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Preserving Distances in Faulty Colored Graphs

Merav Parter, Asaf Petruschka

TL;DR

The paper investigates color fault-tolerant (CFT) network design, where colors model correlated failures, and studies distance- and reachability-preserving subgraphs under color faults. It delivers nearly-tight bounds for 1-CFT sourcewise distance preservers, showing a fundamental dependency on the color-class size Δ via Õ(n^{2-1/(Δ+1)} · σ^{1/(Δ+1)}). It also provides a new inductive construction that reproduces the Δ-EFT upper bounds and clarifies when colors help or do not help sparsification across several preservers, including single-pair and single-source reachability. The results reveal a nuanced landscape: colors can dramatically reduce sparsity for some preservers (e.g., sourcewise distances) while offering no improvement for others (e.g., reachability and bounded flow), and they highlight tiebreaking robustness as a key factor distinguishing 1-CFT from Δ-EFT. Overall, the work advances understanding of how color structure interacts with fault tolerance to shape sparse certificates and has implications for distributed and SRRG-aware network design.

Abstract

We study color fault-tolerant (CFT) network design problems: Given an $n$-vertex graph $G$ whose edges are arbitrarily colored (with no ``legality'' restrictions), the goal is to find a sparse subgraph $H$ such that, when any color fault causes all edges of some color $c$ to crash, the surviving subgraph $H-c$ remains ``similar'' to the surviving graph $G-c$. The similarity is problem-dependent, usually pertaining to distance preserving. If each color class has size $Δ$ or less, a brute-force approach can disregard the colors and take $H$ to be $Δ$-edge fault-tolerant ($Δ$-EFT), so that $H-F$ is similar to $G-F$ for every set $F$ of $\leq Δ$ edges. We ask if the colors can be utilized to provide a sparser $H$. Our main results concern CFT sourcewise distance preservers, where there is a given set $S \subseteq V$ of $σ$ sources, and all $S \times V$ distances should be exactly equal in $H-c$ and in $G-c$. We give nearly-tight upper and lower bounds of $\tildeΘ (n^{2-1/(Δ+1)} \cdot σ^{1/(Δ+1)})$ on the worst-case size of such preservers. The corresponding $Δ$-EFT problem admits the same lower bound, but the state-of-the-art upper bound for $Δ\geq 3$ is $\tilde{O}(n^{2-1/2^Δ} \cdot σ^{1/2^Δ})$. Our approach also leads to new and arguably simpler constructions that recover these $Δ$-EFT bounds and shed some light on their current gaps. We provide additional results along these lines, showcasing problems where the color structure helps or does not help sparsification. For preserving the distance between a single pair of vertices after a color fault, the brute-force approach via $Δ$-EFT is shown to be suboptimal. In contrast, for preserving reachability from a single source in a directed graph, it is (worst-case) optimal.

Preserving Distances in Faulty Colored Graphs

TL;DR

The paper investigates color fault-tolerant (CFT) network design, where colors model correlated failures, and studies distance- and reachability-preserving subgraphs under color faults. It delivers nearly-tight bounds for 1-CFT sourcewise distance preservers, showing a fundamental dependency on the color-class size Δ via Õ(n^{2-1/(Δ+1)} · σ^{1/(Δ+1)}). It also provides a new inductive construction that reproduces the Δ-EFT upper bounds and clarifies when colors help or do not help sparsification across several preservers, including single-pair and single-source reachability. The results reveal a nuanced landscape: colors can dramatically reduce sparsity for some preservers (e.g., sourcewise distances) while offering no improvement for others (e.g., reachability and bounded flow), and they highlight tiebreaking robustness as a key factor distinguishing 1-CFT from Δ-EFT. Overall, the work advances understanding of how color structure interacts with fault tolerance to shape sparse certificates and has implications for distributed and SRRG-aware network design.

