Lettericity 是 NP-完全的
Lettericity Is NP-Complete
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中文总结 AI 辅助
该论文证明了任意图的 lettericity 问题是 NP-完全的，并进一步证明了着色扩展和单词扩展问题也是 NP-完全的，同时给出了指数时间下界。
中文摘要 AI 辅助
图 $G$ 的 lettericity 是使得存在 $w_1, \ldots, w_{|V(G)|} \in \Sigma$ 和一个解码器 $D \subseteq \Sigma^2$，且 $G$ 同构于字母图 $(\{1, \ldots, |V(G)|\}, \{ij : 1 \le i < j \le |V(G)|, w_iw_j \in D\})$ 的最小集合 $\Sigma$ 的大小。在 lettericity 的研究中，大约经过二十年的时间，才在更简单的路径情形下推导出其 lettericity 的闭式表达式；这表明任意图的 lettericity 是否能在多项式时间内计算的问题是非平凡的。事实上，这个问题在近期的文献中已被反复提出作为开放问题。\n我们通过证明任意图上的 lettericity 问题是 \textsf{NP}-完全的（定理 10）来解决这个问题。我们还证明了着色扩展问题——即与 lettericity 相同的问题，但附加条件为：若 $f$ 是从 $G$ 到字母图的同构映射，则对于给定的 $G$ 的着色 $\chi$，有 $w_{f(v)} = \chi(v)$——是 \textsf{NP}-完全的（定理 12）。我们还解决了对单词扩展问题复杂性的分类这一开放问题，该问题与 lettericity 相同，只是 $w_i$ 是固定的；我们证明它是 \textsf{NP}-完全的（定理 13），这连同我们对着色扩展的 \textsf{NP}-完全性结果，与已知结果形成对比：当着色扩展问题的约束和单词扩展问题的约束同时应用于 lettericity 时，lettericity 可以在多项式时间内判定。\n此外，我们利用 \textsf{NP}-完全性证明中的归约来表明，除非指数时间假说是错误的，否则不存在确定性算法在 $2^{o(n)}$ 时间内判定一个 $n$ 顶点图的 lettericity 是否至多为 $k$，即使当 $n = 6k$ 时也是如此（定理 11）。
英文摘要
The lettericity of a graph $G$ is the smallest size of a set $Σ$ such that there exist $w_1, \ldots, w_{|V(G)|} \in Σ$ and a decoder $D \subseteq Σ^2$ for which $G$ is isomorphic to the letter graph $(\{1, \ldots, |V(G)|\}, \{ij : 1 \le i < j \le |V(G)|, w_iw_j \in D\})$. It took around two decades of the study of lettericity for, in the simpler case of paths, a closed-form expression for its lettericity to be derived; this suggests that the question of whether the lettericity of an arbitrary graph can be computed in polynomial time is nontrivial. Indeed, this question has been raised repeatedly as an open problem in recent literature. We solve this problem by showing that the lettericity problem on arbitrary graphs is NP-complete (Theorem 10). We also prove that the coloring extension problem (the same problem as lettericity, with the added condition that if $f$ is the isomorphism mapping from $G$ to the letter graph, $w_{f(v)} = χ(v)$ for a given coloring $χ$ of $G$) is NP-complete (Theorem 12). We also resolve the open problem of classifying the complexity of the word extension problem, which is the same problem as lettericity except that the $w_i$ are fixed; we show it to be NP-complete (Theorem 13), which, in tandem with our NP-completeness result for coloring extension, contrasts with the known result that when the constraint of the coloring extension problem and the constraint of the word extension problem are both applied to lettericity, lettericity can be decided in polynomial time. Additionally, we use the reduction in the NP-completeness proof to show that unless the Exponential Time Hypothesis is false, there cannot exist a deterministic algorithm to decide whether the lettericity of an $n$-vertex graph is at most $k$ in time $2^{o(n)}$, even when $n = 6k$ (Theorem 11).
发表机构
- Universität Trier（特里尔大学）
- University of California San Diego（加利福尼亚大学圣迭戈分校）
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