A polynomial algorithm for thep-centdian problem on a tree

Abstract

The most common problems studied in network location theory are the p-median and the p-center models. The p-median problem on a network is concerned with the location of p points (medians) on the network, such that the total (weighted) distance of all the nodes to their respective nearest points is minimized. The p center problem is concerned with the location of p-points (centers) on the network, such that the maximum (weighted) distance of all the nodes to their respective nearest points is minimized.To capture more real world problems, and obtain a good way to trade-off minisum (efficiency) and minimax (equity) approaches, Halpern [5,6,7], introduced the centdian model, where the objective is to minimize a convex combination of the objective functions of the center and the median problems. In this paper we study the p-centdian problem on tree networks, and present the first polynomial time algorithm for this problem.

Multifacility ordered median problems on networks: A further analysis

“…To do that, we adapt and modify the dynamic algorithm for the p-centdian model on a tree proposed in Tamir et al [24]. Assume that we are given a tree -with ͉ᐂ͉ ϭ M.…”

“…Once the treea has been obtained, a second preprocessing phase similar to the used in Tamir et al [24] is performed. For each node v j , we compute and sort the distances from v j to all seminodes ina .…”

“…We can assume w.l.o.g. that there is a one-to-one correspondence between the elements in L j and the seminodes in Y (Tamir et al [24]). The seminode corresponding to r j i is denoted by y j i , i ϭ 1, .…”

“…It is also worth noting that for a Ͻ b and s ϭ 1 the algorithm reduces to the one given in Tamir et al [24] and the complexity is O( pM 6 ). Moreover, if s ϭ 0, the problem reduces to the p-median problem and the complexity is O( pM 2 ) by the algorithm by Tamir [22].…”

Improved complexity results for several multifacility location problems on trees

“…The node-restricted p-facility k-centrum problem was solved in O(min(k, p)kpn 5 ) and the absolute problem in O(min(k, p) Tamir (2000). For the p-centdian problem, Tamir et al (1998) show how to solve the problem in min{O(pn 6 ), O(n p )} time. Finally, Kalcsics et al (2003) generalize the method in Tamir et al (1998) to the ordered median problem with at most two different values for the modeling weights, solving the problem in O(pn(sn 4 ) 2 ) time, where s ≤ n is a constant that is specific for the ordered median problem.…”