Streaming Algorithms for Connectivity Augmentation

TL;DR

This work introduces streaming algorithms for connectivity augmentation, notably a $(2+\varepsilon)$-approximation for $k$-CAP in link-arrival streams using $O(\varepsilon^{-1} n \log n)$ space and a nearly matching lower bound, establishing a gap with offline results. It extends to the fully streaming model with a near-optimal $O(t)$-approximation for $k$-CAP by leveraging compact $k$-connectivity certificates and an improved weighted spanner construction that avoids dependence on $\log W$ via even-odd bucketing. A core technical contribution is an efficient spanner algorithm in the streaming setting that achieves a $(2t-1+\varepsilon)$-approximation with space $O(\varepsilon^{-1} n^{1+1/t} \log n)$, which in turn enables near-optimal fully streaming solutions for STAP, SNDP, and $k$-ECSS. The paper also develops lower bounds via INDEX-based reductions and demonstrates broad applicability of the techniques across network-design problems, highlighting a fundamental distinction between streaming and offline regimes for these augmentation tasks. The results advance the state of streaming graph algorithms by providing tight space-approximation tradeoffs for fundamental augmentation problems and their network-design relatives.

Abstract

We study the $k$-connectivity augmentation problem ($k$-CAP) in the single-pass streaming model. Given a $(k-1)$-edge connected graph $G=(V,E)$ that is stored in memory, and a stream of weighted edges $L$ with weights in $\{0,1,\dots,W\}$, the goal is to choose a minimum weight subset $L'\subseteq L$ such that $G'=(V,E\cup L')$ is $k$-edge connected. We give a $(2+ε)$-approximation algorithm for this problem which requires to store $O(ε^{-1} n\log n)$ words. Moreover, we show our result is tight: Any algorithm with better than $2$-approximation for the problem requires $Ω(n^2)$ bits of space even when $k=2$. This establishes a gap between the optimal approximation factor one can obtain in the streaming vs the offline setting for $k$-CAP. We further consider a natural generalization to the fully streaming model where both $E$ and $L$ arrive in the stream in an arbitrary order. We show that this problem has a space lower bound that matches the best possible size of a spanner of the same approximation ratio. Following this, we give improved results for spanners on weighted graphs: We show a streaming algorithm that finds a $(2t-1+ε)$-approximate weighted spanner of size at most $O(ε^{-1} n^{1+1/t}\log n)$ for integer $t$, whereas the best prior streaming algorithm for spanner on weighted graphs had size depending on $\log W$. Using our spanner result, we provide an optimal $O(t)$-approximation for $k$-CAP in the fully streaming model with $O(nk + n^{1+1/t})$ words of space. Finally we apply our results to network design problems such as Steiner tree augmentation problem (STAP), $k$-edge connected spanning subgraph ($k$-ECSS), and the general Survivable Network Design problem (SNDP). In particular, we show a single-pass $O(t\log k)$-approximation for SNDP using $O(kn^{1+1/t})$ words of space, where $k$ is the maximum connectivity requirement.

Figures (6)

• Figure 1: This figure illustrates the process of edge sparsification in a cycle for connectivity augmentation.
• Figure 2: An example of 3-edge-connected components $Q_k$ of the graph $G_k$. Thin black edges denote the base cycle, and thick blue edges denote the links from the set $\bigcup_{i\ge 0}E_{k-2i}$; together they form $G_k$. The dashed green lines describe the 3-edge-connected components of graph $G_k$.
• Figure 3: A picture of the cactus $H_k$ (middle) produced by shrinking $G_k$ (left). The tree $T_k$ (right) is produced from cactus $H_k$.
• Figure 4: After adding an edge from $E_{k+2}$ (depicted in red), the partition $\mathcal{C}_{k+2}$ refines the old partition $\mathcal{C}_{k}$: $\{1,3,9\}$ breaks into $\{1\}$ and $\{3,9\}$.
• Figure 5: A lower bound instance of tree augmentation with weighted edges. The black edges form the base tree. The red links are added by Bob with zero weight. The blue link is added by Alice with weight $1$.

Theorems & Definitions (53)

• Theorem 2.1
• Definition 2.2: Cactus Graph
• Lemma 2.3: DinitsKL76
• Corollary 2.4
• Remark 2.5
• Lemma 2.6: Theorem 3 in galvez2021cycle; see also Lemma 2.2 in traub20231
• Theorem 2.7
• proof
• Theorem 2.8
• proof