变边长树上的$p$ -Maxian逆问题
The inverse $p$-maxian problem on trees with variable edge lengths
摘要：
我们关心的问题，修改边的长度在最小的总成本，使预先指定的$p$顶点成为$p$ -maxian关于新的边的长度。这个问题称为树上的逆$p$ -Maxian问题。\textbf{Gassner}在2008年提出了一种有效的组合算法来解决树上的逆1-maxian问题。对于含$p \geq 2$的问题，我们认为该问题可化为多个逆$2$ -Maxian问题.然后，我们开发的算法来解决各种目标函数的逆$2$ -马克西问题。在$l_1$ -范数下的问题可以表示为线性规划，因此可以在多项式时间内求解。特别地，如果底层树是星型树，那么问题可以在线性时间内解决。我们还设计了$O(n\log n)$算法来解决Chebyshev范数和瓶颈Hamming距离下的问题，其中$n$是树的顶点数。最后，加权和Hamming距离下的问题是$NP$ -困难的.
英文摘要：
We concern the problem of modifying the edge lengths of a tree in minimum total cost so that the prespecified $p$ vertices become the $p$-maxian with respect to the new edge lengths. This problem is called the inverse $p$-maxian problem on trees. \textbf{Gassner} proposed efficient combinatorial alogrithm to solve the the inverse 1-maxian problem on trees in 2008. For the problem with $p \geq 2$, we claim that the problem can be reduced to finitely many inverse $2$-maxian problem. We then develop algorithms to solve the inverse $2$-maxian problem for various objective functions. The problem under $l_1$-norm can be formulated as a linear program and thus can be solved in polynomial time. Particularly, if the underlying tree is a star, then the problem can be solved in linear time. We also devised $O(n\log n)$ algorithms to solve the problems under Chebyshev norm and bottleneck Hamming distance, where $n$ is the number of vertices of the tree. Finally, the problem under weighted sum Hamming distance is $NP$-hard.
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