重探路径收缩与环收缩问题
Revisiting Path Contraction and Cycle Contraction
AI总结：
该研究重探NP完全的路径收缩与环收缩问题，提出改进的参数化算法及带约束端点的通用变体算法，还给出平面图的多项式算法和弦图的下界结果。
摘要：
路径收缩（Path Contraction）和环收缩（Cycle Contraction）问题以含$n$个顶点、$m$条边的无向图$G$以及整数$k$为输入，分别判定能否通过在$G$中执行至多$k$次边收缩操作得到一条路径或一个环。我们重新研究了这些NP完全问题并证明了以下结果：路径收缩问题存在运行时间为$\boldsymbol{\textup{O}}^*(2^k)$的算法，该结果优于该问题目前已知的最优算法[Algorithmica 2014]；环收缩问题存在运行时间为$\boldsymbol{\textup{O}}^*((2 + ε_{\text{ℓ}})^k)$的算法，其中$0 < ε_{\text{ℓ}} \text{≤} 0.5509$，与$ℓ = n - k$成反比。\n这些结果的核心是针对路径收缩的一个通用变体——即带约束端点的路径收缩（Path Contraction With Constrained Ends）——的算法。我们还提出了一种$\boldsymbol{\textup{O}}^*(2.5191^n)$时间的算法，用于求解环收缩的优化版本。\n随后，我们将研究转向受限图类并证明了以下结果：平面图上的路径收缩问题存在多项式时间算法；弦图上的路径收缩问题不存在运行时间为$\boldsymbol{\textup{O}}(n^{2-ε} \text{·} 2^{o(\text{tw})})$的算法（对任意$ε>0$），除非正交向量猜想（Orthogonal Vectors Conjecture）不成立，其中$\text{tw}$为输入图的树宽。第二个结果补充了该问题已知的$\boldsymbol{\textup{O}}(nm)$时间（即$\boldsymbol{\textup{O}}(n^2 \text{·} \text{tw})$时间）算法[Discret. Appl. Math. 2014]。
英文摘要：
The Path Contraction and Cycle Contraction problems take as input an undirected graph $G$ with $n$ vertices, $m$ edges and an integer $k$ and determine whether one can obtain a path or a cycle, respectively, by performing at most $k$ edge contractions in $G$. We revisit these NP-complete problems and prove the following results. Path Contraction admits an algorithm running in $\mathcal{O}^*(2^{k})$ time. This improves over the current algorithm known for the problem [Algorithmica 2014]. Cycle Contraction admits an algorithm running in $\mathcal{O}^*((2 + ε_{\ell})^k)$ time where $0 < ε_{\ell} \leq 0.5509$ is inversely proportional to $\ell = n - k$. Central to these results is an algorithm for a general variant of Path Contraction, namely, Path Contraction With Constrained Ends. We also give an $\mathcal{O}^*(2.5191^n)$-time algorithm to solve the optimization version of Cycle Contraction. Next, we turn our attention to restricted graph classes and show the following results. Path Contraction on planar graphs admits a polynomial-time algorithm. Path Contraction on chordal graphs does not admit an algorithm running in time $\mathcal{O}(n^{2-ε} \cdot 2^{o(tw)})$ for any $ε> 0$, unless the Orthogonal Vectors Conjecture fails. Here, $tw$ is the treewidth of the input graph. The second result complements the $\mathcal{O}(nm)$-time, i.e., $\mathcal{O}(n^2 \cdot tw)$-time, algorithm known for the problem [Discret. Appl. Math. 2014].
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