# An Algorithmic Approach to Network Location Problems. I: The p-Centers

## Abstract

Problems of finding p-centers and dominating sets of radius r in networks are discussed in this paper. Let n be the number of vertices and $| E |$ be the number of edges of a network. With the assumption that the distance-matrix of the network is available, we design an $O(| E | \cdot n \cdot \lg n)$ algorithm for finding an absolute 1-center of a vertex-weighted network and an $O(| E | \cdot n + n^2 \cdot \lg n)$ algorithm for finding an absolute 1-center of a vertex-unweighted network (the problem of finding a vertex 1-center of a network is trivial). We show that the problem of finding a (vertex or absolute) p-center (for $1 < p < n$) of a (vertex-weighted or vertex-unweighted) network, and the problem of finding a dominating set of radius r are $NP$-hard even in the case where the network has a simple structure (e.g., a planar graph of maximum vertex degree 3). However, we describe an algorithm of complexity ${O[(| E |^p \cdot n^{2p - 1} / (p - 1)! ) \lg n]}$ (respectively, ${O[| E |^p \cdot n^{2p - 1} / (p - 1) !]}$ for finding an absolute p-center in a vertex-weighted (respectively, vertex-unweighted) network. We proceed by discussing the problems of finding p-centers and dominating sets of networks whose underlying graphs are trees. When the network is a vertex-weighted tree, we obtain the following algorithms: An $O(n \cdot \lg n)$ algorithm for finding the (vertex or absolute) 1-center; an $O(n)$ algorithm for finding a (vertex or absolute) dominating set of radius r; an $O(n^2 \cdot \lg n)$ algorithm for finding a (vertex or absolute) p-center for any $1 < p < n$. Some generalizations of these problems are discussed. When the network is a vertex-unweighted tree, $O(n)$ algorithms for finding the (vertex or absolute) 1-center and an absolute 2-center are already known; we extend these results by giving an $O(n \cdot \lg ^{p - 2} n)$ algorithm for finding an absolute p-center (where $3 \leqq p < n$) and an $O(n \cdot \lg ^{p - 1} n)$ algorithm for finding a vertex p-center (where $2 \leqq p < n$).
In part II we treat the p-median problem.

## References

1.
S. L. Hakimi, Optimum distribution of switching centers in a communication network and some related graph theoretic problems, Operations Res., 13 (1965), 462–475
2.
S. L. Hakimi, Optimum locations of switching centers and the absolute centers and medians of a graph, Operations Res., 12 (1964), 450–459
3.
Edward Minieka, The m-center problem, SIAM Rev., 12 (1970), 138–139
4.
P. M. Dearing, R. L. Francis, A minimax location problem on a network, Transportation Sci., 8 (1974), 333–343
5.
G. Y. Handler, Minimax network location theory and algorithms, Technical Rep., 107, Oper. Res. Center, Mass. Inst. of Tech., Cambridge, Mass., 1974, Nov.
6.
A. J. Goldman, Minimax location of a facility in a network, Transportation Sci., 6 (1972), 407–418
7.
G. Y. Handler, Minimax location of a facility in an undirected tree graph, Transportation Sci., 7 (1973), 287–293
8.
S. Halfin, On finding the absolute and vertex centers of a tree with distances, Transportation Sci., 8 (1974), 75–77
9.
S. L. Hakimi, E. F. Schmeichel, On p-centers in networks, Transportation Sci., 12 (1978), 1–15
10.
Richard M. Karp, R. E. Miller, J. W. Thatcher, Reducibility among combinatorial problemsComplexity of computer computations (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972), Plenum, New York, 1972, 85–103
11.
E. W. Dijkstra, A note on two problems in connexion with graphs, Numer. Math., 1 (1959), 269–271
12.
Alfred V. Aho, John E. Hopcroft, Jeffrey D. Ullman, The design and analysis of computer algorithms, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975x+470
13.
Frank Harary, Graph theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London, 1969ix+274
14.
A. J. Goldman, Optimal center location in simple networks, Transportation Sci., 5 (1971), 212–221
15.
Peter J. Slater, R-domination in graphs, J. Assoc. Comput. Mach., 23 (1976), 446–450
16.
D. W. Matula, R. Kolde, Efficient multi-median location in acyclic networks, presented at ORSA-TIMS, Miami, FL, 1976, Nov. 4
17.
M. R. Garey, D. S. Johnson, The rectilinear Steiner tree problem is NP-complete, SIAM J. Appl. Math., 32 (1977), 826–834
18.
D. S. Johnson, Personal communication
19.
E. J. Cockayne, S. E. Goodman, S. T. Hedetniemi, A linear algorithm for the domination number of a tree, Information Processing Letters, 4 (1975), 41–44
20.
Edward Minieka, The centers and medians of a graph, Operations Res., 25 (1977), 641–650
21.
J. Halpern, The location of a center-median convex combination on an undirected tree, J. Regional Science, 16 (1976), 237–245

## Information & Authors

### Information

#### Published In SIAM Journal on Applied Mathematics
Pages: 513 - 538
ISSN (online): 1095-712X

#### History

Submitted: 5 January 1977
Published online: 12 July 2006