Approximation method improves flexible graph connectivity solutions
A $(p+q)^{O(pq)}$-approximation for $(p, q)$-Flexible Graph Connectivity
Data Structures and Algorithms
Summary
The paper studies a graph problem where connections must be reliable even if some edges are unsafe. The authors design an algorithm that finds nearly optimal ways to connect the graph while meeting safety and connectivity rules. Their solution works especially well when two key parameters, p and q, are fixed numbers. They rely on a structural insight discovered with the help of an AI tool, ChatGPT Astra, to handle complex cuts in the graph efficiently.
What this means in practice
- •For network engineers: Design more cost-efficient resilient communication networks that maintain connectivity despite uncertain or risky connections.
- •For infrastructure planners: Improve planning of transportation or utility networks ensuring redundancy with controlled risk and cost trade-offs.
A theory result. No direct application yet.
Authors
Karthekeyan Chandrasekaran, Raymond Jiang, Krishna Kalathur
Abstract
In the $(p,q)$-Flexible Graph Connectivity problem, the input consists of non-negative integers $p$ and $q$ and a graph $G=(V, E)$ whose edges are classified into safe and unsafe edges with non-negative edge costs. A subgraph H of G is $(p,q)$-Flex-Connected if every non-empty proper subset of vertices has either at least $p$ safe edges or at least $p+q$ total edges crossing it. The goal is to find a minimum cost subset $F\subseteq E$ of edges such that the subgraph $(V, F)$ is $(p,q)$-Flex-Connected. We give a $(p+q)^{O(pq)}$-approximation for this problem, which in particular implies a constant approximation for every fixed constants $p$ and $q$. We achieve this by designing a $(p+q)^{O(pq)}$-approximation for the augmentation problem of finding a minimum cost subset of edges to add to make a (p,q-1)-Flex-Connected graph into a (p,q)-Flex-Connected graph. Underlying the augmentation algorithm is a structural result showing that all deficient cuts can be represented by min rooted-cuts in a $(p+q)^{pq}$-sized collection of digraphs. This structural result was discovered by ChatGPT Astra.