European Journal of Operational Research 206 (2010) 522–527

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Discrete Optimization

A mixed integer linear programming formulation of the maximumbetweenness 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, Serbiab Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36/III, 11 000 Belgrade, Serbia

a r t i c l e i n f o

Article history:Received 6 July 2009Accepted 19 February 2010Available online 25 February 2010

Keywords:Integer programmingLinear programmingBetweenness problem

0377-2217/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.ejor.2010.02.028

q This research was partially supported by Serbiangrant no. 144007. We thank to Jelena Kojic, forcomments.

* Corresponding author.E-mail addresses: aleks3rd@gmail.com (A. Savic), j

mi.sanu.ac.rs (J. Kratica), marija.milanovic@gmail.commatf.bg.ac.rs (D. Dugošija).

URL: http://www.geocities.com/jkratica (J. Kratica

a b s t r a c t

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 theproposed MILP formulation, are compared with results obtained by total enumeration technique. Theresults show that CPLEX optimally solves instances of up to 30 elements and 60 triples in a short periodof time.

� 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 ofdistinct elements from A. Let f : A! Rþ. Let Objðf Þ be a number ofbetweenness 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; . . . ; jAjgwhich are 1–1. Explicitly, to find

maxf Objðf Þ; f 2 fhjh : A! f1; . . . ; jAjg;h is 1—1g

Let us demonstrate this on one small illustrative example.

Example 1. Let A ¼ f1;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 theoptimal solution, obtained by total enumeration, is the function f

given with 1 2 3 4 54 1 2 5 3

� �. Optimal value is Objðf Þ ¼ 4, where

function f satisfies betweenness constraints in triples 1, 2, 4 and 5.

The maximum betweenness problem first arose in the late1970s in the design of circuits (Opatrny, 1979). This problem alsocomes up in questions related to physical mapping in molecular

ll rights reserved.

Ministry of Science under theher useful suggestions and

kratica@gmail.com, jkratica@(M. Milanovic), dugosija@

).

biology. For example, it arises when trying to order markers on achromosome, given the results of a radiation hybrid experiment.In radiation hybrid mapping, a high dose of X-rays is used to breakthe human chromosome of interest into several fragments. The fur-ther apart two markers are on the chromosome, the more likely agiven dose of X-rays will break the chromosome between them,placing the markers on two separate chromosomal fragments. Byestimating the frequency of breakage, and thus the distance, be-tween markers (data about two markers and the correspondingX-ray form a triple), it is possible to determine their order in amanner analogous to meiotic mapping. For more details aboutradiation hybrid experiment, see Cox et al. (1990), Goss and Harris(1975).

A computational task of practical significance in this context isto find a total ordering of the markers that maximizes the numberof satisfied constraints. Indeed, betweenness is central in the soft-ware package RHMAPPER (Slonim et al., 1996, 1997). That packageproduces the order of framework markers based on betweennessconstraints. It employs two greedy heuristics for solving thebetweenness problem.

In Savic (2009), the author describes a genetic algorithm (GA)approach for solving the MBP. The maximum of objective functionis obtained by finding a permutation that satisfies the maximumnumber of betweenness constraints. Every considered permutationis encoded with integer representation. Genetic algorithm wastested on randomly generated instances of up to 50 elementsand 1,000 triples. Unfortunately, genetic algorithm could not verifyoptimality of obtained solutions or give any estimation about solu-tion quality. The advantage of that approach is a quite short run-ning time.

A. Savic et al. / European Journal of Operational Research 206 (2010) 522–527 523

In 1979, Opatrny showed that the problem of deciding whetherthe n points can be totally ordered while satisfying the nt between-ness constraints is NP-complete (Opatrny, 1979). Furthermore, theproblem is MAX SNP complete (and also APX-complete), and forevery a > 47=48 finding a total order that at least a � nt of the nt tri-ples satisfies betweenness constraints is NP-hard (even if all theconstraints are satisfiable) (Chor and Sudan, 1998).

