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We present an $\tilde{O}(m^{10/7})=\tilde{O}(m^{1.43})$-time algorithm for the maximum s-t flow and the minimum s-t cut problems in directed graphs with unit capacities

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

foundations of computer science, (2013): 253-262

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Abstract

We present an 脮m(10/7)= 脮(m1.43-time We recall that 脮f denotes O(f poly(log f)). 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, n2/3) running time bound due to Even and Tarjan EvenT75. By w...More

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Introduction
• The maximum s-t flow problem and its dual, the minimum s-t cut problem, are two of the most fundamental and extensively studied graph problems in combinatorial optimization [Sch[03], AMO93].

They have a wide range of applications, are often used as subroutines in other algorithms, and a number of other important problems – e.g., bipartite matching problem [CLRS09] – can be reduced to them.
• The other component is based on using the interior-point method framework in conjunction with nearly-linear time electrical flow computations, to develop a faster algorithm for the bipartite matching problem.
Highlights
• The maximum s-t flow problem and its dual, the minimum s-t cut problem, are two of the most fundamental and extensively studied graph problems in combinatorial optimization [Sch[03], AMO93].

They have a wide range of applications, are often used as subroutines in other algorithms, and a number of other important problems – e.g., bipartite matching problem [CLRS09] – can be reduced to them
• We show that by a careful composition of these techniques, one is able to ensure that the guiding electrical flows align better with the primal solution – allowing taking larger progress steps and guaranteeing faster convergence – while keeping the unwanted impact of these modifications on the quality of final solution minimal
• We begin the technical part of the paper in Section 2 where we present some preliminaries on maximum flow problem, electrical flows, and bipartite (b-)matching problem, as well as, introduce some theorems we will need in the sequel
• We view an undirected graph G = (V, E, u) as a directed one in which the ordered pair (u, v) ∈ E does not denote an arc anymore, but an edge (u, v) and the order just specifies an orientation of that edge from u to v. (Even though we use the same notation for these two different types of graphs, we will always make sure that it is clear from the context whether we deal with directed graph that has arcs, or with undirected graph that has edges.) From this perspective, the definitions of σ-flow f that we introduced above for directed graphs transfer over to undirected setting almost immediately
• )-time algorithm, we show how one can appropriately “shape” these guiding electrical flows to make their guidance more effective and guarantee faster convergence
• We show that for any integral value of F, we can setup, in O(m) time, a balanced bipartite bmatching problem instance, for some demands b and bipartite graph G = (P ∪ Q, E), such that: (1) there will be a perfect b-matching in Gif there is a feasible s-t flow of value F in G; and (2) given a perfect b-matching in Gone can recover in O(m) time an s-t flow of value F that is feasible in G
Results
• For any ε > 0, any graph G with n vertices and m edges, any demand vector σ, and any resistances r , one can compute in O(m log m log ε−1) time vertex potentials φsuch that φ− φ∗ L ≤ ε φ∗ L, where L is the Laplacian of G, φ∗ are potentials inducing the electrical σ-flow determined by resistances r , and φ L := φT Lφ.
• One can go the other way – there is a simple, combinatorial reduction from the maximum flow problem to the task of finding a perfect bipartite b -matching.[6]
• If one can solve a balanced instance of a perfect bipartite b-matching problem in a graph with nvertices and medges in T (n, m , |b|1) time, one can solve the maximum s-t flow problem in a graph G = (V, E, u) with m arcs and capacity vector u in O((m + T (Θ(m), 4m, 4|u|1)) log |u|1) time.
• It turns out there is a way of changing the maintained solution to make it essentially the same from the point of view of the b-matching instance, while dramatically improving the quality of corresponding electrical flows that guide it.
• Using the reduction from Theorem 3.1, this gives them the analogous algorithm for the maximum s-t flow problem in unit-capacity graphs and that, in turn, results in an algorithm for the bipartite matching problem.
• The authors show how to reduce the maximum s-t flow problem in a directed capacitated graph G = (V, E, u) to solving O(log |u|1) balanced instances of the perfect bipartite b-matching problem, i.e., the authors prove Theorem 3.1.
Conclusion
• Motivated by this path-following approach, the algorithm for computing near-optimal solution to the minimum-cost σ -flow problem will start with some 0-centered solution (f 0, s0, ν0) that has fairly large value of μ(f 0, s0, ν0).
• (This implementation can be viewed as a direct analogue of the update steps of path-following interior-point methods.) The result of this analysis is presented in the following theorem, which, in particular, ties the congestion vector ρ(ˆf t, f t) inflicted by the electrical flowf t with respect to the primal solution f t, to an upperbound on the size δt of the improvement step.
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