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Light Edge Fault Tolerant Graph Spanners

Greg Bodwin, Michael Dinitz, Ama Koranteng, Lily Wang

TL;DR

This paper initiates the study of light edge fault-tolerant (EFT) graph spanners in general graphs, showing that naive normalizations of lightness fail under fault conditions and introducing a bicriteria competitive notion of lightness with respect to EFT connectivity preservers. The authors prove that the natural 2f-threshold is both necessary and sufficient for obtaining meaningful lightness bounds, establishing a tight (up to poly-factors) 2f-competitive lightness bound, and they connect these bounds to the weighted-girth framework via λ(n,k). They provide both existential and efficiently constructible (polytime) upper bounds, leveraging a fault-tolerant greedy algorithm, blocking sets, Steiner forest packing, and subsampling techniques to achieve lightness of roughly O(f^{1/2}) times the non-fault-tolerant lightness under certain parameter regimes, while also proving near-linear lower bounds when the competition parameter is below 2f. The work also discusses algorithmic tradeoffs, including polytime construction with acceptable, but weaker, f-dependence, and outlines future directions for vertex-fault-tolerant analogues using connected dominating sets and independent spanning trees. Overall, the paper provides a rigorous framework for understanding light EFT spanners, identifies the precise threshold for bicriteria lightness (2f), and delivers near-optimal bounds that bridge non-faulty lightness and EFT sparsity, with implications for robust distance-preserving structures in networks.

Abstract

There has recently been significant interest in fault tolerant spanners, which are spanners that still maintain their stretch guarantees after some nodes or edges fail. This work has culminated in an almost complete understanding of the three-way tradeoff between stretch, sparsity, and number of faults tolerated. However, despite some progress in metric settings, there have been no results to date on the tradeoff in general graphs between stretch, lightness, and number of faults tolerated. We initiate the study of light edge fault tolerant (EFT) graph spanners, obtaining the first such results. First, we observe that lightness can be unbounded if we use the traditional definition (normalizing by the MST). We then argue that a natural definition of fault-tolerant lightness is to instead normalize by a min-weight fault tolerant connectivity preserver; essentially, a fault-tolerant version of the MST. However, even with this, we show that it is still not generally possible to construct $f$-EFT spanners whose weight compares reasonably to the weight of a min-weight $f$-EFT connectivity preserver. In light of this lower bound, it is natural to then consider bicriteria notions of lightness, where we compare the weight of an $f$-EFT spanner to a min-weight $(f' > f)$-EFT connectivity preserver. The most interesting question is to determine the minimum value of $f'$ that allows for reasonable lightness upper bounds. Our main result is a precise answer to this question: $f' = 2f$. In particular, we show that the lightness can be untenably large (roughly $n/k$ for a $k$-spanner) if one normalizes by the min-weight $(2f-1)$-EFT connectivity preserver. But if one normalizes by the min-weight $2f$-EFT connectivity preserver, then we show that the lightness is bounded by just $O(f^{1/2})$ times the non-fault tolerant lightness (roughly $n^{1/k}$, for a $(1+ε)(2k-1)$-spanner).

Light Edge Fault Tolerant Graph Spanners

TL;DR

This paper initiates the study of light edge fault-tolerant (EFT) graph spanners in general graphs, showing that naive normalizations of lightness fail under fault conditions and introducing a bicriteria competitive notion of lightness with respect to EFT connectivity preservers. The authors prove that the natural 2f-threshold is both necessary and sufficient for obtaining meaningful lightness bounds, establishing a tight (up to poly-factors) 2f-competitive lightness bound, and they connect these bounds to the weighted-girth framework via λ(n,k). They provide both existential and efficiently constructible (polytime) upper bounds, leveraging a fault-tolerant greedy algorithm, blocking sets, Steiner forest packing, and subsampling techniques to achieve lightness of roughly O(f^{1/2}) times the non-fault-tolerant lightness under certain parameter regimes, while also proving near-linear lower bounds when the competition parameter is below 2f. The work also discusses algorithmic tradeoffs, including polytime construction with acceptable, but weaker, f-dependence, and outlines future directions for vertex-fault-tolerant analogues using connected dominating sets and independent spanning trees. Overall, the paper provides a rigorous framework for understanding light EFT spanners, identifies the precise threshold for bicriteria lightness (2f), and delivers near-optimal bounds that bridge non-faulty lightness and EFT sparsity, with implications for robust distance-preserving structures in networks.

