The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order $\sigma$, the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering $\sigma$, i.e., the maximum number of colors that the first-fit coloring requires over all possible vertex orderings. Since its inception by Grundy in 1939, Grundy Coloring has been examined for its structural and algorithmic aspects. A brute-force $f(k)n^{2^{k-1}}$-time algorithm for Grundy Coloring on general graphs is not difficult to obtain, where $k$ is the number of colors required by the most adversarial vertex ordering. It was asked several times whether the dependency on $k$ in the exponent of $n$ can be avoided or reduced, and its answer seemed elusive until now. We prove that Grundy Coloring is W[1]-hard and the brute-force algorithm is essentially optimal under the Exponential Time Hypothesis, thus settling this question by the negative. The key ingredient in our W[1]-hardness proof is to use so-called half-graphs as a building block to transmit a color from one vertex to another. Leveraging the half-graphs, we also prove that b-Chromatic Core is W[1]-hard, whose parameterized complexity was posed as an open question by Panolan et al. [JCSS '17]. A natural follow-up question is, how the parameterized complexity changes in the absence of (large) half-graphs. We establish fixed-parameter tractability on $K_{t,t}$-free graphs for b-Chromatic Core and Partial Grundy Coloring, making a step toward answering this question. The key combinatorial lemma underlying the tractability result might be of independent interest.

中文翻译：

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Grundy Coloring & Friends, Half-Graphs, Bicliques

首次拟合着色是一种启发式方法，它分配给每个顶点，以指定的顺序 $\sigma$ 到达，即最小的可用颜色。问题 Grundy Coloring 询问最对抗性顶点排序 $\sigma$ 需要多少颜色，即首次拟合着色在所有可能的顶点排序中需要的最大颜色数。自 1939 年由 Grundy 创立以来，Grundy Coloring 一直在对其结构和算法方面进行检查。在一般图上进行 Grundy 着色的蛮力 $f(k)n^{2^{k-1}}$-time 算法不难获得，其中 $k$ 是最对抗性所需的颜色数量顶点排序。有人多次询问是否可以避免或减少对 $n$ 指数中的 $k$ 的依赖，直到现在它的答案似乎都难以捉摸。我们证明了 Grundy Coloring 是 W[1]-hard 并且蛮力算法在指数时间假设下本质上是最优的，从而以否定的方式解决了这个问题。我们的 W[1] 硬度证明的关键要素是使用所谓的半图作为构建块，将颜色从一个顶点传输到另一个顶点。利用半图，我们还证明了 b-Chromatic Core 是 W[1]-hard，Panolan 等人将其参数化复杂性作为一个开放性问题提出。[JCSS '17]。一个自然的后续问题是，在没有（大）半图的情况下，参数化的复杂性如何变化。我们为 b-Chromatic Core 和 Partial Grundy Coloring 在 $K_{t,t}$-free 图上建立固定参数易处理性，朝着回答这个问题迈出了一步。