For better explanation, in complexity theory the class APX is theset of NP optimization problems that allow polynomial-timeapproximation algorithms with approximation ratio bounded bya constant. A problem is said to be APX-hard if there is a polyno-mial-time approximation reduction scheme from every problemin APX to that problem, and to be APX-complete if the problemis APX-hard and also in APX. The MAX SNP class of problems cap-tures the core of hardest problems in APX. Namely, APX is the clo-sure of MAX SNP with respect to some approximation preservingreductions. For more detailed information about MAX SNP andAPX complexity classes, see Papadimitriou and Yannakakis (1991).

The probability that a specific constraint ða; b; cÞ is satisfied byrandomly chosen order of elements is 1/3, since exactly two ofthe six permutations on fa; b; cg have b in the middle. So, asmentioned in Chor and Sudan (1998), the expected number ofconstraints satisfied by a random order is at least 1/3 of nt

constraints.Chor and Sudan (1998) presented a polynomial-time algorithm

for solving a related problem that either determines that there isno feasible solution or finds a total order that satisfies at least1/2 of the nt triples. The algorithm transforms that problem intoa set of quadratic inequalities and solves a semidefinite relaxationof those inequalities in Rn. The n solution points are then projectedon a random line through the origin. The claimed performanceguarantee is shown using simple geometric properties of the semi-definite programming solution.

Similarly, the cyclic ordering problem asks for a total order < ofA such that for each ða; b; cÞ 2 C, either a < b < c or b < c < a orc < a < b (Guttmann and Maucher, 2006). The cyclic orderingproblem is also NP-complete (Galil and Megiddo, 1977) and itsapplication can be found in qualitative spatial reasoning (Isli andCohn, 2000). Note that the problem where a < b < c or a < c < bcan be solved with linear time complexity by using topologicalsorting (Knuth, 2006).

A specific approach to a problem where all betweenness con-straints should be satisfied can be found in Goerdt (2009). Ifnt=n > 2:55 then the MBP has an optimal solution less then nt withhigh probability. Also, if nt=n < 1=6 then the MBP has an optimalsolution equal to nt with high probability.

As can be seen from previous works, this problem is shown tobe very hard. Up to now, there was no linear or quadratic formula-tion of the problem in general sense. Furthermore, there was nomethod for solving the MBP to optimality. This work represents amajor step in that direction.

In the following section a new MILP formulation of the problemwill be presented. This formulation is implemented by using CPLEXsolver and then applied on randomly generated instances from theliterature. On the same instances the total enumeration method istested and the obtained results are compared with the results ofCPLEX. The detailed description of this comparison is presentedin Section 3.

2. A mixed integer linear programming formulation

It is useful to represent discrete optimization problems as inte-ger programming problems in order to use different well-knownoptimization techniques for their solving (Caumond et al., 2009;Conde, 2009; Hertz et al., 2010). Following that idea we have used

the CPLEX solver on the new mixed integer linear programmingformulation for the MBP described below.

Let n be a number of elements in finite set A. Without loss of thegenerality we can assume A ¼ f1; . . . ;ng. Let nt be a number of tri-ples in collection C. Let us denote the ith triple from the collection Cwith ðai; bi; ciÞ; i ¼ 1; . . . ;nt . Also, let a 2 ð0;1�.

Suppose the function f is known in advance, f : f1; . . . ;ng !f1; . . . ;ng. Let us define vector of variables ðx; y; z;uÞ.

xj ¼f ðjÞ � 1

nj ¼ 1; . . . ; n; ð1Þ

and for every i ¼ 1; . . . ; nt let us define

yi ¼1; f ðaiÞ < f ðbiÞ < f ðciÞ;0; otherwise;

�ð2Þ

zi ¼1; f ðaiÞ > f ðbiÞ > f ðciÞ;0; otherwise;

�ð3Þ

ui ¼ yi _ zi ¼1; f ðaiÞ > f ðbiÞ > f ðciÞ _ f ðaiÞ < f ðbiÞ < f ðciÞ;0; otherwise:

