发表机构
Indian Institute of Technology, Gandhinagar（印度技术学院甘地纳加尔分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于(p,q)-宽度和模式多项式的算法，在稀疏宿主图中更快检测小规模诱导模式图，对部分模式达到最优，并优于现有算法。
AI 中文摘要
我们研究在宿主图中检测与固定模式图相对应的诱导子图的算法。我们证明，在五个顶点的21个连通图中，至少有5个可以在时间约为顶点数与边数乘积的范围内被检测到；在六个顶点的112个连通图中，至少有65个可以在时间接近边数平方的范围内被检测到。我们还给出了检测七个顶点上的诱导路径和环的算法，其运行时间约为顶点数乘以边数的平方。我们的主要技术工具是树分解宽度的广义概念，称为(p,q)-宽度。它产生的算法的运行时间同时依赖于顶点数和边数，并且从不比现有界限差。每当宿主图的边数少于其顶点数的近似平方倍时，我们的界限严格更快。对于某些模式，包括七顶点环，我们的算法在标准复杂性理论假设下是最优的。我们进一步利用基于模式的、利用树分解结构（而不仅仅是其宽度）的多项式来发展这种方法。这给出了在二分图中检测偶数个顶点上的诱导路径和环的算法，对于2k个顶点的路径，运行时间约为边数的(k-1)次幂；对于2k个顶点的环，运行时间为该界限乘以顶点数。这些算法比已知的最佳通用图算法更快。
英文摘要
We study algorithms for detecting induced subgraphs corresponding to fixed pattern graphs in host graphs. We show that at least five of the 21 connected graphs on five vertices can be detected in time roughly the product of the number of vertices and the number of edges, and that at least 65 of the 112 connected graphs on six vertices can be detected in time nearly quadratic in the number of edges. We also give algorithms for detecting induced paths and cycles on seven vertices, running in time roughly the number of vertices times the square of the number of edges. Our main technical tool is a generalized notion of tree decomposition width, called (p, q)-width. It yields algorithms whose running times depend on both the number of vertices and the number of edges, and are never worse than existing bounds. Whenever the host graph has fewer than roughly quadratically many edges in its number of vertices, our bounds are strictly faster. For some patterns, including the seven-vertex cycle, our algorithms are optimal under standard complexity-theoretic assumptions. We further develop this approach using pattern-based polynomials that exploit the structure of tree decompositions, not just their width. This gives algorithms for detecting induced paths and cycles on an even number of vertices in bipartite graphs, running in time roughly the (k-1)-th power of the number of edges for paths on 2k vertices, and that same bound times the number of vertices for cycles on 2k vertices. These are faster than the best known algorithms for general graphs.
Comments38 pages, 70 figures
