Fast Shortest Path in Graphs With Sparse Signed Tree Models and Applications

2026-02-18

AI Summary

This work addresses the efficient computation of single-source and all-pairs shortest paths on graphs represented by sparse signed tree models. By introducing signed tree models into shortest path computation for the first time, the authors propose a single-source shortest path algorithm with time complexity $O(p \log n)$, where $p$ denotes the size of the model. Building on this, they derive an $O(n^2 \log n)$ all-pairs shortest path algorithm for graphs of bounded merge-width and an $O(n^2 \log^2 n)$ algorithm for graphs of bounded twin-width. These results substantially accelerate fundamental tasks such as model checking and Boolean matrix multiplication, thereby extending the class of graph instances amenable to efficient algorithmic treatment.

Technology Category

Constraint Satisfaction and Optimization: Satisfiability Modulo TheoriesSearch and Optimization: Combinatorial OptimizationKnowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsSocial Networks and Social Media: Social mining and social search on the Web

Abstract

A signed tree model of a graph $G$ is a compact binary structure consisting of a rooted binary tree whose leaves are bijectively mapped to the vertices of $G$, together with 2-colored edges $xy$, called transversal pairs, interpreted as bicliques or anti-bicliques whose sides are the leaves of the subtrees rooted at $x$ and at $y$. We design an algorithm that, given such a representation of an $n$-vertex graph $G$ with $p$ transversal pairs and a source $v \in V(G)$, computes a shortest-path tree rooted at $v$ in $G$ in time $O(p \log n)$. A wide variety of graph classes are such that for all $n$, their $n$-vertex graphs admit signed tree models with $O(n)$ transversal pairs: for instance, those of bounded symmetric difference, more generally of bounded sd-degeneracy, as well as interval graphs. As applications of our Single-Source Shortest Path algorithm and new techniques, we - improve the runtime of the fixed-parameter algorithm for first-order model checking on graphs given with a witness of low merge-width from cubic [Dreier and Toruńczyk, STOC '25] to quadratic; - give an $O(n^2 \log n)$-time algorithm for All-Pairs Shortest Path (APSP) on graphs given with a witness of low merge-width, generalizing a result known on twin-width [Twin-Width III, SICOMP '24]; - extend and simplify an $O(n^2 \log n)$-time algorithm for multiplying two $n \times n$ matrices $A, B$ of bounded twin-width in [Twin-Width V, STACS '23]: now $A$ solely has to be an adjacency matrix of a graph of bounded twin-width and $B$ can be arbitrary; - give an $O(n^2 \log^2 n)$-time algorithm for APSP on graphs of bounded twin-width, bypassing the need for contraction sequences in [Twin-Width III, SICOMP '24; Bannach et al. STACS '24]; - give an $O(n^{7/3} \log^2 n)$-time algorithm for APSP on graphs of symmetric difference $O(n^{1/3})$.

Problem

• shortest path
• signed tree model
• sparse graph
• merge-width
• twin-width

Innovation

• signed tree model
• shortest path
• merge-width
• twin-width
• transversal pairs