Discrete Optimization A mixed integer linear programming formulation of the maximum betweenness problem q Aleksandar Savic ´ a , Jozef Kratica b, * , Marija Milanovic ´ a , Djordje Dugošija a a Faculty of Mathematics, University of Belgrade, Studentski trg 16/IV 11 000 Belgrade, Serbia b Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36/III, 11 000 Belgrade, Serbia article info Article history: Received 6 July 2009 Accepted 19 February 2010 Available online 25 February 2010 Keywords: Integer programming Linear programming Betweenness problem abstract This paper considers the maximum betweenness problem. A new mixed integer linear programming (MILP) formulation is presented and validity of this formulation is given. Experimental results are per- formed on randomly generated instances from the literature. The results of CPLEX solver, based on the proposed MILP formulation, are compared with results obtained by total enumeration technique. The results show that CPLEX optimally solves instances of up to 30 elements and 60 triples in a short period of time. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction Let A be a finite set and let C be a collection of triples ða; b; cÞ of distinct elements from A. Let f : A ! R þ . Let Objðf Þ be a number of betweenness constraints f ðaÞ < f ðbÞ < f ðcÞ or f ðaÞ > f ðbÞ > f ðcÞ sat- isfied by function f. Now, the maximum betweenness problem (MBP) can be formu- lated as finding MaxðObjðf ÞÞ over all functions f : A !f1; ... ; jAjg which are 1–1. Explicitly, to find max f Objðf Þ; f 2fhjh : A !f1; ... ; jAjg; h is 1—1g Let us demonstrate this on one small illustrative example. Example 1. Let A ¼f1; 2; 3; 4; 5g and let collection C has 5 triples. Let them be (1,3,2), (4,1,5), (5,3,4), (2,1,4), (3,5,1). Then the optimal solution, obtained by total enumeration, is the function f given with 1 2 3 4 5 4 1 2 5 3   . Optimal value is Objðf Þ¼ 4, where function f satisfies betweenness constraints in triples 1, 2, 4 and 5. The maximum betweenness problem first arose in the late 1970s in the design of circuits (Opatrny, 1979). This problem also comes up in questions related to physical mapping in molecular biology. For example, it arises when trying to order markers on a chromosome, given the results of a radiation hybrid experiment. In radiation hybrid mapping, a high dose of X-rays is used to break the human chromosome of interest into several fragments. The fur- ther apart two markers are on the chromosome, the more likely a given dose of X-rays will break the chromosome between them, placing the markers on two separate chromosomal fragments. By estimating the frequency of breakage, and thus the distance, be- tween markers (data about two markers and the corresponding X-ray form a triple), it is possible to determine their order in a manner analogous to meiotic mapping. For more details about radiation hybrid experiment, see Cox et al. (1990), Goss and Harris (1975). A computational task of practical significance in this context is to find a total ordering of the markers that maximizes the number of satisfied constraints. Indeed, betweenness is central in the soft- ware package RHMAPPER (Slonim et al., 1996, 1997). That package produces the order of framework markers based on betweenness constraints. It employs two greedy heuristics for solving the betweenness problem. In Savic ´ (2009), the author describes a genetic algorithm (GA) approach for solving the MBP. The maximum of objective function is obtained by finding a permutation that satisfies the maximum number of betweenness constraints. Every considered permutation is encoded with integer representation. Genetic algorithm was tested on randomly generated instances of up to 50 elements and 1,000 triples. Unfortunately, genetic algorithm could not verify optimality of obtained solutions or give any estimation about solu- tion quality. The advantage of that approach is a quite short run- ning time. 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.02.028 q This research was partially supported by Serbian Ministry of Science under the grant no. 144007. We thank to Jelena Kojic ´ , for her useful suggestions and comments. * Corresponding author. E-mail addresses: aleks3rd@gmail.com (A. Savic ´), jkratica@gmail.com, jkratica@ mi.sanu.ac.rs (J. Kratica), marija.milanovic@gmail.com (M. Milanovic ´), dugosija@ matf.bg.ac.rs (D. Dugošija). URL: http://www.geocities.com/jkratica (J. Kratica). European Journal of Operational Research 206 (2010) 522–527 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor