Almost Tight Additive Guarantees for k-Edge-Connectivity

We consider the $\boldsymbol{k}$-edge connected spanning subgraph (k-ECSS) problem, where we are given an undirected graph $G=(V, E)$ with nonnegative edge costs $\left\{c_{e}\right\}_{e \in E}$, and the goal is to find a minimum-cost subgraph H of G that is k edge connected, i.e., there exist at least k edge-disjoint paths between every pair of vertices in H. For even k, we present a polynomial time algorithm that computes a ($k-2$)-edge connected subgraph of cost at most that of the optimal k-edge connected subgraph of G; for odd k, we obtain a $(k-3)$ edge connected subgraph of cost at most the optimum. In fact, the cost of our solution does not exceed the optimal value, $\mathbf{L P}_{\boldsymbol{k} \text {-ECSSLP }}^{\boldsymbol{*}}$ of the natural LP-relaxation for $\boldsymbol{k}$-ECSS. Since k-ECSS is $A P X$-hard for all values of $k \geq 2$, our results are nearly optimal. They also significantly improve upon the recent work of Hershkowitz, Klein, and Zenklusen [1], both in terms of solution quality and the simplicity of algorithm and its analysis. Interestingly, our techniques also yield an alternate guarantee, where we obtain a($k-1$)-edge connected subgraph of cost at most $1.5 \cdot \mathrm{LP}_{\boldsymbol{k}-\mathrm{ECSSLP}}^{*}$; with unit edge costs, the cost guarantee improves to $\left(1+\frac{4}{3 k}\right) \cdot$ LP $_{\boldsymbol{k} \text {-ECSSLP }}^{\boldsymbol{*}}$, which improves upon the state-of-the-art approximation guarantee for unit edge costs [2], albeit with a unit loss in edge connectivity. Our k-ECSS-result also yields results for the k-edge connected spanning multigraph (k-ECSM) problem, where multiple copies of an edge can be selected. For $\boldsymbol{k}$-ECSM, we obtain a $\left(1+\frac{2}{k}\right)$-approximation algorithm for even k, and $\mathbf{a}\left(1+\frac{3}{k}\right)$ approximation algorithm for odd $\boldsymbol{k}$. Finally, our techniques extend to the degree-bounded versions of k-ECSS and k-ECSM, wherein we also impose degree lower- and upper- bounds on the nodes. Our results for k-ECSS and k-ECSM extend to yield 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 $\{k$-ECSS, k-ECSM $\}$ 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. Work done while N. Kumar was a postdoc in the $C \& O$ department at the University of Waterloo. Supported in part by C. Swamy’s NSERC Discovery grant.