�ð4Þ

Obviously, yi and zi cannot simultaneously have value 1 becauseof ui ¼ yi þ zi for every i. Now, a mathematical model can be formu-lated as follows:

maxXnt

i¼1

ui ð5Þ

subject to

ui ¼ yi þ zi; i ¼ 1; . . . ;nt; ð6Þ

xai� xbi

þ yi 6 1� an; i ¼ 1; . . . ;nt ; ð7Þ

xbi� xci

þ yi 6 1� an; i ¼ 1; . . . ;nt ; ð8Þ

� xaiþ xbi

þ zi 6 1� an; i ¼ 1; . . . ;nt ; ð9Þ

� xbiþ xci

þ zi 6 1� an; i ¼ 1; . . . ;nt ; ð10Þ

xj 2 ½0;1�; j ¼ 1; . . . ;n; ð11Þyi; zi;ui 2 f0;1g; i ¼ 1; . . . ; nt: ð12Þ

As can be seen, there are n real variables, 3 � nt binary variablesand 5 � nt constraints. Parameter a is introduced in order to a

n begreater than a round-off error so a must be greater than 0. Anupper bound on a;a 6 1, is imposed so that the system (6)–(12) al-ways has a nonempty solution set.

Now, we can define ObjMILPðx; y; z;uÞ ¼Pnt

i¼1ui subject to (6)–(12).

Let us show that the solution of this MILP formulation is thesolution of the MBP.

Lemma 1. Let f be a 1–1 function f : f1; . . . ;ng ! f1; . . . ;ng. Thenthere is a solution ðx; y; z;uÞ of system (6)–(12) such thatObjMILPðx; y; z;uÞP Objðf Þ.

Proof. Let vector of variables ðx; y; z; uÞ be defined as (1)–(4). Wewill prove that this vector satisfies system (6)–(12) andObjMILPðx; y; z;uÞP Objðf Þ.

As we have seen earlier, the definition of variables ui ¼ yi _ zi,implies the constraint (6). It is easy to see that xj ¼ f ðjÞ�1

n 6n�1

n 6 1and f ðjÞP 1 implies xj ¼ f ðjÞ�1

n P 0. Now, constraint (11) is satis-fied. From (2)–(4) it is obvious that yi; zi;ui are binary variables asrequired by constraint (12).

From f ðaiÞ < f ðbiÞ < f ðciÞ follows xai < xbi< xci . From the defi-

nition of the variables x there follows xai þ 1n 6 xbi

6 xci � 1n. Then

xbi� xai P 1

n P an, and multiplied by �1 gives xai � xbi

6 � an. Con-

sidering yi 6 1, we have xai � xbiþ yi 6 � a

nþ 1, which provesconstraint (7).

524 A. Savic et al. / European Journal of Operational Research 206 (2010) 522–527

If f ðaiÞP f ðbiÞ _ f ðbiÞP f ðciÞ, then, since yi ¼ 0 and xai 6n�1

n ¼1� 1

n 6 1� an. From xbi

P 0 follows �xbi6 0. Now, xai � xbi

þ yi ¼xai � xbi

6 1� an, which proves constraint (7).

The constraints (8)–(10) can be proved in a similar way.

From (4), we have f ðaiÞ > f ðbiÞ > f ðciÞ _ f ðaiÞ < f ðbiÞ < f ðciÞ )ui ¼ 1. Then, there exist at least Objðf Þ variables ui that equal 1.Since variables ui are binary,

Pnti¼1ui is equal to

Pnti¼1;ui¼11. Now,

ObjMILPðx; y; z;uÞP Objðf Þ. h

Lemma 2. Let ðx; y; z;uÞ be a solution to (5)–(12). Then there is a func-tion f : f1; . . . ;ng ! f1; . . . ;ng such that Objðf ÞP ObjMILPðx; y; z;uÞ.

Proof. Note that in this lemma we do not require that function f is1–1. From the proposition of the lemma we know that ðx; y; z;uÞsatisfies all constraints (6)–(12). For every i 2 f1;2;3; . . . ;ng, letf ðiÞ ¼ k if xi is greater than exactly k� 1 other elements xj. It is easyto see that if xp > xq, then f ðpÞ > f ðqÞ.

