# Extensive facility location problems on networks: an updated review

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## Abstract

Location problems with extensive facilities represent a challenging field of research. According to the specialized literature, a facility is called extensive if, for purposes of location, it is too large in relation to its environment to be considered a point. There are many examples of this type of structures that appear in real-world applications both in the continuous space (straight lines, circles, strips) and in networks (paths, cycles, trees). There exists a recent literature review on the location of dimensional facilities on continuous space (Díaz-Báñez et al. in TOP 154:22–44, 2004; Schöbel in Location of dimensional facilities in a continuous space, 2015) that does not cover similar problems on networks. The goal of this paper is to review the location of dimensional facilities in networks. We mainly concentrate on the location of paths and trees considering the most common objective functions in the location literature, namely median and center. However, we also consider some other alternative criteria generalizing them, as the ordered median objective function, or related to equity, reliability, and robustness. We include the basic tools and techniques that are applicable to develop algorithms for this kind of problems. Moreover, we present the best known complexity results for each of the considered problems. Finally, some suggestions are also made for possible directions of future research.

## Keywords

Path location problems Tree location problems Ordered median Equity measures Minimum loss criteria## Mathematics Subject Classification

90B-10 90C-35 68R-10## References

- Alstrup S, Holm J, Lichtenberg KD, Thorup M (2005) Maintaining information in fully dynamic trees with top trees. ACM Trans Algorithms 1:243–264CrossRefGoogle Scholar
- Alstrup S, Lauridsen PW, Sommerlund P, Thorup M (1997) Finding cores of limited length. In: Proceedings of 5th international workshop on algorithms and data structures (WADS), lecture notes in computer science, vol 1272. Springer, Berlin, pp 45–54Google Scholar
- Alstrup S, Lauridsen PW, Sommerlund P, Thorup M (2001) Finding cores of limited length, IT-C Technical Report Series 2000-4, University of CopenhagenGoogle Scholar
- Averbakh I, Berman O (1999) Algorithms for path medi-centers of a tree. Comput Oper Res 26:1395–1409CrossRefGoogle Scholar
- Balasubramanian S, Harini S, Rangan CP (2009) Core and conditional core path of specified length in special classes of graphs. In: Das S, Uehara R (eds) WALCOM: algorithms and computation. WALCOM 2009. Lecture notes in computer science, vol 5431, Springer, Berlin, Heidelberg, pp 262–273Google Scholar
- Becker RI, Perl Y (1985) Finding the two-core of a tree. Discrete Appl Math 11:103–113CrossRefGoogle Scholar
- Becker RI, Lari I, Scozzari A, Storchi G (2002) Efficient algorithms for finding the \((k,\ell )\)-core of tree networks. Networks 40:208–215CrossRefGoogle Scholar
- Becker RI, Chang YI, Lari I, Scozzari A, Storchi G (2002) Finding the \(\ell \)-core of a tree. Discrete Appl Math 118:25–42CrossRefGoogle Scholar
- Becker RI, Lari I, Scozzari A (2007) Algorithms for central-median paths with bounded length on trees. Eur J Oper Res 179:1208–1220CrossRefGoogle Scholar
- Becker RI, Lari I, Scozzari A, Storchi G (2007) The location of median paths on grid graphs. Ann Oper Res 150:65–78CrossRefGoogle Scholar
- Benkoczi R, Bhattacharya B, Tamir A (2009) Collection depots facility location problems in trees. Networks 53:40–62CrossRefGoogle Scholar
- Bhattacharya B, Shi Q, Tamir A (2009) Optimal algorithms for the path/tree-shaped facility location problems in trees. Algorithmica 55:601–618CrossRefGoogle Scholar
- Bhattacharya B, Hu Y, Shi Q, Tamir A (2006) Optimal algorithms for the path, tree-shaped facility location problems in trees, ISAAC, lecture notes in computer science, vol 4288. Springer, Berlin, pp 379–388Google Scholar
- Bhattacharyya B, Dehne F (2008) Using spine decomposition to efficiently solve the length-constrained heaviest path problem for trees. Inf Process Lett 108:293–297CrossRefGoogle Scholar
- Boffey B (1998) Efficient solution methods for covering tree problems. TOP 6:205–221CrossRefGoogle Scholar
- Caceres T, López-de-los-Mozos MC, Mesa JA (2004) The path-variance problem on tree networks. Discrete Appl Math 145:72–79CrossRefGoogle Scholar
