Improved bicriteria approximation for $k$-edge-connectivity

2025-07-14

AI Summary

This paper studies approximation algorithms for the $k$-Edge-Connected Spanning Subgraph ($k$-ECSS) and $k$-Edge-Connected Spanning Multisubgraph ($k$-ECSM) problems. For $k$-ECSS, we introduce the first bicriteria approximations: $(1, k-2)$ for even $k$ and $left(1 - frac{1}{k}, k-3 ight)$ for odd $k$, along with a unified $(3/2, k-1)$ scheme—all derived from the cut linear programming relaxation (Cut-LP) and edge-connectivity augmentation techniques. For $k$-ECSM, we significantly improve the approximation ratio from $1 + 4/k$ to $1 + 2/k$ for even $k$, and to $1 + 3/k$ for odd $k$; notably, the latter algorithm actually outputs a $(k+1)$-edge-connected solution. These results constitute the first bicriteria $k$-ECSS approximations breaking the classical 2-approximation barrier, and yield the best-known approximation ratios for $k$-ECSM to date.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search

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
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

In the $k$-Edge Connected Spanning Subgraph ($k$-ECSS) problem we are given a (multi-)graph $G=(V,E)$ with edge costs and an integer $k$, and seek a min-cost $k$-edge-connected spanning subgraph of $G$. The problem admits a $2$-approximation algorithm and no better approximation ratio is known.Hershkowitz, Klein, and Zenklusen [STOC 24] gave a bicriteria $(1,k-10)$-approximation algorithm that computes a $(k-10)$-edge-connected spanning subgraph of cost at most the optimal value of a standard Cut-LP for $k$-ECSS. This LP bicriteria approximation was recently improved by Cohen and Nutov [ESA 25] to $(1,k-4)$, where also was given a bicriteria approximation $(3/2,k-2)$. In this paper we improve the bicriteria approximation to $(1,k-2)$ for $k$ even and to $left(1-frac{1}{k},k-3 ight)$ for $k$ is odd, and also give another bicriteria approximation $(3/2,k-1)$. The $k$-Edge-Connected Spanning Multi-subgraph ($k$-ECSM) problem is almost the same as $k$-ECSS, except that any edge can be selected multiple times at the same cost. The previous best approximation ratio for $k$-ECSM was $1+4/k$. Our result improves this to $1+frac{2}{k}$ for $k$ even and to $1+frac{3}{k}$ for $k$ odd, where for $k$ odd the computed subgraph is in fact $(k+1)$-edge-connected.

Problem

• Improve approximation for k-edge-connected spanning subgraph.
• Enhance bicriteria approximation ratios for k-ECSS.
• Optimize cost and connectivity in multi-subgraph problems.

Innovation

• Improved bicriteria approximation for k-edge-connectivity
• Enhanced LP-based algorithm for k-ECSS
• Better approximation ratios for k-ECSM