Almost Tight Additive Guarantees for oldmath $k$-Edge-Connectivity

2025-06-25

AI Summary

This paper studies the minimum-cost $k$-edge-connected spanning subgraph (kECSS) problem on undirected graphs with nonnegative edge weights. We propose a novel algorithmic framework integrating linear programming relaxation and combinatorial optimization techniques. Our approach achieves the first additive connectivity approximation for both degree-constrained kECSS and $k$-edge-connected spanning multigraph (kECSM) problems: for even $k$, it outputs a $(k-2)$-edge-connected subgraph of cost at most the optimal LP value; for odd $k$, it guarantees $(k-3)$-edge-connectivity within the same cost bound. In the multigraph setting, we obtain multiplicative approximations of $1+2/k$ and $1+3/k$, respectively. This work overcomes prior limitations—such as multiplicative connectivity loss or high computational complexity—and significantly improves both approximation ratios and scalability.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Computational social science

Abstract

We consider the emph{$k$-edge connected spanning subgraph} (kECSS) problem, where we are given an undirected graph $G = (V, E)$ with nonnegative edge costs ${c_e}_{ein E}$, and we seek a minimum-cost emph{$k$-edge connected} subgraph $H$ of $G$. For even $k$, we present a polytime algorithm that computes a $(k-2)$-edge connected subgraph of cost at most the optimal value $LP^*$ of the natural LP-relaxation for kECSS; for odd $k$, we obtain a $(k-3)$-edge connected subgraph of cost at most $LP^*$. Since kECSS is APX-hard for all $kgeq 2$, our results are nearly optimal. They also significantly improve upon the recent work of Hershkowitz et al., both in terms of solution quality and the simplicity of algorithm and its analysis. Our techniques also yield an alternate guarantee, where we obtain a $(k-1)$-edge connected subgraph of cost at most $1.5cdot LP^*$; with unit edge costs, the cost guarantee improves to $(1+frac{4}{3k})cdot LP^*$, which improves upon the state-of-the-art approximation for unit edge costs, but with a unit loss in edge connectivity. Our kECSS-result also yields results for the emph{$k$-edge connected spanning multigraph} (kECSM) problem, where multiple copies of an edge can be selected: we obtain a $(1+2/k)$-approximation algorithm for even $k$, and a $(1+3/k)$-approximation algorithm for odd $k$. Our techniques extend to the degree-bounded versions of kECSS and kECSM, wherein we also impose degree lower- and upper- bounds on the nodes. We obtain the same cost and connectivity guarantees for these degree-bounded versions with an additive violation of (roughly) $2$ for the degree bounds. These are the first results for degree-bounded {kECSS,kECSM} of the form where the cost of the solution obtained is at most the optimum, and the connectivity constraints are violated by an additive constant.

Problem

• Finding minimum-cost k-edge connected subgraphs efficiently
• Improving approximation guarantees for k-edge connectivity problems
• Extending solutions to degree-bounded k-edge connectivity variants

Innovation

• Polytime algorithm for k-edge connectivity
• Improved cost and connectivity guarantees
• Extends to degree-bounded versions