Since ui are binary variables, there are exactly ObjMILPðx; y; z;uÞof them equal to 1. For each i such that ui ¼ 1 ¼ yi þ zi, it followsthat yi ¼ 1 or zi ¼ 1. If yi ¼ 1, from constraints (7) and (8) we have�xai þ xbi

P an > 0 and �xbi

þ xci P an > 0, respectively. Then,

xai < xbi< xci so f ðaiÞ < f ðbiÞ < f ðciÞ, which means that the ith

betweenness constraint is satisfied. Analogously, if zi ¼ 1, thenf ðaiÞ > f ðbiÞ > f ðciÞ, which also means that the ith betweennessconstraint is satisfied.

Previously defined function f has at least ObjMILPðx; y; z;uÞsatisfied betweenness constraints, so Objðf ÞP ObjMILPðx; y; z;uÞ.h

Lemma 3. Let f be function f : f1; . . . ;ng ! f1; . . . ;ng. Then there is a1–1 function g : f1; . . . ;ng ! f1; . . . ;ng such that ObjðgÞP Objðf Þ.

Proof. Let s be a permutation of {1,. . .,n} such thatf ðsð1ÞÞ 6 f ðsð2ÞÞ 6 . . . 6 f ðsðnÞÞ and g ¼ s�1. Then g is a permuta-tion. Let f ðiÞ < f ðjÞ and i ¼ sðkÞ; j ¼ sðlÞ. From f ðsðkÞÞ < f ðsðlÞÞ it fol-lows k < l, so that s�1ðiÞ < s�1ðjÞ and finally gðiÞ < gðjÞ. It meansthat for every triple ðai; bi; ciÞ; f ðaiÞ < f ðbiÞ < f ðciÞ implies gðaiÞ <gðbiÞ < gðciÞ and f ðaiÞ > f ðbiÞ > f ðciÞ implies gðaiÞ > gðbiÞ > gðciÞ soObjðgÞP Objðf Þ. h

Theorem 1. Function f satisfies the maximum number of between-ness constraints if and only if there is an optimal solution ðx; y; z;uÞof (5)–(12).

Proof. The direction ð)Þ can be easily deduced from Lemma 1. Theother direction ð(Þ directly follows from Lemmas 2 and 3. h

Let us demonstrate CPLEX behavior on the MBP instance fromExample 1 with a ¼ 0:5 and a ¼ 0:9.

Example 2. Let set A and collection C be as in Example 1. Then onepossible optimal solution, after solving MILP (5)–(12), obtained byCPLEX, for a ¼ 0:5, is:

x1 ¼ 0:3; x2 ¼ 0:0; x3 ¼ 0:1; x4 ¼ 0:4; x5 ¼ 0:2;

y1 ¼ 0; y2 ¼ 0; y3 ¼ 0; y4 ¼ 1; y5 ¼ 1;

z1 ¼ 1; z2 ¼ 1; z3 ¼ 0; z4 ¼ 0; z5 ¼ 0;

u1 ¼ 1; u2 ¼ 1; u3 ¼ 0; u4 ¼ 1; u5 ¼ 1:

Objective value is 4.

Example 3. Let set A and collection C be as in Example 1. Then onepossible optimal solution, with objective value 4, after solvingMILP (5)–(12), obtained by CPLEX, for a ¼ 0:9, is:

x1 ¼ 0:18; x2 ¼ 0:54; x3 ¼ 0:36; x4 ¼ 0:0; x5 ¼ 0:54;y1 ¼ 1; y2 ¼ 1; y3 ¼ 0; y4 ¼ 0; y5 ¼ 0;z1 ¼ 0; z2 ¼ 0; z3 ¼ 1; z4 ¼ 1; z5 ¼ 0;u1 ¼ 1; u2 ¼ 1; u3 ¼ 1; u4 ¼ 1; u5 ¼ 0:

Note that, in order to simplify formulation and reduce numbers ofvariables and constraints, we can omit variables ui. Therefore, thecompact formulation is:

maxXnt

i¼1

ðyi þ ziÞ ð13Þ

subject to

xai� xbi

þ yi 6 1� an; i ¼ 1; . . . ;nt ; ð14Þ

xbi� xci

þ yi 6 1� an; i ¼ 1; . . . ;nt ; ð15Þ

� xaiþ xbi

þ zi 6 1� an; i ¼ 1; . . . ;nt ; ð16Þ

� xbiþ xci

þ zi 6 1� an; i ¼ 1; . . . ;nt ; ð17Þ

xj 2 ½0;1�; j ¼ 1; . . . ;n; ð18Þyi; zi 2 f0;1g; i ¼ 1; . . . ;nt: ð19Þ

All numerical experiments are performed on MILP formulation(13)–(19). Now, there are n real variables, 2 � nt binary variablesand 4 � nt constraints.

