Asaf Petruschka

Yonggang Jiang, Merav Parter, Asaf Petruschka

We study the succinct representations of vertex cuts by centralized oracles and labeling schemes. For an undirected $n$-vertex graph $G = (V,E)$ and integer parameter $f \geq 1$, the goal is supporting vertex cut queries: Given $F \subseteq V$ with $|F| \leq f$, determine if $F$ is a vertex cut in $G$. In the centralized data structure setting, it is required to preprocess $G$ into an $f$-vertex cut oracle that can answer such queries quickly, while occupying only small space. In the labeling setting, one should assign a short label to each vertex in $G$, so that a cut query $F$ can be answered by merely inspecting the labels assigned to the vertices in $F$. While the ``$st$ cut variants'' of the above problems have been extensively studied and are known to admit very efficient solutions, the basic (global) ``cut query'' setting is essentially open (particularly for $f > 3$). This work achieves the first significant progress on these problems: [$f$-Vertex Cut Labels:] Every $n$-vertex graph admits an $f$-vertex cut labeling scheme, where the labels have length of $\tilde{O}(n^{1-1/f})$ bits (when $f$ is polylogarithmic in $n$). This nearly matches the recent lower bound given by Long, Pettie and Saranurak (SODA 2025). [$f$-Vertex Cut Oracles:] For $f=O(\log n)$, every $n$-vertex graph $G$ admits $f$-vertex cut oracle with $\tilde{O}(n)$ space and $\tilde{O}(2^f)$ query time. We also show that our $f$-vertex cut oracles for every $1 \leq f \leq n$ are optimal up to $n^{o(1)}$ factors (conditioned on plausible fine-grained complexity conjectures). If $G$ is $f$-connected, i.e., when one is interested in \emph{minimum} vertex cut queries, the query time improves to $\tilde{O}(f^2)$, for any $1 \leq f \leq n$.

Merav Parter, Asaf Petruschka

We revisit the problem of fault-tolerant (FT) distance preservers, when failure events in the network admit a form of correlation modeled as color faults. FT distance preservers are sparse subgraphs that preserve distances between specified pairs of vertices, even after some edge or vertex failures occur. In the classical fault model, any set of at most $k$ edges or vertices might fail (where $k \geq 1$ is a given parameter). Despite extensive research, the classical model admits significant and tantalizing gaps, both in terms of sparsity bounds and of algorithmic efficiency. In this work, we study the problem in the recently introduced color fault-tolerant (CFT) model: the given graph $G=(V,E)$ has arbitrary colors on its edges/vertices where each color appears at most $k$ times, and is susceptible to color faults, where the failure of color $c$ causes all the $c$-colored elements to crash. Our main contribution is in the multi-source setting, where $G$ has a source-set $S \subseteq V$, and the CFT preserver should preserve $S \times V$ distances under any single color fault. We show the following results (where $n = |V|$, $m = |E|$): - There exists a CFT distance preserver $H$ of $G$ with $\tilde{O}(n^{2 - \frac{1}{k+1}} \cdot |S|^{\frac{1}{k+1}} )$ edges. - The above sparsity bound is worst-case optimal up to polylogarithmic terms. - There is a combinatorial randomized algorithm that produces a preserver $H$ whose size meets the above optimal sparsity bound, with running time of $\tilde{O}(m \cdot n^{1 - \frac{1}{k+1}} \cdot |S|^{\frac{1}{k+1}})$. - The above running time is conditionally optimal: a polynomial improvement would refute the combinatorial Boolean Matrix Multiplication (BMM) conjecture. Furthermore, the running time remains optimal even if we only require mild sparsification to $m^{1-ε}$ edges.