Yonggang Jiang

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 $1 \leq f \leq n$.

Yi-Jun Chang, Yanyu Chen, Dipan Dey et al.

We investigate the \emph{minimum weight cycle (MWC)} problem in the $\mathsf{CONGEST}$ model of distributed computing. For undirected weighted graphs, we design a randomized algorithm that achieves a $(k+1)$-approximation, for any \emph{real} number $k \ge 1$. The round complexity of algorithm is \[ \tilde{O}\!\Big( n^{\frac{k+1}{2k+1}} + n^{\frac{1}{k}} + D\, n^{\frac{1}{2(2k+1)}} + D^{\frac{2}{5}} n^{\frac{2}{5}+\frac{1}{2(2k+1)}} \Big). \] where $n$ denotes the number of nodes and $D$ is the unweighted diameter of the graph. This result yields a smooth trade-off between approximation ratio and round complexity. In particular, when $k \geq 2$ and $D = \tilde{O}(n^{1/4})$, the bound simplifies to \[ \tilde{O}\!\left( n^{\frac{k+1}{2k+1}} \right) \] On the lower bound side, assuming the Erdős girth conjecture, we prove that for every \emph{integer} $k \ge 1$, any randomized $(k+1-ε)$-approximation algorithm for MWC requires \[ \tildeΩ\!\left( n^{\frac{k+1}{2k+1}} \right) \] rounds. This lower bound holds for both directed unweighted and undirected weighted graphs, and applies even to graphs with small diameter $D = Θ(\log n)$. Taken together, our upper and lower bounds \emph{match up to polylogarithmic factors} for graphs of sufficiently small diameter $D = \tilde{O}(n^{1/4})$ (when $k \geq 2$), yielding a nearly tight bound on the distributed complexity of the problem. Our results improve upon the previous state of the art: Manoharan and Ramachandran (PODC~2024) demonstrated a $(2+ε)$-approximation algorithm for undirected weighted graphs with round complexity $\tilde{O}(n^{2/3}+D)$, and proved that for any arbitrarily large number $α$, any $α$-approximation algorithm for directed unweighted or undirected weighted graphs requires $Ω(\sqrt{n}/\log n)$ rounds.

Bernhard Haeupler, Yonggang Jiang, Thatchaphol Saranurak

We show that every directed graph $G$ with $n$ vertices and $m$ edges admits a directed acyclic graph (DAG) with $m^{1+o(1)}$ edges, called a DAG projection, that can either $(1+1/\text{polylog} (n))$-approximate distances between all pairs of vertices $(s,t)$ in $G$, or $n^{o(1)}$-approximate maximum flow between all pairs of vertex subsets $(S,T)$ in $G$. Previous similar results suffer a $Ω(\log n)$ approximation factor for distances [Assadi, Hoppenworth, Wein, STOC'25] [Filtser, SODA'26] and, for maximum flow, no prior result of this type is known. Our DAG projections admit $m^{1+o(1)}$-time constructions. Further, they admit almost-optimal parallel constructions, i.e., algorithms with $m^{1+o(1)}$ work and $m^{o(1)}$ depth, assuming the ones for approximate shortest path or maximum flow on DAGs, even when the input $G$ is not a DAG. DAG projections immediately transfer results on DAGs, usually simpler and more efficient, to directed graphs. As examples, we improve the state-of-the-art of $(1+ε)$-approximate distance preservers [Hoppenworth, Xu, Xu, SODA'25] and single-source minimum cut [Cheung, Lau, Leung, SICOMP'13], and obtain simpler construction of $(n^{1/3},ε)$-hop-set [Kogan, Parter, SODA'22] [Bernstein, Wein, SODA'23] and combinatorial max flow algorithms [Bernstein, Blikstad, Saranurak, Tu, FOCS'24] [Bernstein, Blikstad, Li, Saranurak, Tu, FOCS'25]. Finally, via DAG projections, we reduce major open problems on almost-optimal parallel algorithms for exact single-source shortest paths (SSSP) and maximum flow to easier settings: (1) From exact directed SSSP to exact undirected ones, (2) From exact directed SSSP to $(1+1/\text{polylog}(n))$-approximation on DAGs, and (3) From exact directed maximum flow to $n^{o(1)}$-approximation on DAGs.