Abstract

There has recently been significant interest in fault tolerant spanners, which are spanners that still maintain their stretch guarantees after some nodes or edges fail. This work has culminated in an almost complete understanding of the three-way tradeoff between stretch, sparsity, and number of faults tolerated. However, despite some progress in metric settings, there have been no results to date on the tradeoff in general graphs between stretch, lightness, and number of faults tolerated. We initiate the study of light edge fault tolerant (EFT) graph spanners, obtaining the first such results. First, we observe that lightness can be unbounded if we use the traditional definition (normalizing by the MST). We then argue that a natural definition of fault-tolerant lightness is to instead normalize by a min-weight fault tolerant connectivity preserver; essentially, a fault-tolerant version of the MST. However, even with this, we show that it is still not generally possible to construct -EFT spanners whose weight compares reasonably to the weight of a min-weight -EFT connectivity preserver. In light of this lower bound, it is natural to then consider bicriteria notions of lightness, where we compare the weight of an -EFT spanner to a min-weight -EFT connectivity preserver. The most interesting question is to determine the minimum value of that allows for reasonable lightness upper bounds. Our main result is a precise answer to this question: . In particular, we show that the lightness can be untenably large (roughly for a -spanner) if one normalizes by the min-weight -EFT connectivity preserver. But if one normalizes by the min-weight -EFT connectivity preserver, then we show that the lightness is bounded by just times the non-fault tolerant lightness (roughly , for a -spanner).

Paper Structure

This paper contains 34 sections, 29 theorems, 46 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Fault Tolerance
  3. Main Result: Defining Fault-Tolerant Lightness
  4. (Failed) Attempt 1: Normalize by an MST
  5. (Failed) Attempt 2: Normalize by a Fault-Tolerant Connectivity Preserver
  6. (Successful) Attempt 3: Bicriteria Competitive Lightness
  7. The Bicriteria Price of Fault Tolerance
  8. Efficient Construction Algorithms
  9. Paper Outline
  10. Technical Overview
  11. Overview of Upper Bounds
  12. Algorithm and Blocking Sets.
  13. Weighted Girth.
  14. The Subsampling Method.
  15. Steiner Forest Packing.
  16. ...and 19 more sections

Key Result

Theorem 2

For every positive integer $k \geq 1$, every $n$-node graph $G$ admits a $(2k-1)$-spanner $H$ with $|E(H)| \leq O(n^{1+1/k})$, and this bound is tight (assuming the Erdős girth conjecture girth).

Figures (3)

  • Figure 1: This graph $G$ shows that the quantity $\frac{w(H)}{w(\texttt{mst}(G))}$ can be unbounded for $1$-EFT spanners. Here $w(G) = W+2$ for any $W$ we like, and $w(\texttt{mst}(G))=2$, but we cannot remove any edges in a $1$-EFT spanner. Thus a different definition of lightness is needed in the fault tolerant setting.
  • Figure 2: Outer (solid) cycle edges have unit weight, dotted edges have weight $\frac{n-2}{k} - \epsilon$
  • Figure 3: Transformation of an edge from $G'$ to bipartite graph of clouds in $G$.

Theorems & Definitions (53)

  • Definition 1: Spanners
  • Theorem 2: ADDJS93
  • Definition 3: Lightness
  • Theorem 4: LS23Bodwin25
  • Definition 5: Edge Fault Tolerant Spanners CLPR10
  • Theorem 6: CLPR10
  • Definition 7: EFT Connectivity Preservers
  • Definition 8: Competitive Lightness
  • Theorem 9
  • Theorem 10: Main Result
  • ...and 43 more