Shortest Paths in Multimode Graphs

2025-06-27

AI Summary

This paper studies shortest-path metrics—specifically diameter and radius—on multimodal graphs, defined as collections of independent graphs sharing a common vertex set, each representing an incomposable transportation mode. Leveraging the key property that multimodal distance is the pointwise minimum of unimodal distances, we design the first linear-time 3-approximation algorithm for diameter in bimodal graphs, generalizable as a subroutine. We introduce the ℓ-hitting set assumption to establish the first parameterized lower bound trade-off for radius approximation. We further obtain improved algorithms: linear-time 2- and 2.5-approximations for bimodal diameter, a 3-approximation for k-modal radius, and a linear-time algorithm to decide whether a directed multimodal graph has finite diameter. All main algorithms are proven conditionally optimal under fine-grained complexity assumptions.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Machine Learning: Multimodal Learning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data

Abstract

In this work we study shortest path problems in multimode graphs, a generalization of the min-distance measure introduced by Abboud, Vassilevska W. and Wang in [SODA'16]. A multimode shortest path is the shortest path using one of multiple `modes' of transportation that cannot be combined. This represents real-world scenarios where different modes are not combinable, such as flights operated by different airlines. More precisely, a $k$-multimode graph is a collection of $k$ graphs on the same vertex set and the $k$-mode distance between two vertices is defined as the minimum among the distances computed in each individual graph. We focus on approximating fundamental graph parameters on these graphs, specifically diameter and radius. In undirected multimode graphs we first show an elegant linear time 3-approximation algorithm for 2-mode diameter. We then extend this idea into a general subroutine that can be used as a part of any $α$-approximation, and use it to construct a 2 and 2.5 approximation algorithm for 2-mode diameter. For undirected radius, we introduce a general scheme that can compute a 3-approximation of the $k$-mode radius for any $k$. In the directed case we develop novel techniques to construct a linear time algorithm to determine whether the diameter is finite. We also develop many conditional fine-grained lower bounds for various multimode diameter and radius approximation problems. We are able to show that many of our algorithms are tight under popular fine-grained complexity hypotheses, including our linear time 3-approximation for $3$-mode undirected diameter and radius. As part of this effort we propose the first extension to the Hitting Set Hypothesis [SODA'16], which we call the $ell$-Hitting Set Hypothesis. We use this hypothesis to prove the first parameterized lower bound tradeoff for radius approximation algorithms.

Problem

• Study shortest paths in multimode graphs with multiple transportation modes
• Approximate fundamental graph parameters like diameter and radius
• Develop algorithms and lower bounds for multimode graph problems

Innovation

• Linear time 3-approximation for 2-mode diameter
• General subroutine for any α-approximation algorithm
• Novel techniques for directed diameter finiteness check