Navigating Central Path with Electrical Flows: From Flows to Matchings, and Back

Abstract

We present an O(m 10 7 ) = O(m 1.43 )-time 1 algorithm for the maximum s-t flow and the minimum s-t cut problems in directed graphs with unit capacities. This is the first improvement over the sparse-graph case of the long-standing O(m min{ √ m, n 2/3 }) running time bound due to Even and Tarjan [ET75] and Karzanov [Kar73]. By well-known reductions, this also establishes an O(m 10 7 )-time algorithm for the maximum-cardinality bipartite matching problem. That, in turn, gives an improvement over the celebrated O(m √ n) running time bound of Hopcroft and Karp [HK73] and Karzanov [Kar73] whenever the input graph is sufficiently sparse.At a very high level, our results stem from acquiring a deeper understanding of interior-point methods -a powerful tool in convex optimization -in the context of flow problems, as well as, utilizing certain interplay between maximum flows and bipartite matchings.The core of our approach comprises a primal-dual algorithm for (near-)perfect bipartite bmatching problem. This algorithm is inspired by path-following interior-point methods and employs electrical flow computations to gradually improve the quality of maintained solution by advancing it toward (near-)optimality along so-called central path. To analyze this process, we establish a formal connection that ties its convergence rate to the structure of corresponding electrical flows. Then, we exploit that connection to obtain a convergence guarantee for our algorithm that improves upon the well-known barrier of Ω( √ m) iterations corresponding to the generic worst-case performance bounds for interior-point-method-based algorithms. This improvement is based on refining certain insights into behavior of electrical flows that stem from the work of Christiano et al. [CKM + 11] and combining them with a new technique for preconditioning primal-dual solutions.The final ingredient of our approach is a simple reduction of the maximum s-t flow problem to the bipartite b-matching problem. This reduction is then composed with the recent sublinear-time algorithm for finding perfect matchings in regular graphs of Goel et al. [GKK10], to derive an efficient procedure for rounding fractional s-t flows and bipartite matchings.

Approximate Gaussian Elimination for Laplacians - Fast, Sparse, and Simple

“…+ 04], and computing maximum flows in graphs [DS08, CKM + 11, Mad13,LS14]. It has also been used as a primitive in the design of several fast algorithms [KM09,OSV12,KMP12,LKP12,KRS15].…”

Randomized Online Matching in Regular Graphs

“…(In contrast, for general bipartite graphs, the best current maximum matching algorithms require ω(m) time [38,49].) Subsequent work, including [5,16,17,18,58], finally culminated in an O(m)-time perfect matching algorithm for all d. This linear-time bound was later proven to be optimal for deterministic algorithms by Goel, Kapralov and Khanna [34].…”

Fully polynomial-time parameterized computations for graphs and matrices of low treewidth

“…Before that, Edmonds [23] gave the first polynomial time algorithm, then bested by combinatorial algorithms of Micali and Vazirani [54,62], Blum [7], and Gabow and Tarjan [31], each running in O(m √ n) time. Recently M adry [46] gave an O(m 10/7 ) algorithm for the unweighted bipartite case, then generalized to the weighted bipartite case by Cohen et al [18]. For graphs of treewidth k, simply because their number of edges is m = O(kn), the above gives O(kn 1.5 ) and O(k 1.42 n 1.42 ) algorithms, respectively.…”

“…The running times of these algorithms vary depending on the variants they solve, but all of them are far larger than linear. In particular, the fastest known algorithm in the directed unit-weight setting, which is the case considered in this work, is due to M adry [46,47] and works in time O(m 10/7 ). For this reason, recently there was a line of work on finding nearlinear (1+ε)-approximation algorithms for the maximum flow problem [17,40,42,52,59], culminating in a (1 + ε)-approximation algorithm working in undirected graphs in time O(ε −2 · m log 11 n), proposed by Peng [52].…”

“…Starting with the first algorithm of Ford and Fulkerson [30], which works in time O(F · (n + m)) for integer capacities, where F is the maximum size of the flow, a long chain of improvements and generalizations was proposed throughout the years; see e.g. [18,21,24,25,33,39,46,47,50]. The running times of these algorithms vary depending on the variants they solve, but all of them are far larger than linear.…”