3. Experimental study

All computations were executed on 2.8 GHz PC with 1 GB RAMunder the Windows operating system. For experimental testings,we used randomly generated instances from Savic (2009). Thoseinstances include different numbers of elements in setAðn ¼ 10;11;12;15;20;30Þ and different numbers of triples in C(ranging from 20 to 300). To gain more reliability of the testingprocess, for every pair of n and nt we generated three additional in-stances by using different random seeds. To instances’ names fromSavic (2009), character ‘‘a” is appended, while newly generated in-stances’ names are ended by characters ‘‘b”, ‘‘c” and ‘‘d”. Randomseeds used for ‘‘a”, ‘‘b”, ‘‘c”, and ‘‘d” instances are 11111, 33333,55555, and 77777, respectively. Instances were generated on thefollowing principles. At the beginning of the generation process,collection C was empty. The number of elements ðnÞ and numberof triples ðntÞwere selected a priori. After this, in every step a triple(i.e. a variation of 3 different elements from set f1; . . . ;ng) was ran-domly generated. After the triple generation, it was checked if thattriple already existed in the collection of generated triples. If therewas no such a triple, the generated triple was included in C and theprocess was continued until the number of generated triples be-came equal to the a priori selected number nt .

In order to show effectiveness of the proposed MILP formula-tion, we tested it on those instances by using CPLEX 10.1 solver.For evaluation of the CPLEX solutions quality, we implementedthe total enumeration technique. To do this, it was necessary togenerate all permutations of set f1; . . . ;ng and, after that, to findwhich one of them satisfied the maximum number of betweennessconstraints ðpðaÞ < pðbÞ < pðcÞ _ pðaÞ > pðbÞ > pðcÞÞ for triplesða; b; cÞ from the collection C. For any specific permutation it wasnot time consuming to check how many betweenness constraintsit satisfied, but the number of permutations grew rapidly with in-crease of n. Already for n ¼ 12 the number of permutations was12! ¼ 479001600, so, accordingly, total enumeration solved tooptimality only those instances where the number of elements ofset A was smaller than or equal to 12.

Table 1Experimental results for different values of parameter a.

Instance name opt CPLEX a ¼ 0:01 CPLEX a ¼ 0:1 CPLEX a ¼ 0:5 CPLEX a ¼ 0:9

sol t (seconds) sol t (seconds) sol t (seconds) sol t (seconds)

mbp-10-20a 16 16 0.406 16 0.61 16 0.312 16 0.407mbp-10-50a 29 29 7200.11 29 7200.67 29 7200.34 29 5480.14mbp-10-100a 50 47 7200.875 48 7200.937 45 7200.984 46 7200.984mbp-11-20a 14 14 0.64 14 0.703 14 0.906 14 0.718mbp-11-50a 33 33 7200.53 32 7200.52 33 7200.75 33 1457.09mbp-11-100a 55 55 7200.937 52 7200.984 55 7200.906 52 7200.984mbp-12-20a 17 17 0.265 17 0.246 17 0.375 17 0.234mbp-12-50a 34 34 7200.58 33 7200.48 32 6295.25 34 7200.7mbp-12-100a 56 56 7201.015 56 7201.25 48 7201.015 47 7201.062mbp-15-30a 26 26 14.328 26 3.765 26 1.218 26 2.015mbp-15-70a – 46 7201.359 46 7201.421 42 7201.812 44 7201.281mbp-15-200a – 100 7200.703 76 7200.656 100 7200.735 89 7200.75mbp-20-40a 37 37 0.781 37 1.078 37 0.671 37 0.343mbp-20-100a – 60 7201.031 54 7200.921 63 7201 55 7200.906mbp-20-200a – 114 7200.08 110 7200.89 111 7200.56 92 7200.55mbp-30-60a 55 55 2238.75 54 7201.531 54 7201.453 55 1704.265mbp-30-150a – 104 7200.75 106 7200.781 98 7200.781 105 7200.843mbp-30-300a – 173 7200.63 169 7200.83 168 7200.88 165 7200.547

