We address the problem of modifying vertex weights of a block graph at minimum total cost so that a predetermined set of p connected vertices becomes a connected p-median on the perturbed block graph. This problem is the so-called inverse connected p-median problem on block graphs. We consider the problem on a block graph with uniform edge lengths under various cost functions, say rectilinear norm, Chebyshev norm, and bottleneck Hamming distance. To solve the problem, we first find an optimality criterion for a set that is a connected p-median. Based on this criterion, we can formulate the problem as a convex or quasiconvex univariate optimization problem. Finally, we develop combinatorial algorithms that solve the problems under the three cost functions in \(O(n\log n)\) time, where n is the number of vertices in the underlying block graph.

我们解决了以最小的总成本修改框图的顶点权重的问题，以使预定的p个连接顶点集成为扰动框图上的p个中位数。这个问题是框图上的所谓的逆p-中值问题。我们在各种成本函数（例如直线范数，切比雪夫范数和瓶颈汉明距离）下在具有统一边长的框图上考虑该问题。为了解决该问题，我们首先找到一个连通p的最优准则。-中位数。基于此准则，我们可以将该问题公式化为凸或拟凸单变量优化问题。最后，我们开发组合算法来解决\（O（n \ log n）\）时间内三个成本函数下的问题，其中n是基础框图中的顶点数。