Almost Tight Additive Guarantees for $k$-Edge-Connectivity

TL;DR

This work delivers near-optimal additive guarantees for the $k$-edge-connected spanning problems by a streamlined LP-rounding approach that preserves a laminar, tightly uncrossed set of cuts. For even $k$, it produces a cost-at-most-$LP^*$ subgraph that is $(k-2)$-edge-connected; for odd $k$, the guarantee is $(k-3)$, with a complementary bicriteria tradeoff yielding $(k-1)$-edge connectivity at $1.5\cdot LP^*$. The results extend to the $k$-edge-connected spanning multigraph problem with improved approximations and to degree-bounded variants with only additive degree-violations, representing a significant step toward optimal additive guarantees in survivable network design. The core technique hinges on maintaining a full-rank laminar family of tight cut-constraints via a robust uncrossing property, enabling progress in each refinement step without relying on ghost-edge augmentations. Overall, the paper sharpens the understanding of the tradeoff between connectivity and cost in ${k}\text{-ECSS}$ and ${k}\text{-ECSM}$ and broadens the toolkit for degree-bounded network design using iterative rounding and laminarity.

Abstract

We consider the \emph{$k$-edge connected spanning subgraph} (kECSS) problem, where we are given an undirected graph $G = (V, E)$ with nonnegative edge costs $\{c_e\}_{e\in E}$, and we seek a minimum-cost \emph{$k$-edge connected} subgraph $H$ of $G$. For even $k$, we present a polytime algorithm that computes a $(k-2)$-edge connected subgraph of cost at most the optimal value $LP^*$ of the natural LP-relaxation for kECSS; for odd $k$, we obtain a $(k-3)$-edge connected subgraph of cost at most $LP^*$. Since kECSS is APX-hard for all $k\geq 2$, our results are nearly optimal. They also significantly improve upon the recent work of Hershkowitz et al., both in terms of solution quality and the simplicity of algorithm and its analysis. Our techniques also yield an alternate guarantee, where we obtain a $(k-1)$-edge connected subgraph of cost at most $1.5\cdot LP^*$; with unit edge costs, the cost guarantee improves to $(1+\frac{4}{3k})\cdot LP^*$, which improves upon the state-of-the-art approximation for unit edge costs, but with a unit loss in edge connectivity. Our kECSS-result also yields results for the \emph{$k$-edge connected spanning multigraph} (kECSM) problem, where multiple copies of an edge can be selected: we obtain a $(1+2/k)$-approximation algorithm for even $k$, and a $(1+3/k)$-approximation algorithm for odd $k$. Our techniques extend to the degree-bounded versions of kECSS and kECSM, wherein we also impose degree lower- and upper- bounds on the nodes. We obtain 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 \{kECSS,kECSM\} 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.

Figures (3)

• Figure 1: The approximation vs. connectivity tradeoff for (a) ${k}\text{-}\mathsf{ECSS}$, and (b) unweighted ${k}\text{-}\mathsf{ECSS}$. The shaded circles in (a) and the shaded squares in (b) denote our results, as stated in Theorems \ref{['kecss-thm']} and \ref{['bicriteria-kecss']}. The ? denotes the point $(k-1,1)$, which is the tightest polytime-achievable guarantee of the form $(\cdot,1)$. The results in NutovC25 were obtained independently and concurrently; neither group was aware of the others' work when they obtained their respective results. For unweighted ${k}\text{-}\mathsf{ECSS}$: Gabow and Gallagher GabowG12 actually obtain a $\bigl(1+\frac{1.91}{k}\bigr)$-approximation; the results in GabowGTW09GabowG12 are not stated as yielding the $(k-2,1)$-guarantee, but they can be translated to obtain this guarantee; see Section \ref{['relwork']}.
• Figure 3: The blobs denote the sets in the laminar family $\mathcal{L}$, and the dashed and solid lines represent the edges in $F$. The dashed (red) edges have $z_e=0.5$, and the solid (green) edges have $z_e=0.75$. Note that $z\bigl(\delta(S)\bigr)$ is an integer for all $S\in\mathcal{L}$.
• Figure 4: An extreme-point solution to ${6}\text{-ECSSLP}$. The dashed (red) edges have $\overline x_e=0.5$, the solid (green) edges have $\overline x_e=0.75$, and the thick (blue) edges have $\overline x_e=1$. After dropping integral edges, we obtain the residual instance shown in Fig. \ref{['ex1-fig']}.

Theorems & Definitions (30)

• Theorem 1.1: Proved in Section \ref{['kecss']}
• Theorem 1.2: Proved in Section \ref{['bicriteria']}
• Theorem 1.3: Proved in Section \ref{['kecsm']}
• Definition 2.1
• Lemma 2.2
• proof
• proof
• Claim 2.3
• proof
• Lemma 2.4