计算第二个Hamilton圈的精确算法和近似算法
Exact and Approximate Algorithms for Computing a Second Hamiltonian Cycle
摘要：
本文考虑如下全泛函问题：给定一个三次Hamilton图$G$和$G$的一个Hamilton圈$C_0$，如何计算$G$的第二个Hamilton圈$C_1 \neq C_0$？塞德里克·史密斯在1946年用非构造性的宇称论证证明了这样的第二个哈密尔顿循环总是存在的。我们的主要结果是一个算法，计算时间$O(n \cdot 2^{(0.3-\varepsilon)n})$时间的第二个哈密顿周期，为一些积极的常数$\varepsilon>0$，并在多项式空间，从而提高了国家的最先进的运行时间解决这个问题。我们的算法是基于一个基本的结构性质的Mesquason的棒棒糖算法，我们在这里证明的第一次。在近似的第二个周期的长度在一个给定的哈密尔顿圈$C_0$的哈密尔顿图$G$（其中我们可能不保证存在第二个哈密尔顿圈）的方向上，我们提供了一个线性时间算法计算第二个周期的长度至少为$n - 4α(\sqrt{n}+2α)+8$，其中$α= \frac{Δ-2}{δ-2}$和$δ,Δ$是最小和最大程度的图，分别。该近似结果也改进了现有技术。
英文摘要：
In this paper we consider the following total functional problem: Given a cubic Hamiltonian graph $G$ and a Hamiltonian cycle $C_0$ of $G$, how can we compute a second Hamiltonian cycle $C_1 \neq C_0$ of $G$? Cedric Smith proved in 1946, using a non-constructive parity argument, that such a second Hamiltonian cycle always exists. Our main result is an algorithm which computes the second Hamiltonian cycle in time $O(n \cdot 2^{(0.3-\varepsilon)n})$ time, for some positive constant $\varepsilon>0$, and in polynomial space, thus improving the state of the art running time for solving this problem. Our algorithm is based on a fundamental structural property of Thomason's lollipop algorithm, which we prove here for the first time. In the direction of approximating the length of a second cycle in a Hamiltonian graph $G$ with a given Hamiltonian cycle $C_0$ (where we may not have guarantees on the existence of a second Hamiltonian cycle), we provide a linear-time algorithm computing a second cycle with length at least $n - 4α(\sqrt{n}+2α)+8$, where $α= \frac{Δ-2}{δ-2}$ and $δ,Δ$ are the minimum and the maximum degree of the graph, respectively. This approximation result also improves the state of the art.
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