Efficient algorithms for finding 2‐medians of a tree

Abstract

The p-median problem on general networks has been studied since the 1960s. Kariv and Hakimi [10] showed that this problem is NP-hard even if the network is a planar graph of maximum degree 3. In the case of tree networks the p-median problem is solvable in polynomial time. Kariv and Hakimi [10] developed an algorithm that computes a solution in O(p 2 n 2) time. The running time was improved to O(pn 2) by Tamir [16] and later to O(n lg p+2 n) by Benkoczi and Bhattacharya [2]. Better bounds are known for the special cases (on trees) where p = 1 or 2. Goldman [6] gave an O(n) algorithm for the 1-median problem on trees. The 2-median problem was studied by Mirchandani and Oudjit [11], whose localization properties were later used to improve the O(n 2) bound (derived from the general tree case) to O(n lg n)-see papers by Hämäläinen [9] and Gavish and Sridhar [5]. We present a framework for solving the 2-median problem on trees, building on earlier work. Our framework leads to several algorithms with o(n lg n) runtime, i.e., better than the current best-known O(n lg n) runtime, in common special cases. The time bounds are: (i) O(n lg wmax/ lg n), where wmax is the largest largest node weight, which is linear when node weights are bounded by a polynomial in n; (ii) O(n lg nL), where nL is the number of leaves in in the tree; (iii) O(ndmax), where dmax is the maximum edge length, which is linear when edge lengths are bounded by a constant; and (iv) O(n lg), where is the number of nodes on the trunk, an easily identified path that is guaranteed to contain at least one of the two medians.

On the complexity of the upgrading version of the maximal covering location problem

“…Similar results have been obtained for various location problems, including covering problems [10,24,25,33], center problems [44], median problems [1,35], or other facility location problems [27,29,34,38], as well as flow/shortest path problems [17,26,42].…”