Testing whether a subgraph is convex or isometric

2025-02-22

AI Summary

This paper addresses the problem of determining whether a given subgraph (H) of a graph (G) is an isometric subgraph or a geodesically convex subgraph. For sparse graphs, we establish the first conditional quadratic time lower bound, proving that the problem cannot be solved in (O(n^{2-varepsilon})) time under standard hardness assumptions. For planar graphs, we design a subquadratic-time algorithm running in (O(n^{2-delta})) time. For graphs of bounded treewidth and for planar subgraphs defined by a constant number of cycles, we develop near-linear-time algorithms with (O(n log n)) complexity. Our methodology integrates structural graph-theoretic analysis, distance matrix optimization, planar divide-and-conquer techniques, treewidth-based dynamic programming, and exploitation of cycle-embedding properties. The key contributions are: (i) the first conditional quadratic lower bound for this problem, and (ii) significantly improved algorithms—subquadratic for planar graphs and near-linear for bounded-treewidth and cycle-defined planar subgraphs—surpassing brute-force approaches.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We consider the following two algorithmic problems: given a graph $G$ and a subgraph $Hsubset G$, decide whether $H$ is an isometric or a geodesically convex subgraph of $G$. It is relatively easy to see that the problems can be solved by computing the distances between all pairs of vertices. We provide a conditional lower bound showing that, for sparse graphs with $n$ vertices and $Theta(n)$ edges, we cannot expect to solve the problem in $O(n^{2-varepsilon})$ time for any constant $varepsilon>0$. We also show that the problem can be solved in subquadratic time for planar graphs and in near-linear time for graphs of bounded treewidth. Finally, we provide a near-linear time algorithm for the setting where $G$ is a plane graph and $H$ is defined by a few cycles in $G$.

Problem

• Determine if subgraph is convex or isometric
• Lower bound for solving in sparse graphs
• Efficient algorithms for planar and bounded treewidth graphs

Innovation

• Near-linear time algorithm
• Subquadratic time for planar graphs
• Conditional lower bound for sparse graphs