Bicriteria approximation for $k$-edge-connectivity

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.