Fault-Tolerant Spanners against Bounded-Degree Edge Failures: Linearly More Faults, Almost For Free
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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 ⊆ G, the faulty-degree (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 (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 ⊆ G with |E(H)| = O(fn) edges that has the following guarantee: for any edge set F with faulty-degree (F)≤ f and every vertex pair u,v, it holds that u and v are connected in H ∖ F iff they are connected in G ∖ 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 polylog n factor in spanner size.
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