参数大于保值的线性排列问题
The Linear Arrangement Problem Parameterized Above Guaranteed Value
摘要：
线性排列（LA）是将不同的整数分配给图的顶点。LA的成本是图的边的长度之和，其中边的长度被定义为分配给其端点的整数之差的绝对值。对于许多应用，人们希望找到具有小成本的LA。然而，决定给定的图$G$是否允许以给定整数为界的LA是一个经典的NP完全问题。由于$G$的每条边对任何LA的成本至少有一个贡献，因此如果通过成本的上界进行参数化，则问题变得平凡固定参数易处理（FPT）。Fernau问，如果用成本减去给定图的边数的上限进行参数化，问题是否仍然是FPT;因此，问题是否在保证值以上进行了FPT参数化。我们通过导出一个算法来肯定地回答这个问题，该算法在时间上决定一个具有$m$边和$n$顶点的给定图$O(m+n+5.88^k)$是否允许成本最多为$m+k$的LA（如果存在这样的LA，则该算法计算该LA）。我们的算法是基于一个程序，生成一个问题的线性大小的连接图$G$在线性时间的核心。我们还证明了Serna和Thilikos提出的更一般的参数化LA问题不是FPT，除非P=NP。
英文摘要：
A linear arrangement (LA) is an assignment of distinct integers to the vertices of a graph. The cost of an LA is the sum of lengths of the edges of the graph, where the length of an edge is defined as the absolute value of the difference of the integers assigned to its ends. For many application one hopes to find an LA with small cost. However, it is a classical NP-complete problem to decide whether a given graph $G$ admits an LA of cost bounded by a given integer. Since every edge of $G$ contributes at least one to the cost of any LA, the problem becomes trivially fixed-parameter tractable (FPT) if parameterized by the upper bound of the cost. Fernau asked whether the problem remains FPT if parameterized by the upper bound of the cost minus the number of edges of the given graph; thus whether the problem is FPT ``parameterized above guaranteed value.'' We answer this question positively by deriving an algorithm which decides in time $O(m+n+5.88^k)$ whether a given graph with $m$ edges and $n$ vertices admits an LA of cost at most $m+k$ (the algorithm computes such an LA if it exists). Our algorithm is based on a procedure which generates a problem kernel of linear size in linear time for a connected graph $G$. We also prove that more general parameterized LA problems stated by Serna and Thilikos are not FPT, unless P=NP.
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