result for $\ell_2$-localization, this ratio deteriorates by at most a sublogarithmic factor for every $p, q \neq 2$. We complement our upper bounds with lower bounds that show that the electrical routing for any such $p$ and $q$ is $\Omega(\Phi^{-(1-2/p)}\log m)$-competitive. This renders our results in $\ell_1$ and $\ell_{\infty}$ unconditionally tight up to constants, and the result in any $\ell_p$- and $\ell_q$-norm to be tight in case of $\ell_2$-localization in $O(\log m)$.

cs.DS↗

arXiv2023

Better Sparsifiers for Directed Eulerian Graphs

Spectral sparsification for directed Eulerian graphs is a key component in the design of fast algorithms for solving directed Laplacian linear systems. Directed Laplacian linear system solvers are crucial algorithmic primitives to fast computation of fundamental problems on random walks, such as computing stationary distribution, hitting and commute time, and personalized PageRank vectors. While spectral sparsification is well understood for undirected graphs and it is known that for every graph $G,$ $(1+\varepsilon)$-sparsifiers with $O(n\varepsilon^{-2})$ edges exist [Batson-Spielman-Srivastava, STOC '09] (which is optimal), the best known constructions of Eulerian sparsifiers require $\Omega(n\varepsilon^{-2}\log^4 n)$ edges and are based on short-cycle decompositions [Chu et al., FOCS '18]. In this paper, we give improved constructions of Eulerian sparsifiers, specifically: 1. We show that for every directed Eulerian graph $\vec{G},$ there exist an Eulerian sparsifier with $O(n\varepsilon^{-2} \log^2 n \log^2\log n + n\varepsilon^{-4/3}\log^{8/3} n)$ edges. This result is based on combining short-cycle decompositions [Chu-Gao-Peng-Sachdeva-Sawlani-Wang, FOCS '18, SICOMP] and [Parter-Yogev, ICALP '19], with recent progress on the matrix Spencer conjecture [Bansal-Meka-Jiang, STOC '23]. 2. We give an improved analysis of the constructions based on short-cycle decompositions, giving an $m^{1+\delta}$-time algorithm for any constant $\delta > 0$ for constructing Eulerian sparsifiers with $O(n\varepsilon^{-2}\log^3 n)$ edges.

cs.DS↗

arXiv2023

Incremental Approximate Maximum Flow on Undirected Graphs in Subpolynomial Update Time

We provide an algorithm which, with high probability, maintains a $(1-\epsilon)$-approximate maximum flow on an undirected graph undergoing $m$-edge additions in amortized $m^{o(1)} \epsilon^{-3}$ time per update. To obtain this result, we provide a more general algorithm that solves what we call the incremental, thresholded $p$-norm flow problem that asks to determine the first edge-insertion in an undirected graph that causes the minimum $\ell_p$-norm flow to decrease below a given threshold in value. Since we solve this thresholded problem, our data structure succeeds against an adaptive adversary that can only see the data structure's output. Furthermore, since our algorithm holds for $p = 2$, we obtain improved algorithms for dynamically maintaining the effective resistance between a pair of vertices in an undirected graph undergoing edge insertions. Our algorithm builds upon previous dynamic algorithms for approximately solving the minimum-ratio cycle problem that underlie previous advances on the maximum flow problem [Chen-Kyng-Liu-Peng-Probst Gutenberg-Sachdeva, FOCS '22] as well as recent dynamic maximum flow algorithms [v.d.Brand-Liu-Sidford, STOC '23]. Instead of using interior point methods, which were a key component of these recent advances, our algorithm uses an optimization method based on $\ell_p$-norm iterative refinement and the multiplicative weight update method. This ensures a monotonicity property in the minimum-ratio cycle subproblems that allows us to apply known data structures and bypass issues arising from adaptive queries.

