Title :
Nearly Maximum Flows in Nearly Linear Time
Author_Institution :
Comput. Sci. Div., Univ. of California at Berkeley, Berkeley, CA, USA
Abstract :
We introduce a new approach to the maximum flow problem in undirected, capacitated graphs using congestion-approximators: easy-to-compute functions that approximate the congestion required to route single-commodity demands in a graph to within some factor α. Our algorithm maintains an arbitrary flow that may have some residual excess and deficits, while taking steps to minimize a potential function measuring the congestion of the current flow plus an over-estimate of the congestion required to route the residual demand. Since the residual term over-estimates, the descent process gradually moves the contribution to our potential function from the residual term to the congestion term, eventually achieving a flow routing the desired demands with nearly minimal congestion after Õ(α2ε-2 log2 n) iterations. Our approach is similar in spirit to that used by Spielman and Teng (STOC 2004) for solving Laplacian systems, and we summarize our approach as trying to do for ℓ∞-flows what they do for ℓ∞-flows. Together with a nearly linear time construction of a no(1)-congestion-approximator, we obtain 1 + ε-optimal singlecommodity flows undirected graphs in time m1+o(1)ε-2, yielding the fastest known algorithm for that problem. Our requirements of a congestion-approximator are quite low, suggesting even faster and simpler algorithms for certain classes of graphs. For example, an α-competitive oblivious routing tree meets our definition, even without knowing how to route the tree back in the graph. For graphs of conductance φ, a trivial φ-1-congestionapproximator gives an extremely simple algorithm for finding Õ(mφ-1).
Keywords :
computational complexity; graph theory; iterative methods; optimisation; α-competitive oblivious routing tree; Laplacian system; congestion-approximators; descent process; flow congestion; flow routing; function minimization; graph conductance; iteration; maximum flow problem; nearly linear time construction; nearly maximum flows; nearly minimal congestion; optimal single-commodity flow undirected graph; residual demand; residual excess; single-commodity demand routing; undirected capacitated graph; Approximation algorithms; Approximation methods; Equations; Laplace equations; Routing; Vectors; Vegetation;
Conference_Titel :
Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium on
Conference_Location :
Berkeley, CA
DOI :
10.1109/FOCS.2013.36