Ghost Value Augmentation for $k$-Edge-Connectivity

TL;DR

The paper tackles the $k$-ECSS problem by introducing ghost value augmentations into an iterative-relaxation framework, enabling a poly-time rounding that achieves a cost no larger than the optimal $(k+10)$-ECSS LP optimum and yielding a near-$1$-approximation for the related $k$-ECSM problem. A central contribution is the Main Rounding Theorem, which guarantees a cost-preserving integral rounding with at most a $k-9$ decrease in cut-connectivity, enabling strong guarantees despite limited slack. The work also settles a conjecture by Pritchard by proving a $(1+O(1/k))$-approximation for $k$-ECSM and establishes a matching hardness of approximation via a reduction from unweighted TAP, thereby delineating the asymptotic limits of current techniques. Overall, the method advances resource-augmented approaches for connectivity problems and provides the first asymptotically tight LP-based approximation for $k$-ECSM, with implications for subset/Steiner variants.

Abstract

We give a poly-time algorithm for the $k$-edge-connected spanning subgraph ($k$-ECSS) problem that returns a solution of cost no greater than the cheapest $(k+10)$-ECSS on the same graph. Our approach enhances the iterative relaxation framework with a new ingredient, which we call ghost values, that allows for high sparsity in intermediate problems. Our guarantees improve upon the best-known approximation factor of $2$ for $k$-ECSS whenever the optimal value of $(k+10)$-ECSS is close to that of $k$-ECSS. This is a property that holds for the closely related problem $k$-edge-connected spanning multi-subgraph ($k$-ECSM), which is identical to $k$-ECSS except edges can be selected multiple times at the same cost. As a consequence, we obtain a $\left(1+O\left(\frac{1}{k}\right)\right)$-approximation algorithm for $k$-ECSM, which resolves a conjecture of Pritchard and improves upon a recent $\left(1+O\left(\frac{1}{\sqrt{k}}\right)\right)$-approximation algorithm of Karlin, Klein, Oveis Gharan, and Zhang. Moreover, we present a matching lower bound for $k$-ECSM, showing that our approximation ratio is tight up to the constant factor in $O\left(\frac{1}{k}\right)$, unless $P=NP$.

Figures (5)

• Figure 1: An example where skew supermodularity of the requirement function fails. The solid blue edges represent collections of frozen edges between sets. The dashed edges represent collections of non-frozen edges. Here, $S \smallsetminus T$ and $S \cap T$ have been dropped as they have at least $k-c$ frozen edges. However, $S,T,T \smallsetminus S$, and $S \cup T$ have fewer than $k-c$ frozen edges and thus remain. So, $f(S)=f(T)=k$ but $f(S \smallsetminus T)=f(S \cap T)=0$.
• Figure 2: An example of where uncrossing breaks down, where in the top figure we let $\ell=\lceil \frac{c+1}{2} \rceil$ and in the bottom figure we assume $c=10$ for concreteness. The blue edges represent collections of frozen edges and the dotted edges represent collections of fractional edges. To ensure feasibility we let $0 < a < 1$. In this example, we drop cuts when there are $k-c$ frozen edges (or $k-10$ in the bottom picture). Then, in both examples we have dropped $S \smallsetminus T$ and $S \cap T$, but we have not yet dropped $S$ and $T$.
• Figure 3: The blue edges are frozen, and the dotted edges are fractional. Consider the red set $S$ with $2$ incident fractional edges. Since $\delta(S)$ has only $2$ fractional edges, and the edges inside $S$ are all frozen, any set contained in $S$ is already safe, as is the case with the blue vertex.
• Figure 4: A situation in which ghost value augmentation is necessary. All dotted edges have value 1/2, so $S$ and $T$ are tight but not dropped. $S \cap T$ and $S \smallsetminus T$, however, have been dropped and thus are allowed to drop below connectivity $k$. Therefore it is not possible to uncross $S$ and $T$. Note that the integrality of the rightmost blue edge leaving $T$ is not necessary. In particular, this rightmost blue edge could be any number of fractional edges and the instance would have the same behavior.
• Figure 5: Our reduction from TAP to $k$-ECSM. \ref{['sfig:tap1']}: TAP instance on $G$ with links $L$ dashed and edge $e$ labeled. \ref{['sfig:tap2']}: feasible TAP solution in blue. \ref{['sfig:tap3']}: a tree cut $S$. \ref{['sfig:tap4']}: $k$-ECSM instance on $H$ with $w_e$, $w_e'$ labeled; $V_E$ are the small nodes. \ref{['sfig:tap5']}: feasible $k$-ECSM solution for $H$ where non-zero links in blue and values of edges incident to $w_e$ and $w_e'$ labeled. \ref{['sfig:tap6']}: cut $S'$ in $H$ corresponding to $S$ in $G$.

Theorems & Definitions (58)

• Theorem 1.1
• Conjecture 1.2: Pri11
• Theorem 1.3
• proof : Proof of \ref{['thm:mainECSM']} assuming $k$ is given in unary
• Theorem 1.4
• Theorem 2.1: Main Rounding Theorem
• Theorem 2.1
• proof
• Theorem 2.1
• proof : Proof of \ref{['thm:mainECSM']} without assuming that $k$ is given in unary