Improved 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. 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 $\left(1-\frac{1}{k},k-3\right)$ for $k$ is odd, and also give another bicriteria approximation $(3/2,k-1)$. After this paper was written, we became aware that the same result was achieved earlier by Kumar and Swamy. 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.