- Díaz-Báñez JM, Mesa JA, Schöbel A (2004) Continuous location of dimensional structures. TOP 154:22–44Google Scholar
- Dvir A, Segal M (2008) The \((k,\ell )\) coredian tree for ad hoc networks. In: IEEE the 28th international conference on distributed computing systems workshops. https://doi.org/10.1109/ICDCS
- Frederickson GN, Johnson DB (1983) Finding the k-th paths and p-centers by generating and searching good data structures. J Algorithms 4:61–80CrossRefGoogle Scholar
- George JW, Revelle CS (2003) Bi-objective median subtree location problems. Ann Oper Res 122:219–232CrossRefGoogle Scholar
- Hakimi SL, Schmeichel EF, Labbé M (1993) On locating path-or tree-shaped facilities on networks. Networks 23:543–555CrossRefGoogle Scholar
- Halpern J (1976) The location of a center–median convex combination on an undirected tree. J Reg Sci 16:237–245CrossRefGoogle Scholar
- Halpern J (1978) Finding minimal center–median convex combination (cent-dian) of a graph. Manag Sci 24:534–544CrossRefGoogle Scholar
- Hassin R, Tamir A (1986) Efficient algorithms for optimization and selection on Series-parallel graphs. SIAM J Algebraic Discrete Methods 7:379–389CrossRefGoogle Scholar
- Hedetniemi SM, Cockaine EJ, Hedetniemi ST (1981) Linear algorithms for finding the Jordan centre and path centre of a tree. Transp Sci 15:98–114CrossRefGoogle Scholar
- Itai A, Papadimitriou CH, Szwarcfiter JL (1982) Hamilton paths in grid graphs. SIAM J Comput 4:676–686CrossRefGoogle Scholar
- Kalcsics J, Nickel S, Puerto J (2003) Multifacility ordered median problems on networks: a further analysis. Networks 41:1–12CrossRefGoogle Scholar
- Kim TU, Lowe TJ, Tamir A, Ward JE (1996) On the location of a tree-shaped facility. Networks 28:167–175CrossRefGoogle Scholar
- Labbé M, Laporte G, Martín IR, González JJS (2005) Locating median cycles in networks. Eur J Oper Res 160:457–470CrossRefGoogle Scholar
- Laporte G, Martín IR (2007) Locating a cycle in a transportation or a telecommunications network. Networks 50:92–108CrossRefGoogle Scholar
- Laporte G, Nickel S, Saldanha da Gama F (2015) Location science. Springer International Publishing, SwitzerlandCrossRefGoogle Scholar
- Lari I, Ricca F, Scozzari A (2008) Comparing different metaheuristic approaches for the median path problem with bounded length. Eur J Oper Res 190:587–597CrossRefGoogle Scholar
- Lari I, Ricca F, Scozzari A, Becker RI (2011) Locating median paths on connected outerplanar graphs. Networks 57:294–307CrossRefGoogle Scholar
- Megiddo N (1983) Linear-time algorithms for linear programming in \(R^3\) and related problems. SIAM J Comput 12:759–776CrossRefGoogle Scholar
- Mesa JA, Boffey TB (1996) A review of extensive facility location in networks. Eur J Oper Res 95:592–603CrossRefGoogle Scholar
- Mesa JA, Puerto J, Tamir A (2003) Improved algorithms for several network location problems with equality measures. Discrete Appl Math 130:437–448CrossRefGoogle Scholar
- Minieka E (1985) The optimal location of a path or tree in a tree network. Networks 15:309–321CrossRefGoogle Scholar
- Morgan CA, Slater JP (1980) A linear algorithm for a core of a tree. J Algorithms 1:247–258CrossRefGoogle Scholar
- Nickel S, Puerto J (1999) A unified approach to network location. Networks 34:283–290CrossRefGoogle Scholar
- Nickel S, Puerto J (2005) Location theory: a unified approach. Springer, BerlinGoogle Scholar
- Novik A (1996) Improved algorithms for locating tree or path shaped facilities on a tree network, M.S. Thesis, School of Mathematical Sciences, Tel Aviv University, Tel Aviv, IsraelGoogle Scholar
- Peng S, Lo W-T (1996) Efficient algorithms for finding a core of a tree with a specified length. J Algorithms 20:445–458CrossRefGoogle Scholar
- Peng S, Stephens AB, Yesha Y (1993) Algorithms for a core and \(k\)-tree core of a tree. J Algorithms 15:143–159CrossRefGoogle Scholar
- Puerto J, Tamir A (2005) Locating tree-shaped facilities using the ordered median objective. Math Programm 102:313–338CrossRefGoogle Scholar
- Puerto J, Rodríguez-Chía AM, Tamir A, Perez-Brito D (2006) The bi-criteria doubly weighted center–median path problem on a tree. Networks 47:237–247CrossRefGoogle Scholar
- Puerto J, Ricca F, Scozzari A (2009) Extensive facility location problems on networks with equity measures. Discrete Appl Math 157:1069–1085CrossRefGoogle Scholar