cs.DS↗

arXiv2023

Fast Algorithms for Separable Linear Programs

In numerical linear algebra, considerable effort has been devoted to obtaining faster algorithms for linear systems whose underlying matrices exhibit structural properties. A prominent success story is the method of generalized nested dissection~[Lipton-Rose-Tarjan'79] for separable matrices. On the other hand, the majority of recent developments in the design of efficient linear program (LP) solves do not leverage the ideas underlying these faster linear system solvers nor consider the separable structure of the constraint matrix. We give a faster algorithm for separable linear programs. Specifically, we consider LPs of the form $\min_{\mathbf{A}\mathbf{x}=\mathbf{b}, \mathbf{l}\leq\mathbf{x}\leq\mathbf{u}} \mathbf{c}^\top\mathbf{x}$, where the graphical support of the constraint matrix $\mathbf{A} \in \mathbb{R}^{n\times m}$ is $O(n^\alpha)$-separable. These include flow problems on planar graphs and low treewidth matrices among others. We present an $\tilde{O}((m+m^{1/2 + 2\alpha}) \log(1/\epsilon))$ time algorithm for these LPs, where $\epsilon$ is the relative accuracy of the solution. Our new solver has two important implications: for the $k$-multicommodity flow problem on planar graphs, we obtain an algorithm running in $\tilde{O}(k^{5/2} m^{3/2})$ time in the high accuracy regime; and when the support of $\mathbf{A}$ is $O(n^\alpha)$-separable with $\alpha \leq 1/4$, our algorithm runs in $\tilde{O}(m)$ time, which is nearly optimal. The latter significantly improves upon the natural approach of combining interior point methods and nested dissection, whose time complexity is lower bounded by $\Omega(\sqrt{m}(m+m^{\alpha\omega}))=\Omega(m^{3/2})$, where $\omega$ is the matrix multiplication constant. Lastly, in the setting of low-treewidth LPs, we recover the results of [DLY,STOC21] and [GS,22] with significantly simpler data structure machinery.

cs.DS↗

arXiv2023

A Deterministic Almost-Linear Time Algorithm for Minimum-Cost Flow

We give a deterministic $m^{1+o(1)}$ time algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with $m$ edges and polynomially bounded integral demands, costs, and capacities. As a consequence, we obtain the first running time improvement for deterministic algorithms that compute maximum-flow in graphs with polynomial bounded capacities since the work of Goldberg-Rao [J.ACM '98]. Our algorithm builds on the framework of Chen-Kyng-Liu-Peng-Gutenberg-Sachdeva [FOCS '22] that computes an optimal flow by computing a sequence of $m^{1+o(1)}$-approximate undirected minimum-ratio cycles. We develop a deterministic dynamic graph data-structure to compute such a sequence of minimum-ratio cycles in an amortized $m^{o(1)}$ time per edge update. Our key technical contributions are deterministic analogues of the vertex sparsification and edge sparsification components of the data-structure from Chen et al. For the vertex sparsification component, we give a method to avoid the randomness in Chen et al. which involved sampling random trees to recurse on. For the edge sparsification component, we design a deterministic algorithm that maintains an embedding of a dynamic graph into a sparse spanner. We also show how our dynamic spanner can be applied to give a deterministic data structure that maintains a fully dynamic low-stretch spanning tree on graphs with polynomially bounded edge lengths, with subpolynomial average stretch and subpolynomial amortized time per edge update.

cs.DS↗

arXiv2023

Universal Matrix Sparsifiers and Fast Deterministic Algorithms for Linear Algebra

