固定高度网格上的三色 Free-Flood-It 是多项式时间可解的
Three-Color Free-Flood-It on Fixed-Height Grids Is Polynomial-Time Solvable
- Key Laboratory of System Software (Chinese Academy of Sciences)（系统软件重点实验室（中国科学院））
- State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所计算机科学与技术国家重点实验室）
- University of Chinese Academy of Sciences（中国科学院大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文提出确定性算法，在固定高度网格上多项式时间内求解三色 Free-Flood-It，并给出 EPTAS 近似方案，解决了此前悬而未决的三色情形。
中文摘要 AI 辅助
我们针对至多三色的矩形网格 $P_k\square P_n$ 上的 \textsc{Free-Flood-It} 给出一个确定性算法。对于每个固定高度 $k$，该算法在 $N^{O(k^2)}$ 时间内计算出最小移动次数和最优序列，其中 $N=kn$。这解决了之前悬而未决的完整 $3\times n$ 棋盘上的三色情形。证明使用了一种将洪水策略表示为连通区域着色的方法。我们证明，可以选取一个最优着色，使得其区域形成一棵有根树，并且对祖先路径上的颜色有严格限制。这些限制使得树在任何固定大小的连通窗口中可见的部分仅允许多项式数量的描述。其余的共同祖先可能形成任意长的链。我们将该链保留在栈上，并使用有限的上下文无关递推来最小化总着色成本，而无需枚举所有可能的栈内容。边界处的连通性信息确保所得的局部描述组装成一个有效的全局着色。该论证也适用于具有有界大小的连通层排序的图，前提是每条边位于一层内或连接相邻层。一族三行棋盘表明，算法处理的无界祖先链即使对于最优着色也是必要的。对于任何固定的高度和调色板大小，我们还给出了一个确定性 EPTAS 和一个具有显式失败概率界限的随机采样变体。这些近似结果使用了一个单独的算法，其对移动预算具有单指数依赖性。
英文摘要
We give a deterministic algorithm for \textsc{Free-Flood-It} on rectangular grids $P_k\square P_n$ with at most three colors. For every fixed height $k$, it computes the minimum number of moves and an optimal sequence in $N^{O(k^2)}$ time, where $N=kn$. This resolves the previously open three-color case on complete $3\times n$ boards. The proof uses a representation of flooding strategies as paintings by connected regions. We show that an optimal painting can be chosen so that its regions form a rooted tree with strong restrictions on the colors along ancestral paths. These restrictions make the part of the tree visible in any fixed-size connected window admit only polynomially many descriptions. The remaining common ancestors may form an arbitrarily long chain. We retain that chain on a stack and use a finite context-free recurrence to minimize the total painting cost without enumerating all possible stack contents. Connectivity information at the boundary ensures that the resulting local descriptions assemble into one valid global painting. The argument also applies to graphs supplied with an ordering into connected layers of bounded size, provided every edge lies within one layer or joins consecutive layers. A family of three-row boards shows that the unbounded ancestor chains handled by the algorithm are necessary even for optimal paintings. For any fixed height and palette size, we also give a deterministic EPTAS and a randomized sampling variant with an explicit failure-probability bound. These approximation results use a separate algorithm with single-exponential dependence on the move budget.
