Fast Approximation Algorithms for the Generalized Survivable Network Design Problem

TL;DR

This paper introduces the first strongly polynomial time FPTAS for the LP relaxation of the $f$-connectivity network design problem with proper functions, leading to a near-optimal approximation for the generalized survivable network design problem.

Contribution

It presents a novel strongly polynomial time FPTAS for the LP relaxation of the $f$-connectivity problem, enabling a $(2+ ext{epsilon})$-approximation for the generalized survivable network design problem.

Findings

First strongly polynomial FPTAS for LP relaxation of $f$-connectivity.

Achieves a $(2+ ext{epsilon})$-approximation for the problem.

Applicable to a broad class of NP-hard network design problems.

Abstract

In a standard $f$-connectivity network design problem, we are given an undirected graph $G=(V,E)$, a cut-requirement function $f:2^V \rightarrow {\mathbb{N}}$, and non-negative costs $c(e)$ for all $e \in E$. We are then asked to find a minimum-cost vector $x \in {\mathbb{N}}^E$ such that $x(\delta(S)) \geq f(S)$ for all $S \subseteq V$. We focus on the class of such problems where $f$ is a proper function. This encodes many well-studied NP-hard problems such as the generalized survivable network design problem. In this paper we present the first strongly polynomial time FPTAS for solving the LP relaxation of the standard IP formulation of the $f$-connectivity problem with general proper functions $f$. Implementing Jain's algorithm, this yields a strongly polynomial time $(2+\epsilon)$-approximation for the generalized survivable network design problem (where we consider rounding up of rationals an arithmetic operation).