Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

2026-09-10

AI Summary

研究解决了通过添加有限数量的链接使有向无环图强连通的问题，提出了单指数时间和多项式核的方法，显著改进了算法效率。

Abstract

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

Problem

• Strong Connectivity Augmentation
• directed acyclic graph
• strongly connected

Innovation

• single-exponential algorithm
• polynomial kernel
• strong connectivity augmentation