Bicriteria approximation for $k$-edge-connectivity

2025-07-04

AI Summary

This paper studies approximation algorithms for the $k$-edge-connected spanning subgraph ($k$-ECSS) problem and its multigraph variant ($k$-ECSM). To overcome the long-standing 2-approximation barrier, it introduces a novel framework combining cutting-plane linear programming (Cut-LP) relaxation with combinatorial construction. The method relaxes connectivity constraints to achieve better trade-offs between cost and structural feasibility. This yields improved bicriteria approximations for $k$-ECSS: from $(1, k-10)$ to $(1, k-4)$, and the first $(3/2, k-2)$ guarantee. For $k$-ECSM, it achieves a single-criterion approximation ratio of $1 + 4/k$, significantly improving upon prior bounds. The results advance the theoretical frontier of $k$-edge-connected subgraph problems by establishing tighter approximation guarantees through a more flexible treatment of connectivity requirements.

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: Representation, reconstruction, and subgraph or motif discovery in 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. Recently, 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. We improve the bicriteria approximation to $(1,k-4)$, and also give another non-trivial bicriteria approximation $(3/2,k-2)$. 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. A $(1,k-p)$ bicriteria approximation for $k$-ECSS w.r.t. Cut-LP implies approximation ratio $1+p/k$ for $k$-ECSM, hence our result also improves the approximation ratio for $k$-ECSM.

Problem

• Improve bicriteria approximation for k-edge-connected subgraphs
• Enhance cost and connectivity trade-off in k-ECSS
• Extend results to k-edge-connected multi-subgraph problem

Innovation

• Bicriteria approximation for k-edge-connectivity
• Improved (1, k-4) approximation algorithm
• Non-trivial (3/2, k-2) bicriteria approximation