Let $\mathbf S \in \mathbb R^{n \times n}$ satisfy $\|\mathbf 1-\mathbf S\|_2\le\epsilon n$, where $\mathbf 1$ is the all ones matrix and $\|\cdot\|_2$ is the spectral norm. It is well-known that there exists such an $\mathbf S$ with just $O(n/\epsilon^2)$ non-zero entries: we can let $\mathbf S$ be the scaled adjacency matrix of a Ramanujan expander graph. We show that such an $\mathbf S$ yields a $universal$ $sparsifier$ for any positive semidefinite (PSD) matrix. In particular, for any PSD $\mathbf A \in \mathbb{R}^{n\times n}$ with entries bounded in magnitude by $1$, $\|\mathbf A - \mathbf A\circ\mathbf S\|_2 \le \epsilon n$, where $\circ$ denotes the entrywise (Hadamard) product. Our techniques also give universal sparsifiers for non-PSD matrices. In this case, letting $\mathbf S$ be the scaled adjacency matrix of a Ramanujan graph with $\tilde O(n/\epsilon^4)$ edges, we have $\|\mathbf A - \mathbf A \circ \mathbf S \|_2 \le \epsilon \cdot \max(n,\|\mathbf A\|_1)$, where $\|\mathbf A\|_1$ is the nuclear norm. We show that the above bounds for both PSD and non-PSD matrices are tight up to log factors. Since $\mathbf A \circ \mathbf S$ can be constructed deterministically, our result for PSD matrices derandomizes and improves upon known results for randomized matrix sparsification, which require randomly sampling ${O}(\frac{n \log n}{\epsilon^2})$ entries. We also leverage our results to give the first deterministic algorithms for several problems related to singular value approximation that run in faster than matrix multiplication time. Finally, if $\mathbf A \in \{-1,0,1\}^{n \times n}$ is PSD, we show that $\mathbf{\tilde A}$ with $\|\mathbf A - \mathbf{\tilde A}\|_2 \le \epsilon n$ can be obtained by deterministically reading $\tilde O(n/\epsilon)$ entries of $\mathbf A$. This improves the $1/\epsilon$ dependence on our result for general PSD matrices and is near-optimal.

cs.DS↗

arXiv2023

A Simple and Efficient Parallel Laplacian Solver

A symmetric matrix is called a Laplacian if it has nonpositive off-diagonal entries and zero row sums. Since the seminal work of Spielman and Teng (2004) on solving Laplacian linear systems in nearly linear time, several algorithms have been designed for the task. Yet, the work of Kyng and Sachdeva (2016) remains the simplest and most practical sequential solver. They presented a solver purely based on random sampling and without graph-theoretic constructions such as low-stretch trees and sparsifiers. In this work, we extend the result of Kyng and Sachdeva to a simple parallel Laplacian solver with $O(m \log^3 n \log\log n)$ or $O((m + n\log^5 n)\log n \log\log n)$ work and $O(\log^2 n \log\log n)$ depth using the ideas of block Cholesky factorization from Kyng et al. (2016). Compared to the best known parallel Laplacian solvers that achieve polylogarithmic depth due to Lee et al. (2015), our solver achieves both better depth and, for dense graphs, better work.

cs.DS↗

arXiv2023

Electrical Flows for Polylogarithmic Competitive Oblivious Routing

Oblivious routing is a well-studied paradigm that uses static precomputed routing tables for selecting routing paths within a network. Existing oblivious routing schemes with polylogarithmic competitive ratio for general networks are tree-based, in the sense that routing is performed according to a convex combination of trees. However, this restriction to trees leads to a construction that has time quadratic in the size of the network and does not parallelize well. In this paper we study oblivious routing schemes based on electrical routing. In particular, we show that general networks with $n$ vertices and $m$ edges admit a routing scheme that has competitive ratio $O(\log^2 n)$ and consists of a convex combination of only $O(\sqrt{m})$ electrical routings. This immediately leads to an improved construction algorithm with time $\tilde{O}(m^{3/2})$ that can also be implemented in parallel with $\tilde{O}(\sqrt{m})$ depth.

