论文标题
有限度切割是固定参数
Bounded-Degree Cut is Fixed-Parameter Tractable
论文作者
论文摘要
在有限的削减问题中，我们将为我们提供一个多graph $ g =（v，e）$，两个不连接的顶点子集$ a，b \ subseteq v $，两个函数$ \ mathrm {u} _a，\ mathrm {u} $ k \ geq 0 $。任务是确定是否有最小的$（a，b）$ - cut $（v_a，v_a，v_b）$的大小最多$ k $，以便在诱导的$ g [v_a] $中，每个顶点$ v \ in v_a $ in V_A $的程度最多是每种$ \ nathrm {u} $ v _ $ v_a（v_a）$的$。 $ g [v_b] $最多是$ \ mathrm {u} _b（v）$。在本文中，我们表明，通过给出$ 2^{18K} | g |^{o（1）} $ - 时间算法，有限的剪切问题是固定参数。这是此问题的第一个单个指数fpt算法。该算法的核心是基于重要切割的两个新的引理，这在最小切割的一部分中为顶点子集的候选者数量提供了一些上限，从而满足了某些特性。这些引理可用于设计固定参数可拖动算法，以解决更多相关问题。
In the bounded-degree cut problem, we are given a multigraph $G=(V,E)$, two disjoint vertex subsets $A,B\subseteq V$, two functions $\mathrm{u}_A, \mathrm{u}_B:V\to \{0,1,\ldots,|E|\}$ on $V$, and an integer $k\geq 0$. The task is to determine whether there is a minimal $(A,B)$-cut $(V_A,V_B)$ of size at most $k$ such that the degree of each vertex $v\in V_A$ in the induced subgraph $G[V_A]$ is at most $\mathrm{u}_A(v)$ and the degree of each vertex $v\in V_B$ in the induced subgraph $G[V_B]$ is at most $\mathrm{u}_B(v)$. In this paper, we show that the bounded-degree cut problem is fixed-parameter tractable by giving a $2^{18k}|G|^{O(1)}$-time algorithm. This is the first single exponential FPT algorithm for this problem. The core of the algorithm lies two new lemmas based on important cuts, which give some upper bounds on the number of candidates for vertex subsets in one part of a minimal cut satisfying some properties. These lemmas can be used to design fixed-parameter tractable algorithms for more related problems.
