Incremental Exact Min-Cut in Poly-logarithmic Amortized Update Time

TL;DR

This paper introduces a deterministic incremental algorithm that maintains the exact minimum cut size with near-constant amortized update time, and also proposes space-efficient approximation algorithms with high probability guarantees.

Contribution

It provides the first near-constant amortized update time algorithm for exact incremental minimum cut and develops space-efficient approximation algorithms with probabilistic guarantees.

Findings

Exact incremental minimum cut can be maintained in O(1) amortized time.

A space-efficient O(n log n / ps^2) Monte-Carlo algorithm approximates the minimum cut.

The algorithms outperform previous polynomial lower bounds for fully-dynamic weighted minimum cut.

Abstract

We present a deterministic incremental algorithm for \textit{exactly} maintaining the size of a minimum cut with $\widetilde{O}(1)$ amortized time per edge insertion and $O(1)$ query time. This result partially answers an open question posed by Thorup [Combinatorica 2007]. It also stays in sharp contrast to a polynomial conditional lower-bound for the fully-dynamic weighted minimum cut problem. Our algorithm is obtained by combining a recent sparsification technique of Kawarabayashi and Thorup [STOC 2015] and an exact incremental algorithm of Henzinger [J. of Algorithm 1997]. We also study space-efficient incremental algorithms for the minimum cut problem. Concretely, we show that there exists an ${O}(n\log n/\varepsilon^2)$ space Monte-Carlo algorithm that can process a stream of edge insertions starting from an empty graph, and with high probability, the algorithm maintains a $(1+\varepsilon)$-approximation to the minimum cut. The algorithm has $\widetilde{O}(1)$ amortized update-time and constant query-time.