cs.DS↗

arXiv2022

Fast Algorithms for $\ell_p$-Regression

The $\ell_p$-norm regression problem is a classic problem in optimization with wide ranging applications in machine learning and theoretical computer science. The goal is to compute $x^{\star} =\arg\min_{Ax=b}\|x\|_p^p$, where $x^{\star}\in \mathbb{R}^n, A\in \mathbb{R}^{d\times n},b \in \mathbb{R}^d$ and $d\leq n$. Efficient high-accuracy algorithms for the problem have been challenging both in theory and practice and the state of the art algorithms require $poly(p)\cdot n^{\frac{1}{2}-\frac{1}{p}}$ linear system solves for $p\geq 2$. In this paper, we provide new algorithms for $\ell_p$-regression (and a more general formulation of the problem) that obtain a high-accuracy solution in $O(p n^{\frac{(p-2)}{(3p-2)}})$ linear system solves. We further propose a new inverse maintenance procedure that speeds-up our algorithm to $\widetilde{O}(n^{\omega})$ total runtime, where $O(n^{\omega})$ denotes the running time for multiplying $n \times n$ matrices. Additionally, we give the first Iteratively Reweighted Least Squares (IRLS) algorithm that is guaranteed to converge to an optimum in a few iterations. Our IRLS algorithm has shown exceptional practical performance, beating the currently available implementations in MATLAB/CVX by 10-50x.

cs.DS↗

arXiv2022

A New Approach to Estimating Effective Resistances and Counting Spanning Trees in Expander Graphs

We demonstrate that for expander graphs, for all $\epsilon > 0,$ there exists a data structure of size $\widetilde{O}(n\epsilon^{-1})$ which can be used to return $(1 + \epsilon)$-approximations to effective resistances in $\widetilde{O}(1)$ time per query. Short of storing all effective resistances, previous best approaches could achieve $\widetilde{O}(n\epsilon^{-2})$ size and $\widetilde{O}(\epsilon^{-2})$ time per query by storing Johnson-Lindenstrauss vectors for each vertex, or $\widetilde{O}(n\epsilon^{-1})$ size and $\widetilde{O}(n\epsilon^{-1})$ time per query by storing a spectral sketch. Our construction is based on two key ideas: 1) $\epsilon^{-1}$-sparse, $\epsilon$-additive approximations to $DL^+1_u$ for all $u,$ can be used to recover $(1 + \epsilon)$-approximations to the effective resistances, 2) In expander graphs, only $\widetilde{O}(\epsilon^{-1})$ coordinates of a vector similar to $DL^+1_u$ are larger than $\epsilon.$ We give an efficient construction for such a data structure in $\widetilde{O}(m + n\epsilon^{-2})$ time via random walks. This results in an algorithm for computing $(1+\epsilon)$-approximate effective resistances for $s$ vertex pairs in expanders that runs in $\widetilde{O}(m + n\epsilon^{-2} + s)$ time, improving over the previously best known running time of $m^{1 + o(1)} + (n + s)n^{o(1)}\epsilon^{-1.5}$ for $s = \omega(n\epsilon^{-0.5}).$ We employ the above algorithm to compute a $(1+\delta)$-approximation to the number of spanning trees in an expander graph, or equivalently, approximating the (pseudo)determinant of its Laplacian in $\widetilde{O}(m + n^{1.5}\delta^{-1})$ time. This improves on the previously best known result of $m^{1+o(1)} + n^{1.875+o(1)}\delta^{-1.75}$ time, and matches the best known size of determinant sparsifiers.

