2-Medians in trees with pos/neg weights

The multi-facility median problem with Pos/Neg weights on general graphs

“…The above result is essentially the same as in [5]. They showed, that there always exists an optimal solution where either both solution points are nodes, or where one of the solution points is a node and the other is a point such that the midpoint between the two solution points is a node.…”

“…Burkard et al [5] show that for the 2-median problem on trees, the optimal solution either consists of a pair of nodes, or one solution point is a node v and the other a point y such that the midpoint between v and y is a node. Unfortunately, neither of the two sets are valid for the general problem, as the next example shows.…”

“…If we restrict our search for an optimal solution to the set of nodes and bottleneck points we obtain two optimal solutions: {v 2 , v 4 } and fv 5 , ð½v 1 ,v 2 , 1 4 Þg with objective value À 3. With respect to the FDS of [5], we obtain {v 2 , v 4 } and fv 2 , ð½v 4 ,v 4 , 1 2 Þg again with value À3. However, if we extend our search to the whole graph, a better solution is given by…”

“…Burkard and Krarup [4] present a linear time algorithm for the single facility median problem with pos/neg weights on cactus graphs. For the 2-median problem with pos/neg weights on trees, Burkard et al [5] show that there always exists an optimal solution where at least one of the solution points is a node. Using this property, they derive an O(n 3 ) algorithm for the absolute problem and O(n 2 ) algorithms for the node restricted version and the absolute problem on paths.…”

The two‐median problem on Manhattan meshes

“…For k ≥ 3, the problem is known as the k-median problem. In particular, the twomedian problem was discussed widely on trees [2,5,6,11].…”