发表机构
Max Planck Institute for Informatics; Saarland Informatics Campus; TU Dresden; University of Liverpool（马克斯·普朗克 informatics 研究所; 萨尔兰 informatics 校园; 德累斯顿工业大学; 利物浦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对图问题，提出带同构先验的计算模型，利用先前同构图的预计算结果加速新查询，证明部分NP难问题可多项式时间处理，并突破若干细粒度下界。
AI 中文摘要
如果我们对先前的数据执行了高强度的计算，能否避免对相似的未来输入进行重复计算？受此问题启发，我们为图问题引入了一种新的计算模型，称为带同构先验的算法。在此模型中解决图问题$\Pi$涉及两个阶段：（i）预处理阶段快速分析先前的图$G_1,..., G_k$以及（先前计算出的）精确最优值OPT$(G_i)$。（ii）随后，给定一个新图H，一个快速的查询阶段必须要么（a）输出精确解OPT(H)，要么（b）正确报告H与任何$G_i$不同构。当H与某个$G_i$同构时，我们能否避免从头计算OPT(H)？我们证明对于许多问题情况确实如此；对于其他许多问题，我们建立了条件下界。$\textbf{(1)}$一些NP-困难问题，包括约束最短路径、$\ell_p$-最短路径和约束生成树，在我们的模型中允许多项式时间的预处理和查询时间。相比之下，Karp的21个NP-完全问题中的几乎所有问题以及对于每个固定$\varepsilon>0$的$(2-\varepsilon)$-近似$k$-中心问题，除非图同构（GI）属于P，否则即使只有O(1)个先验，也不存在这样的算法。$\textbf{(2)}$与条件性的$n^{3-o(1)}$细粒度下界相反，我们的框架为负三角形实现了$O(n^\omega)$的查询时间，为替换路径实现了近线性的查询时间。它还为最大流实现了近线性查询时间。$\textbf{(3)}$虽然无限持续时间博弈（奇偶博弈、平均收益博弈、能量博弈和随机博弈）是否存在多项式时间算法仍然是一个重大开放问题，但在我们的模型中它们可以轻松地以近线性时间求解。我们的证明依赖于现有工具的简单组合，并且对于没有专业背景的读者来说也是易于理解的。
英文摘要
If we run a heavy-duty computation on prior data, can we avoid repeated computation for similar future inputs? Inspired by this question, we introduce a new computational model for graph problems called algorithms with isomorphic priors. Solving a graph problem $Π$ in this model involves two phases: (i) The preprocessing phase quickly analyzes prior graphs $G_1, ..., G_k$ along with the (previously computed) exact optimal values OPT$(G_i)$. (ii) Subsequently, given a new graph H, a fast query phase must either (a) output the exact solution OPT(H), or (b) correctly report that H is not isomorphic to any $G_i$. Can we avoid computing OPT(H) from scratch when H is isomorphic to some $G_i$? We show that this is the case for a number of problems; for many others, we establish conditional lower bounds. $\textbf{(1)}$ Some NP-hard problems, including Constrained Shortest Path and $\ell_p$-Shortest Path and Constrained Spanning Tree, admit polynomial preprocessing and query times in our model. In contrast, almost all of Karp's 21 NP-complete problems and $(2-\varepsilon)$-approximate $k$-Center, for every fixed $\varepsilon>0$, admit no such algorithms unless Graph Isomorphism (GI) is in P, even with O(1) priors. $\textbf{(2)}$ In contrast to conditional $n^{3-o(1)}$ fine-grained lower bounds, our framework achieves an $O(n^ω)$ query time for Negative Triangle and a near-linear query time for Replacement Path. It also achieves near-linear query time for Maximum Flow. $\textbf{(3)}$ While it remains a major open problem whether infinite-duration games (Paritiy Game, Mean Payoff Game, Energy Game, and Stochastic Game) admit polynomial-time algorithms, they can be easily solved in near-linear time within our model. Our proofs rely on a simple combination of existing tools and are accessible to readers without specialized background.
CommentsAccepted at FOCS 2026
