Homology flows, cohomology cuts

Abstract

We describe the first algorithm to compute maximum flows in surface-embedded graphs in near-linear time. Specifically, given a graph embedded on a surface of genus g, with two specified vertices s and t and integer edge capacities that sum to C, our algorithm computes a maximum (s, t)-flow in O(g 8 n log 2 n log 2 C) time. We also present a combinatorial algorithm that takes g O(g) n 3/2 arithmetic operations. Except for the special case of planar graphs, for which an O(n log n)-time algorithm has been known for 20 years, the best previous time bounds for maximum flows in surface-embedded graphs follow from algorithms for general sparse graphs. For graphs of any fixed genus, our algorithms improve these time bounds by roughly a factor of n. Our key insight is to optimize the homology class of the flow, rather than directly optimizing the flow itself; two flows are in the same homology class if their difference is a weighted sum of directed facial cycles. A dual formulation of our algorithm computes the minimum-cost circulation in a given (real or integer) homology class.

Shortest Paths in Planar Graphs with Real Lengths in O(nlog2 n/loglogn) Time

“…Chambers et al claimed [1] that the algorithm in [7] generalizes to bounded genus graphs. However, the proof (Theorem 3.2 in [1]) is not detailed, and it seems that our new technique is required for its correctness [2]. The resulting running time for fixed genus is also improved to O(n log 2 n/ log log n).…”

Minimum cuts and shortest homologous cycles

“…In a companion paper [14], we describe algorithms to compute a maximum flow in surface embedded graphs, using very different techniques from this paper. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)-flow in O(g 7 n log 2 n log 2 C) time for integer capacities that sum to C, or in (g log n) O(g) n time for real capacities.…”

“…Like several earlier papers [15,16,23,24], we will consider only one-dimensional cellular homology with coefficients in the finite field 2 ; this restriction allows us to radically simplify our definitions. For a more general treatment of homology, we refer the reader to our companion paper [14] or to standard references on topology [41,68,73].…”

Flows in One-Crossing-Minor-Free Graphs

“…Although flows can be computed in polynomial time for arbitrary graphs, it is of interest to find classes of graphs for which flows can be computed more quickly, and specialized algorithms are known for flows in planar graphs [3,13,17,18,21,22,24,25,27,35], graphs of bounded genus [4,5], graphs with small crossing number [19], and graphs of bounded treewidth [15].…”

“…To handle the general case, we would need to combine clique-sums, bounded-genus, apexes, and vortexes. The problem of flows on bounded genus surfaces has been previously examined [4,5] and the present work focuses on clique-sums, as these are the main feature in the structural decomposition for one-crossing-minor-free graphs. However, it remains unclear how to handle apexes and vortexes.…”