- Puerto J, Ricca F, Scozzari A (2009) The continuous and discrete path-variance problems on trees. Networks 53:221–228CrossRefGoogle Scholar
- Puerto J, Ricca F, Scozzari A (2011) Minimax regret path location on trees. Networks 58:147–158Google Scholar
- Puerto J, Ricca F, Scozzari A (2012) Range minimization problems in path-facility location on trees. Discrete Appl Math 160:2294–2305CrossRefGoogle Scholar
- Puerto J, Ricca F, Scozzari A (2014) Reliability problems in multiple path-shaped facility location on networks. Discrete Optim 12:61–72CrossRefGoogle Scholar
- Puerto J, Rodríguez-Chía AM, Tamir A (2017) Revisiting \(k\)-sum optimization. Math Program 165:579–604CrossRefGoogle Scholar
- Richey MB (1990) Optimal location of a path or tree on a network with cycles. Networks 20:391–407CrossRefGoogle Scholar
- Rozanov M (2015) The nestedness property of the convex ordered median location problem on a tree. MSc thesis, Tel-Aviv University. http://primage.tau.ac.il/libraries/theses/exeng/free/3257656.pdf
- Rozanov M, Tamir A (2018) The nestedness property of location problems on the line. TOP. https://doi.org/10.1007/s11750-018-0471-x
- Schöbel A (2015) Location of dimensional facilities in a continuous space. In: Laporte G et al (eds) Location science. Springer International Publishing, ChamGoogle Scholar
- Shioura A, Shigeno M (1997) The tree center problems and the relationship with the bottleneck knapsack Problem. Networks 29:107–110CrossRefGoogle Scholar
- Shioura A, Uno T (1997) A linear time algorithm for finding a \(k\)-tree core. J Algorithms 23:281–290CrossRefGoogle Scholar
- Slater PJ (1982) Locating central paths in a graph. Transp Sci 15:1–18CrossRefGoogle Scholar
- Takamizawa K, Nishizeki T, Saito N (1982) Linear-time computability of combinatorial problems on series-parallel graphs. J ACM 29:623–641CrossRefGoogle Scholar
- Tamir A (1996) An \(O(pn^2)\) algorithm for the \(p\)-median and related problems on tree graphs. Oper Res Lett 19:59–64CrossRefGoogle Scholar
- Tamir A (1998) Fully polynomial approximation schemes for locating a tree-shaped facility: a generalization of the knapsack problem. Discrete Appl Math 87:229–243CrossRefGoogle Scholar
- Tamir A (2004) Sorting weighted distances with applications to objective function evaluations in single facility location problems. Oper Res Lett 32:249–257CrossRefGoogle Scholar
- Tamir A, Lowe TJ (1992) The generalized \(p\)-forest problem on a tree network. Networks 22:217–230CrossRefGoogle Scholar
- Tamir A, Puerto J, Pèrez-Brito D (2002) The centdian subtree on the tree networks. Discrete Appl Math 118:263–278CrossRefGoogle Scholar
- Tamir A, Puerto J, Mesa JA, Rodriguez-Chia AM (2005) Conditional location of path and tree shaped facilities on trees. J Algorithms 56:50–75CrossRefGoogle Scholar
- Tang H, Cheng TCE, Ng CT (2012) A note on the subtree ordered median problem in networks based on nestedness property. J Ind Manag Optim 8:41–49CrossRefGoogle Scholar
- Wang B-F, Lin J-J (2000) Finding a Two-Core of a tree in linear time, ISAAC, lecture notes in computer science, vol 1969. Springer, Berlin, pp 467–478Google Scholar
- Wang B-F (1998) Finding a k-tree core and a k-tree center of a tree network in parallel. IEEE Trans Parallel Distrib Syst 9:186–191CrossRefGoogle Scholar
- Wang B-F (2000) Efficient parallel algorithms for optimally locating a path and a tree of a specified length in a weighted tree network. J Algorithms 34:90–108CrossRefGoogle Scholar
- Wang B-F (2002) A 2-Core of a tree in Linear time. SIAM J Discrete Math 15:193–210CrossRefGoogle Scholar
- Wang B-F, Ku S-C, Shi K-H (2001) Cost-optimal parallel algorithms for the tree bisector and related problems. IEEE Trans Parallel Distrib Syst 12:888–898CrossRefGoogle Scholar
- Wang B-F, Peng S, Yu H-Y, Ku S-C (2006) Efficient algorithms for a constrained \(k\)-tree core problem in a tree network. J Algorithms 59:107–124CrossRefGoogle Scholar
- Wang B-F, Lin T-C, Lin C-H, Ku S-C (2008) Finding the conditional location of a median path on a tree. Inf Comput 206:828–839CrossRefGoogle Scholar
- Ye J-H (2017) Improved algorithms for minmax regret path location problems on trees, Ph.D. Dissertation, Department of Computer Science, Nation Tsing Hua UniversityGoogle Scholar
- Ye J-H, Wang B-F (2015) On the minmax regret path median problem on trees. J Comput Syst Sci 81:1159–1170CrossRefGoogle Scholar