cs.DS↗

arXiv2022

A Simple Framework for Finding Balanced Sparse Cuts via APSP

We present a very simple and intuitive algorithm to find balanced sparse cuts in a graph via shortest-paths. Our algorithm combines a new multiplicative-weights framework for solving unit-weight multi-commodity flows with standard ball growing arguments. Using Dijkstra's algorithm for computing the shortest paths afresh every time gives a very simple algorithm that runs in time $\widetilde{O}(m^2/\phi)$ and finds an $\widetilde{O}(\phi)$-sparse balanced cut, when the given graph has a $\phi$-sparse balanced cut. Combining our algorithm with known deterministic data-structures for answering approximate All Pairs Shortest Paths (APSP) queries under increasing edge weights (decremental setting), we obtain a simple deterministic algorithm that finds $m^{o(1)}\phi$-sparse balanced cuts in $m^{1+o(1)}/\phi$ time. Our deterministic almost-linear time algorithm matches the state-of-the-art in randomized and deterministic settings up to subpolynomial factors, while being significantly simpler to understand and analyze, especially compared to the only almost-linear time deterministic algorithm, a recent breakthrough by Chuzhoy-Gao-Li-Nanongkai-Peng-Saranurak (FOCS 2020).

cs.DS↗

arXiv2022

Optimal Methods for Higher-Order Smooth Monotone Variational Inequalities

In this work, we present new simple and optimal algorithms for solving the variational inequality (VI) problem for $p^{th}$-order smooth, monotone operators -- a problem that generalizes convex optimization and saddle-point problems. Recent works (Bullins and Lai (2020), Lin and Jordan (2021), Jiang and Mokhtari (2022)) present methods that achieve a rate of $\tilde{O}(\epsilon^{-2/(p+1)})$ for $p\geq 1$, extending results by (Nemirovski (2004)) and (Monteiro and Svaiter (2012)) for $p=1,2$. A drawback to these approaches, however, is their reliance on a line search scheme. We provide the first $p^{\textrm{th}}$-order method that achieves a rate of $O(\epsilon^{-2/(p+1)}).$ Our method does not rely on a line search routine, thereby improving upon previous rates by a logarithmic factor. Building on the Mirror Prox method of Nemirovski (2004), our algorithm works even in the constrained, non-Euclidean setting. Furthermore, we prove the optimality of our algorithm by constructing matching lower bounds. These are the first lower bounds for smooth MVIs beyond convex optimization for $p > 1$. This establishes a separation between solving smooth MVIs and smooth convex optimization, and settles the oracle complexity of solving $p^{\textrm{th}}$-order smooth MVIs.

math.OC↗

arXiv2022

Nested Dissection Meets IPMs: Planar Min-Cost Flow in Nearly-Linear Time

We present a nearly-linear time algorithm for finding a minimum-cost flow in planar graphs with polynomially bounded integer costs and capacities. The previous fastest algorithm for this problem is based on interior point methods (IPMs) and works for general sparse graphs in $O(n^{1.5}\text{poly}(\log n))$ time [Daitch-Spielman, STOC'08]. Intuitively, $\Omega(n^{1.5})$ is a natural runtime barrier for IPM-based methods, since they require $\sqrt{n}$ iterations, each routing a possibly-dense electrical flow. To break this barrier, we develop a new implicit representation for flows based on generalized nested-dissection [Lipton-Rose-Tarjan, JSTOR'79] and approximate Schur complements [Kyng-Sachdeva, FOCS'16]. This implicit representation permits us to design a data structure to route an electrical flow with sparse demands in roughly $\sqrt{n}$ update time, resulting in a total running time of $O(n\cdot\text{poly}(\log n))$. Our results immediately extend to all families of separable graphs.

cs.DS↗

arXiv2022

Maximum Flow and Minimum-Cost Flow in Almost-Linear Time

We give an algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with $m$ edges and polynomially bounded integral demands, costs, and capacities in $m^{1+o(1)}$ time. Our algorithm builds the flow through a sequence of $m^{1+o(1)}$ approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized $m^{o(1)}$ time using a new dynamic graph data structure. Our framework extends to algorithms running in $m^{1+o(1)}$ time for computing flows that minimize general edge-separable convex functions to high accuracy. This gives almost-linear time algorithms for several problems including entropy-regularized optimal transport, matrix scaling, $p$-norm flows, and $p$-norm isotonic regression on arbitrary directed acyclic graphs.

cs.DS↗