Holiest Minimum-Cost Paths and Flows in Surface Graphs

TL;DR

This paper introduces a deterministic perturbation scheme for surface-embedded graphs that guarantees unique shortest paths and flows, leading to faster algorithms for minimum cuts and shortest paths in planar and surface graphs.

Contribution

It presents a simple deterministic perturbation method for surface graphs and applies it to develop faster algorithms for minimum cut and shortest path problems.

Findings

Deterministic $O(n ext{ log log } n)$ time algorithm for directed planar minimum cut.

Deterministic $O(g^2 n ext{ log } n)$ preprocessing for multiple-source shortest paths.

Generalization of linear-time algorithms to arbitrary orientable surfaces.

Abstract

Let $G$ be an edge-weighted directed graph with $n$ vertices embedded on an orientable surface of genus $g$. We describe a simple deterministic lexicographic perturbation scheme that guarantees uniqueness of minimum-cost flows and shortest paths in $G$. The perturbations take $O(gn)$ time to compute. We use our perturbation scheme in a black box manner to derive a deterministic $O(n \log \log n)$ time algorithm for minimum cut in \emph{directed} edge-weighted planar graphs and a deterministic $O(g^2 n \log n)$ time proprocessing scheme for the multiple-source shortest paths problem of computing a shortest path oracle for all vertices lying on a common face of a surface embedded graph. The latter result yields faster deterministic near-linear time algorithms for a variety of problems in constant genus surface embedded graphs. Finally, we open the black box in order to generalize a recent linear-time algorithm for multiple-source shortest paths in unweighted undirected planar graphs to work in arbitrary orientable surfaces. Our algorithm runs in $O(g^2 n \log g)$ time in this setting, and it can be used to give improved linear time algorithms for several problems in unweighted undirected surface embedded graphs of constant genus including the computation of minimum cuts, shortest topologically non-trivial cycles, and minimum homology bases.