顶点着色边加权：核化与推广
Vertex-Coloring Edge-Weighting: Kernelization and Generalization
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中文总结 AI 辅助
本文研究图边加权诱导顶点着色问题，证明其参数化版本有多项式核，预加权版本在顶点覆盖数下为FPT，在树深度下为W[1]-难，并改进运行时间至2^{O(k log k)}·n。
中文摘要 AI 辅助
图的边加权会诱导其顶点的着色，其中顶点的颜色是与该顶点关联的边的总权重。如果相邻顶点总是获得不同的颜色，则这样的边加权是恰当的。对于权重集{0,1}以及{1,2}，判定一个图是否允许恰当加权已知是NP完全的。在最近的工作（arXiv:2604.12363）中，我们证明了这两个问题在参数化为顶点覆盖数k时是FPT的，但据我们所知，这两个参数化问题是否具有多项式核是开放的。在本工作中，我们证明了当参数化为k时，这两个问题都具有多项式核。我们还证明了这两个问题在参数化为树深度时是W[1]-难的，回答了我们先前工作中的另一个问题。然后我们研究了这两个问题的预加权版本，其中某些边的权重是预先固定的，任务是扩展该赋值以得到整个图的恰当加权。我们证明了两个预加权问题在参数化为顶点覆盖数k时都是FPT的。对于{1,2}版本，运行时间为2^{O(k log k)}·n；对于{0,1}版本，当每个预权重为1时，我们获得相同的运行时间，而在一般情况下则得到一个较慢的FPT算法。我们还证明了两个预加权问题在参数化为（i）反馈顶点集数或（ii）输入图的树深度时都是W[1]-难的。由于没有预分配权重的图是特殊情况，我们针对预加权版本的算法也解决了两个原始问题，时间为2^{O(k log k)}·n，显著改善了我们先前工作中2^{O(k^4)}·n^{O(1)}的界限。
英文摘要
An edge weighting of a graph induces a coloring of its vertices in which the color of a vertex is the total weight of the edges incident with it. Such an edge weighting is proper if adjacent vertices always receive distinct colors. Deciding whether a graph admits a proper weighting is known to be NP-complete for the weight set $\{0,1\}$, and also for $\{1,2\}$. In recent work (arXiv:2604.12363) we showed that both problems are FPT parameterized by the vertex cover number $k$, but it was open -- to the best of our knowledge -- whether either parameterized problem had a polynomial kernel. In this work, we show that both problems have polynomial kernels when parameterized by $k$. We also show that both problems are W[1]-hard parameterized by treedepth, answering another question from our earlier work. We then study the pre-weighted versions of the two problems, in which the weights of some edges are fixed in advance, and the task is to extend the assignment to a proper weighting of the whole graph. We show that both pre-weighted problems are FPT parameterized by the vertex cover number $k$. For the $\{1,2\}$ version the running time is $2^{O(k \log k)} \cdot n$; for the $\{0,1\}$ version we obtain the same running time when every pre-weight is $1$, and a slower FPT algorithm in the general case. We also show that both pre-weighted problems are W[1]-hard parameterized by either of (i) the feedback vertex set number or (ii) the treedepth of the input graph. Since a graph with no pre-assigned weights is a special case, our algorithms for the pre-weighted versions solve the two original problems as well, in time $2^{O(k \log k)} \cdot n$, significantly improving on the bound of $2^{O(k^4)} \cdot n^{O(1)}$ from our earlier work.
发表机构
- IIT Hyderabad（印度理工学院海德拉巴分校）
- University of Leeds（利兹大学）
- Chennai Mathematical Institute（金奈数学研究所）
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