Identifying Vertices in Optimal Feedback Vertex Sets through Antler Decompositions
Preprocessing algorithms can identify vertices that belong to an optimal feedback vertex set by finding antler decompositions in the input graph.
Dijkstra's Algorithm is Universally Optimal for Shortest Path Ordering
Dijkstra's algorithm, when combined with a sufficiently efficient heap data structure, is universally optimal for the problem of ordering nodes by their distance from the source.
Efficient Dynamic Algorithms for Maintaining Approximate Maximum Matching in Graphs
We present a new algorithm that maintains a (1-ε)-approximate maximum matching in a fully dynamic graph, where the update-time depends on the density of a certain class of graphs called Ordered Ruzsa-Szemerédi (ORS) graphs. We also provide improved upper bounds on the density of both ORS and Ruzsa-Szemerédi (RS) graphs with linear size matchings.
Efficient Algorithms for Predicting Terminal Configurations of Sandpile Instances on Graphs
This paper presents efficient algorithms for computing the terminal configurations of sandpile instances on various graph structures, including trees, paths, cliques, and general graphs. The key ideas are to directly compute the number of firings at each vertex instead of simulating individual events, and to leverage data structures like splittable binary search trees to accelerate the computation.
Efficient Parallelizable Algorithm for Approximating Sparsest Cut
This paper presents a new algorithm that computes an O(√(log n)/ε)-approximation for the Sparsest Cut problem using O((nε logO(1) n) · Tmaxflow) expected runtime, where Tmaxflow is the runtime of a maxflow algorithm. The algorithm is parallelizable and can be implemented on O(nε) processors in expected parallel runtime O((logO(1) n) · Tmaxflow).
Fine-Grained Complexity of Subgraph Isomorphism: Characterizing Patterns with Subquadratic Algorithms
For any pattern graph H, we determine whether it admits subquadratic algorithms for minimum-weight subgraph, subgraph listing, and subgraph enumeration, and if so, we provide the optimal time complexity.
Computational Complexity of Finding Stable Sets in Graphs with Matroid Constraints
The Independent Stable Set problem seeks to find a stable set of vertices in a graph that is also independent with respect to a given matroid. This problem generalizes several well-studied algorithmic problems, including Rainbow Independent Set, Rainbow Matching, and Bipartite Matching with Separation.
Efficient Algorithms for Minor Containment and Disjoint Paths in Almost-Linear Time
The authors present an algorithm that can solve the Rooted Minor Containment problem in almost-linear fixed-parameter time, which implies the existence of an n^(1+o(1))-time algorithm for deciding membership in every minor-closed class of graphs. They also obtain an Ok(m^(1+o(1)))-time algorithm for the Disjoint Paths problem.
Additive Approximation Algorithm for k-Geodesic Center in δ-Hyperbolic Graphs
The authors provide an additive O(δ)-approximation algorithm for the k-Geodesic Center problem on δ-hyperbolic graphs, where the goal is to find a set of k isometric paths that minimizes the maximum distance between any vertex and the set of paths.
A Simple and Efficient 2-Approximation Algorithm for the Maximum-Leaf Spanning Tree Problem
We present a simple and efficient O(m)-time 2-approximation algorithm for the maximum-leaf spanning tree problem, which is NP-complete even for planar graphs with maximum degree 4.