Fast Approximation Algorithms for Bounded Degree and Crossing Spanning Tree Problems

TL;DR

This paper introduces fast approximation algorithms for the Bounded-Degree Minimum Spanning Tree and Crossing Spanning Tree problems, utilizing LP solutions and swap-rounding techniques to achieve near-linear time complexity and improved efficiency.

Contribution

It presents a near-linear time algorithm for approximating the minimum degree spanning tree and introduces a fast swap-rounding method applicable to various combinatorial problems.

Findings

Achieves a $(1+\epsilon)$ approximation for BD-MST in near-linear time.

Develops a fast implementation of swap-rounding in the spanning tree polytope.

Provides a framework for graph sparsification to accelerate algorithms.

Abstract

We develop fast approximation algorithms for the minimum-cost version of the Bounded-Degree MST problem (BD-MST) and its generalization the Crossing Spanning Tree problem (Crossing-ST). We solve the underlying LP to within a $(1+\epsilon)$ approximation factor in near-linear time via the multiplicative weight update (MWU) technique. This yields, in particular, a near-linear time algorithm that outputs an estimate $B$ such that $B \le B^* \le \lceil (1+\epsilon)B \rceil +1$ where $B^*$ is the minimum-degree of a spanning tree of a given graph. To round the fractional solution, in our main technical contribution, we describe a fast near-linear time implementation of swap-rounding in the spanning tree polytope of a graph. The fractional solution can also be used to sparsify the input graph that can in turn be used to speed up existing combinatorial algorithms. Together, these ideas lead to significantly faster approximation algorithms than known before for the two problems of interest. In addition, a fast algorithm for swap rounding in the graphic matroid is a generic tool that has other applications, including to TSP and submodular function maximization.