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

TL;DR

An algorithm is designed that, given a representation of an n-vertex graph with p transversal pairs and a source v, computes a shortest-path tree rooted at v in time $O(p \log n)$.

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 $O(n^{1/3})$.

Figures (2)

• Figure 1: Classes with sparse signed tree models, or almost signed tree models (for the two on the right column), with the best running time known for undirected unweighted All-Pairs Shortest Path. When relevant, we also indicate the running time when witnessing sequences of low width are given. The results obtained in this paper are highlighted in blue.
• Figure 2: A signed tree model of a 14-vertex graph, with $A(T)$ in amber and $B(T)$ in blue. The topmost amber edge and the dashed red edge cross (so the latter could not be a transversal pair). Vertex 8 is adjacent to 4 because 4,8 is covered by the blue edge $e_1$, and to 2 because 2,8 is covered by the blue edge $e_3$, but 8 is not adjacent to 7 because 7,8 is covered by the amber edge $e_2$.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Corollary 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Lemma 10: Lemma 7.4 in Chan25 or Lemma 13 in Duraj24