单指数算法与强连通增强问题的多项式核
Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation
- The University of Tokyo（东京大学）
- CyberAgent, Inc.（CyberAgent公司）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文为强连通增强问题设计了单指数参数化算法和多项式核，分别达到$O^*(9^k)$时间与$O(k^4)$顶点核，无权情形下改进为$O^*(4^k)$和$O(k^3)$顶点。
AI中文摘要：
强连通增强（SCA）问题询问是否可以通过添加至多$k$条总权重在给定预算内的指定边，使一个有向无环图变为强连通。Klinkby、Misra和Saurabh（SODA 2021）给出了一个$O^*(2^{O(k\log k)})$时间的算法，并询问该问题是否允许单指数参数化算法和多项式核。我们对这两个问题都给出肯定回答：SCA可以在$O^*(9^k)$时间内求解，并承认一个具有$O(k^4)$个顶点和$O(k^{16})$比特的多项式核。对于无权SCA，我们获得$O^*(4^k)$时间和具有$O(k^3)$个顶点的核。我们的算法基于一个特别简单的归约，归约到具有两种边代价的强连通生成子图问题。
英文摘要：
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.
