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- W4386793438 abstract "We study a new and stronger notion of fault-tolerant graph structures whose size bounds depend on the degree of the failing edge set, rather than the total number of faults. For a subset of faulty edges $F subseteq G$, the faulty-degree $deg(F)$ is the largest number of faults in $F$ incident to any given vertex. We design new fault-tolerant structures with size comparable to previous constructions, but which tolerate every fault set of small faulty-degree $deg(F)$, rather than only fault sets of small size $|F|$. Our main results are: - New FT-Certificates: For every $n$-vertex graph $G$ and degree threshold $f$, one can compute a connectivity certificate $H subseteq G$ with $|E(H)| = widetilde{O}(fn)$ edges that has the following guarantee: for any edge set $F$ with faulty-degree $deg(F)leq f$ and every vertex pair $u,v$, it holds that $u$ and $v$ are connected in $H setminus F$ iff they are connected in $G setminus F$. This bound on $|E(H)|$ is nearly tight. Since our certificates handle some fault sets of size up to $|F|=O(fn)$, prior work did not imply any nontrivial upper bound for this problem, even when $f=1$. - New FT-Spanners: We show that every $n$-vertex graph $G$ admits a $(2k-1)$-spanner $H$ with $|E(H)| = O_k(f^{1-1/k} n^{1+1/k})$ edges, which tolerates any fault set $F$ of faulty-degree at most $f$. This bound on $|E(H)|$ optimal up to its hidden dependence on $k$, and it is close to the bound of $O_k(|F|^{1/2} n^{1+1/k} + |F|n)$ that is known for the case where the total number of faults is $|F|$ [Bodwin, Dinitz, Robelle SODA '22]. Our proof of this theorem is non-constructive, but by following a proof strategy of Dinitz and Robelle [PODC '20], we show that the runtime can be made polynomial by paying an additional $text{polylog } n$ factor in spanner size." @default.
- W4386793438 created "2023-09-16" @default.
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- W4386793438 date "2023-09-12" @default.
- W4386793438 modified "2023-09-30" @default.
- W4386793438 title "Fault-Tolerant Spanners against Bounded-Degree Edge Failures: Linearly More Faults, Almost For Free" @default.
- W4386793438 doi "https://doi.org/10.48550/arxiv.2309.06696" @default.
- W4386793438 hasPublicationYear "2023" @default.
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