Cut Query Reachability for DAGs with Subquadratic Queries

Reachability in Directed Acyclic Graphs with Near-Linear Cut Queries

This work begins a systematic study of basic problems in directed \emph{acyclic} graphs (DAGs) and shows that reachability from a single vertex and even topological sorting are both computable in O(n \log^3 n) many cut queries.

Streaming Algorithms for Monotonicity Testing

Consider a poset - or equivalently an $n$-vertex DAG $G=(V, E)$ - and a boolean function $f: V \rightarrow \{0, 1\}$ on its vertex set. We say $f$ is monotone if $f(u) \leq f(v)$ for all $(u, v) \in E$. While there is extensive literature on the query complexity of testing monotonicity, we focus instead on the space complexity and initiate the study of this problem in the streaming setting. Namely, the edges of $G$ arrive in an arbitrary order, and the goal is to estimate distance to monotonicity of a given function $f$ using $\widetilde{O}(n)$ space. Note that while this space allows receiving and storing $f$, it is much smaller than the input graph $G$ which could have up to $\Omega(n^2)$ edges. Our main result is an algorithm that $(1+\epsilon)$-approximates distance to monotonicity in $\sqrt{n}^{1+o(1)}$ passes. We also prove that this is the best pass-complexity one can hope for, for any $O(1)$-approximation, short of improving the state-of-the-art streaming algorithm for $st$-reachability, which is a very well-studied problem. On the technical side, our algorithm approximates the size of maximum matching in (a subgraph of) the transitive closure of $G$. While the maximum matching problem has received significant attention in the streaming setting, the fact that we are computing it in the transitive closure requires very different ideas. In fact, a main contribution of our work is to connect sublinear time algorithms for estimating the maximum matching size to the streaming setting for the first time. While existing off-the-shelf sublinear time algorithms only result in an $n\sqrt{n}^{1+o(1)}$ pass algorithm in our setting, we show how to significantly improve upon them by allowing stronger queries (such as vertex and subset queries) that can be implemented just as efficiently as more standard adjacency matrix and list queries for our problem.

A Linear-Time Approximation Scheme for the Densest Subgraph Problem

This paper provides the first truly linear-time approximation scheme for the Densest Subgraph Problem, and uses assignments arising from a flow-based formulation together with a structural carving lemma to progressively carve "sparse" parts of the graph while nearly preserving the densest subgraph.

Parameterizing the Complexity of Finding Long Paths in DAGs

Instance-Optimality of Bidirectional Dijkstra on Simple Graphs

It is shown that bidirectional Dijkstra is still instance-optimal on simple undirected weighted graphs under the order-oblivious model, where incident edges are given in a random order, and under the order-dependent model, where bidirectional Dijkstra is not instance-optimal.

Structural Tractability Frontiers for Metric Repair

This paper asks what structural properties of the graph itself make metric repair tractable, and gives pseudo-polynomial time algorithms for series-parallel graphs, and by generalization, graphs of bounded treewidth and a new algorithm for the length-bounded multicut problem.