A tournament T=(V,A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T' on O(k) vertices. In fact, given any fixed e>0, the kernelized instance has at most (2+e)k vertices. Our result improves the previous known bound of O(k^2) on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k-FAST.
On the Complexity of the Minimum Cost Homomorphism Problem for Reflexive Multipartite Tournaments
对于有向图$D$和$H$，如果$uv\in A(D)$蕴涵$f(u)f(v)\in A(H).$，则映射$f: V(D)\dom V(H)$是$D$到$H$的同态。对于固定有向图$H$，同态问题是判定输入有向图$D$是否允许与$H$同态，记为HOMP（$H$）。有向图允许有环，但不允许有平行弧。同态问题的自然优化版本定义如下。如果每个顶点$u \in V(D)$与成本$c_i(u), i \in V(H)$相关联，则同态$f$的成本是$\sum_{u\in V(D)}c_{f(u)}(u)$。对于每一个固定有向图$H$，我们有{ \em最小成本同态问题} $H$，并将其表示为MinHOMP（$H$）。问题是对一个代价为$c_i(u),$ $u \in V(D), i\in V(H)$的输入图$D$，判定是否存在一个从$D$到$H$的同态，如果存在，则找出一个最小代价的同态.在最近的一篇论文中，我们提出了一个问题的特征多项式时间可解和NP-困难的情况下的最小成本同态问题的无环多部竞赛可能的循环（w.p.l.）。本文解决了自反多部竞赛图的问题，并证明了这类多部竞赛图w.p.l.问题的一个较好的困难性。使用4阶W.P.L.的非循环3部竞赛图作为例子。\footnote{This paper was submitted to Discrete Mathematics on April 6,…
FAST问题是在竞赛图中寻找最小反馈弧集问题。在本文中，我们提出了一些算法，类似于排序算法的FAST问题，我们分析他们。本文提出了一种求解FAST问题的伪插入排序算法，并证明了该算法的平均后向边数等于（（n^2- 5 n +8）/4）-2 ^（1-n）.我们引入了伪归并排序算法，我们发现的概率是落后的边缘。最后，我们介绍了其他算法这个问题。
