Approximating k-Edge-Connected Spanning Subgraphs via a Near-Linear Time LP Solver

TL;DR

This paper introduces a near-linear time LP solver that approximates the $k$-edge-connected spanning subgraph problem, achieving near-optimal solutions faster than previous algorithms and improving efficiency for network resilience design.

Contribution

It presents a near-linear time algorithm that approximates the LP relaxation for $k$ECSS, leading to faster approximation algorithms with ratios close to 2.

Findings

Achieves a $(1+psilon)$-approximate fractional solution in  O(m/psilon^2) time.

Provides a $(2+psilon)$-approximate integral solution with improved runtime.

Reduces dependence on $k$ in the approximation algorithm's complexity.

Abstract

In the $k$-edge-connected spanning subgraph ($k$ECSS) problem, our goal is to compute a minimum-cost sub-network that is resilient against up to $k$ link failures: Given an $n$-node $m$-edge graph with a cost function on the edges, our goal is to compute a minimum-cost $k$-edge-connected spanning subgraph. This NP-hard problem generalizes the minimum spanning tree problem and is the "uniform case" of a much broader class of survival network design problems (SNDP). A factor of two has remained the best approximation ratio for polynomial-time algorithms for the whole class of SNDP, even for a special case of $2$ECSS. The fastest $2$-approximation algorithm is however rather slow, taking $O(mn k)$ time [Khuller, Vishkin, STOC'92]. A faster time complexity of $O(n^2)$ can be obtained, but with a higher approximation guarantee of $(2k-1)$ [Gabow, Goemans, Williamson, IPCO'93]. Our main contribution is an algorithm that $(1+\epsilon)$-approximates the optimal fractional solution in $\tilde O(m/\epsilon^2)$ time (independent of $k$), which can be turned into a $(2+\epsilon)$ approximation algorithm that runs in time $\tilde O\left(\frac{m}{\epsilon^2} + \frac{k^2n^{1.5}}{\epsilon^2}\right)$ for (integral) $k$ECSS; this improves the running time of the aforementioned results while keeping the approximation ratio arbitrarily close to a factor of two.