A polynomial method for the pos/neg weighted 3-median problem on a tree
Tóm tắt
Let a connected undirected graph G = (V, E) be given. In the classical p-median problem we want to find a set X containing p points in G such that the sum of weighted distances from X to all vertices in V is minimized. We consider the semi-obnoxious case where every vertex has either a positive or negative weight. In this case we have two different objective functions: the sum of the minimum weighted distances from X to all vertices and the sum of the weighted minimum distances. In this paper we show that for the case p = 3 an optimal solution for the second model in a tree can be found in O(n
5) time. If the 3-median is restricted to vertices or if the tree is a path then the complexity can be reduced to O(n
3).
Tài liệu tham khảo
Burkard RE, Çela E, Dollani H (2000) 2-Median in trees with pos/neg weights. Discrete Appl Math 105:51–71
Burkard RE, Fathali J, Taghizadeh KH (2004) The p-maxian problem on a tree. Oper Res Lett (to appear)
Burkard RE, Krarup J, (1998) A linear algorithm for the pos/neg-weighted 1-median problem on a cactus. Computing 60:193–215
Cappanera P, (1999) A survey on obnoxious facility location problems. Technical Report: TR-99-11, Dipartimento di Informatica, Universita di Pisa.
Carrizosa E, Plastria F (1999) Location of semi-obnoxious facilities. Stud Locational Anal 12:1–27
Church RL, Garfinkel RS (1978) Locating an obnoxious facility on a network. Transport Sci 12:107–118
Drezner Z, Hamacher H (2002) Facility location: applications and Theory. Springer, Berlin Heidelberg New York
Eiselt HA, Laporte G (1995). Objectives in location problems. In: Drezner Z (eds). Facility location. A Survey of applications and methods. (Chapter 8). Springer, Berlin Heidelberg New York
Gavish B, Sridhar S (1995) Computing the 2-median on tree networks in O(nlog n) time. Networks 26:305–317
Goldman AJ (1971) Optimal center location in simple networks. Transporta Sci 5:212–221
Hassin R, Tamir A (1991) Improved complexity bounds for location problems on the real line. Oper Res Lett 10:395–402
Hua LK et al. (1962) Applications of mathematical methods to wheat harvesting. Chin Math 2:77–91
Kariv O, Hakimi SL (1979) An algorithmic approach to network location problems. Part II: p-medians. SIAM J Appl Math 37:539–560
Krarup J, Pisinger D, Plastria F (2002) Discrete location problems with push-pull objectives. Discrete Appl Maths 123:363–378
Mirchandani PB, Francis R (1990) Discrete Location Theory. Wiley, New York
Plastria F (1996) Optimal location of undesirable facilities: an overview. Belg J Oper Res Statist Comput Sci 36:109–127
Tamir A (1991) Obnoxious facility location on graphs. SIAM J Discrete Math 4:550–567
Tamir A (1996) An O(pn 2) algorithm for the p-median and related problems on tree graphs. Oper Res Lett 19:59–64
Ting SS (1984) A linear-time algorithm for maxisum facility location on tree networks. Transport Sci 18:76–84
Zelinka B (1968) Medians and peripherians of trees. Arch Math 4:87–95