Abstract

We study color fault-tolerant (CFT) network design problems: Given an -vertex graph whose edges are arbitrarily colored (with no ``legality'' restrictions), the goal is to find a sparse subgraph such that, when any color fault causes all edges of some color to crash, the surviving subgraph remains ``similar'' to the surviving graph . The similarity is problem-dependent, usually pertaining to distance preserving. If each color class has size or less, a brute-force approach can disregard the colors and take to be -edge fault-tolerant (-EFT), so that is similar to for every set of edges. We ask if the colors can be utilized to provide a sparser . Our main results concern CFT sourcewise distance preservers, where there is a given set of sources, and all distances should be exactly equal in and in . We give nearly-tight upper and lower bounds of on the worst-case size of such preservers. The corresponding -EFT problem admits the same lower bound, but the state-of-the-art upper bound for is . Our approach also leads to new and arguably simpler constructions that recover these -EFT bounds and shed some light on their current gaps. We provide additional results along these lines, showcasing problems where the color structure helps or does not help sparsification. For preserving the distance between a single pair of vertices after a color fault, the brute-force approach via -EFT is shown to be suboptimal. In contrast, for preserving reachability from a single source in a directed graph, it is (worst-case) optimal.

Paper Structure

This paper contains 30 sections, 20 theorems, 30 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Color Faults vs. Edge Faults.
  3. $1$-CFT vs. $\Delta$-EFT.
  4. Our Results
  5. Sourcewise Distance Preservers
  6. New $\Delta$-EFT Constructions.
  7. Tiebreaking Robustness.
  8. Applications.
  9. Other Preservers: Do Colors Provably "Help"?
  10. Single-Pair Distance Preservers: Provably "Yes".
  11. Single-Source Reachability and Bounded Flow Preservers: Provably "No".
  12. Remark on Vertex-Color Faults.
  13. Preliminaries
  14. Organization.
  15. Sourcewise Distance Preservers
  16. ...and 15 more sections

Key Result

Theorem 1.2

The worst-case size of a $1$-CFT sourcewise preserver, over $n$-vertex graphs with $\sigma$ source vertices, whose edges are colored such that color classes have size at most $\Delta$, is

Figures (4)

  • Figure 1: Illustration: third inductive step. Faulty edges of color $c$ are red. Distance thresholds from $t$ on the paths denoted by dashed lines. Left: Case \ref{['case1']}, $\pi_G(a_2, t)$ is a suffix of $\pi_G(s,t\mid c)$. Middle: Case \ref{['case2']}, a $c$-colored edge is on the $d_3$-suffix of $\pi_G(a_2, t)$. Right: Neither of the cases \ref{['case1']} and \ref{['case2']}. Then, we find a new $c$-colored edge on $\pi_G(a_2, t)$ with distance $> d_3$ from $t$, add it to $E_2$ to create $E_3$, and deduce that $\pi_G(s,t \mid c)$ must contain some $a_3 \in A_3$ on its $d_3$-suffix.
  • Figure 2: Construction of the tree $T(\Delta=2, q=3)$. $T_1$, $T_2$ and $T_3$ are copies of $T(1,3)$ with disjoint colors. The corresponding color $c_v$ for each leaf $v$ is shown beneath it.
  • Figure 3: The graph $G$ from \ref{['thm:single-pair-lb']}. The graph $G^*$ with pairs $P^* = \{(x_i, y_i)\}_{i=1}^n$ form a worst-case instance for $\mathtt{DP}(n,n)$, reweighted so that all distances in $G^*$ are smaller than $1$.
  • Figure 4: Construction of the tree $T(\Delta=3)$. $T_0$ and $T_1$ are copies of $T(2)$ with disjoint colors. The corresponding color $c_v$ for each leaf $v$ is shown beneath it.

Theorems & Definitions (35)

  • Definition 1.1: CFT Distance/Reachability Preservers
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 3.1
  • proof
  • Theorem 3.3: BodwinGPW17
  • ...and 25 more