Sublinear Edge Fault Tolerant Spanners for Hypergraphs

TL;DR

Problem: construct sublinear, fault-tolerant hyperspanners for hypergraphs under edge faults. Approach: formalize associated-graph reductions, adapt the peel-off baseline, and develop a Parter-inspired clustering method to achieve sublinear EFT hyperspanners with stretch 2k-1; provide a lower bound and an additive hyperspanner pathway. Contributions: sublinear EFT hyperspanner of size $O(k^2 f^{1-1/(rk)} n^{1+1/k} log n)$ in time $ ilde{O}(m r^3 + f n)$, a lower bound $oxed{ ilde{ ext{Omega}}(f^{1-1/r-1/(rk)} n^{1+1/k-o(1)})}$ up to poly(k) factors, and a method to obtain additive EFT hyperspanners by combining multiplicative EFT hyperspanners with additive hyperspanners. Significance: extends fault-tolerant spanner theory to hypergraphs with scalable, parallelizable constructions and establishes foundational bounds and additive extensions.

Abstract

We initiate the study on fault-tolerant spanners in hypergraphs and develop fast algorithms for their constructions. A fault-tolerant (FT) spanner preserves approximate distances under network failures, often used in applications like network design and distributed systems. While classic (fault-free) spanners are believed to be easily extended to hypergraphs such as by the method of associated graphs, we reveal that this is not the case in the fault-tolerant setting: simple methods can only get a linear size in the maximum number of faults $f$. In contrast, all known optimal size of FT spanners are sublinear in $f$. Inspired by the FT clustering technique, we propose a clustering based algorithm that achieves an improved sublinear size bound. For an $n$-node $m$-edge hypergraph with rank $r$ and a sketch parameter $k$, our algorithm constructs edge FT (EFT) hyperspanners of stretch $2k-1$ and size $O(k^2f^{1-1/(rk)}n^{1+1/k}\log n)$ with high probability in time $\widetilde{O}(mr^3+fn)$. We also establish a lower bound of $Ω(f^{1-1/r-1/rk}n^{1+1/k-o(1)})$ edges for EFT hyperspanners, which leaves a gap of poly$(k)f^{1/r}$. Finally, we provide an algorithm for constructing additive EFT hyperspanners by combining multiplicative EFT hyperspanners with additive hyperspanners. We believe that our work will spark interest in developing optimal FT spanners for hypergraphs.

Figures (1)

• Figure 1: Bypasses of class $(v,e)$. $P = SP(s,t, H \setminus F)$, $B_x = SP(x_1, x_2, S)$ and $B_y = SP(y_1, y_2, S)$. Hyperedge $e$ containing $v$ and $u$ belongs to the faulty set.

Theorems & Definitions (30)

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