Thatchaphol Saranurak

Nikhil Bansal, Arun Jambulapati, Thatchaphol Saranurak

We present the first polynomial-time algorithm for computing a near-optimal \emph{flow}-expander decomposition. Given a graph $G$ and a parameter $ϕ$, our algorithm removes at most a $ϕ\log^{1+o(1)}n$ fraction of edges so that every remaining connected component is a $ϕ$-\emph{flow}-expander (a stronger guarantee than being a $ϕ$-\emph{cut}-expander). This achieves overhead $\log^{1+o(1)}n$, nearly matching the $Ω(\log n)$ graph-theoretic lower bound that already holds for cut-expander decompositions, up to a $\log^{o(1)}n$ factor. Prior polynomial-time algorithms required removing $O(ϕ\log^{1.5}n)$ and $O(ϕ\log^{2}n)$ fractions of edges to guarantee $ϕ$-cut-expander and $ϕ$-flow-expander components, respectively.

Bernhard Haeupler, Yaowei Long, Antti Roeyskoe et al.

A fault-tolerant distance labeling scheme assigns a label to each vertex and edge of an undirected weighted graph $G$ with $n$ vertices so that, for any edge set $F$ of size $|F| \leq f$, one can approximate the distance between $p$ and $q$ in $G \setminus F$ by reading only the labels of $F \cup \{p,q\}$. For any $k$, we present a deterministic polynomial-time scheme with $O(k^{4})$ approximation and $\tilde{O}(f^{4}n^{1/k})$ label size. This is the first scheme to achieve a constant approximation while handling any number of edge faults $f$, resolving the open problem posed by Dory and Parter [DP21]. All previous schemes provided only a linear-in-$f$ approximation [DP21, LPS25]. Our labeling scheme directly improves the state of the art in the simpler setting of distance sensitivity oracles. Even for just $f = Θ(\log n)$ faults, all previous oracles either have super-linear query time, linear-in-$f$ approximation [CLPR12], or exponentially worse $2^{{\rm poly}(k)}$ approximation dependency in $k$ [HLS24].

Xizhe Li, Yaowei Long, David Pidugu et al.

We give an improved connectivity oracle under vertex failures. After a set of $k$ vertices fails, our oracle performs an $O(k^{6})$-time update independent of the graph size $n$, and then answers pairwise connectivity queries in optimal $O(k)$ time. For constant $k$, it uses near-linear space and can be built in near-linear preprocessing time. In contrast, all prior oracles with $n$-independent update time[PSS+22, vdBS19] either require $Ω(n^{2})$ space or incur $2^{2^{O(k)}}$ update and query time. Moreover, their preprocessing time is polynomially large in $n$, far from near-linear. Our oracle builds on the unbreakable decomposition framework of[PSS+22], but introduces three new ingredients: (i) shortcutting over the tree decomposition to reduce space from quadratic to near-linear, (ii) bootstrapping that leverages $n$-dependent oracles internally to obtain near-linear preprocessing, and (iii) a new patch set mechanism that yields conditionally optimal $O(k)$ query time.

Vikrant Ashvinkumar, Aaron Bernstein, Maximilian Probst Gutenberg et al.

We present parallel algorithms for computing single-source reachability and shortest paths on directed $n$-vertex $m$-edge graphs using near-linear $\tilde{O}(m)$ work and $o(\sqrt{n})$ depth whenever $m\ge n^{1+o(1)}$. At the extreme of $m=Ω(n^{2})$, our reachability and shortest path algorithms have depth only $n^{0.136}$ and $n^{0.25+o(1)}$, respectively. The state-of-the-art parallel algorithms with near-linear work for both problems require $Ω(\sqrt{n})$ depth in all density regimes.

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.