0.01 0.1 0.5 0.9smaller instances 1 4 3 3larger instances 0 1 1 2

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

num

ber o

f ins

tanc

es

alpha

smaller instances larger instances

Fig. 1. Number of instances where enumeration technique was better than CPLEX.

A. Savic et al. / European Journal of Operational Research 206 (2010) 522–527 525

In formulation (13)–(19) there exists parameter a. Conse-quently, we tested how different values of a influenced the CPLEXperformance. For tested values a ¼ 0:01;0:1;0:5;0:9, results are gi-ven in Table 1. For a ¼ 0:001 CPLEX could not solve the problembecause of round-off errors.

The first column in Table 1 contains the names of the instances.The instances’ names carry information about the number of ele-ments n and the number of triples nt . For example, instancembp� 11� 100 is the instance with n ¼ 11 elements in the set Aand nt ¼ 100 triples in the collection C. The second column con-tains optimal solutions which were obtained by either CPLEX or to-tal enumeration in case when the method finished its work. Thethird and fourth columns contain solution values and runningtimes of CPLEX ða ¼ 0:01Þ. In the same way, results for values0.1, 0.5, and 0.9 of parameter a are presented.

As can be seen from Table 1, CPLEX ða ¼ 0:01Þ outperforms totalenumeration. CPLEX ða ¼ 0:01Þ reached all optimal solutions ex-cept one on instances where total enumeration obtained the opti-mum. On larger instances where total enumeration did not reachthe optimum, CPLEX ða ¼ 0:01Þ gives better results in the sametime limit and in some cases it even reaches the optimum. For

other tested values of the parameter a CPLEX gave significantlyworse results, as can be seen from Fig. 1. Consequently, further tes-tings were performed only for a ¼ 0:01.

Results obtained by total enumeration, CPLEX, and GA are givenin Tables 2 and 3. In the first column the names of the instances aregiven. The second column contains optimal solutions which wereobtained by either CPLEX or total enumeration in case when themethod finished its work. In the third and fourth columns, thesolution value and running time of total enumeration are given.In the fifth and sixth columns, solution values and running timesof CPLEX ða ¼ 0:01Þ are given, respectively. There was a time lim-itation of 7,200 seconds, approximately. The last column containssolution values obtained by GA approach from Savic (2009). Allrunning times for GA are short (up to 0.5 seconds) so they are omit-ted. The mark ‘‘opt” is written if optimal value was reached. If nei-ther CPLEX nor total enumeration produced an optimal result, thesign ‘‘–” is written in the second column.

As we have already mentioned, heuristic methods could notverify optimality of obtained solutions or give a useful estimationof solution quality. Thus, results of GA could not be directly com-pared with results of exact methods.

Table 2Experimental results on smaller instances.

Instance name opt enum CPLEX a ¼ 0:01 GA

sol t (seconds) sol t (seconds)

mbp-10-20a 16 opt 2.234 opt 0.406 16mbp-10-20b 16 opt 1.828 opt 0.344 16mbp-10-20c 14 opt 1.859 opt 0.578 14mbp-10-20d 16 opt 1.765 opt 0.375 16mbp-10-50a 29 opt 3.671 opt 7200.11 29mbp-10-50b 29 opt 3.14 opt 7203.312 29mbp-10-50c 28 opt 3.14 opt 7202.593 28mbp-10-50d 30 opt 3.078 opt 7202.687 30mbp-10-100a 50 opt 6.359 47 7200.875 50mbp-10-100b 50 opt 5.375 49 7201 50mbp-10-100c 53 opt 5.406 opt 7201 53mbp-10-100d 57 opt 5.343 opt 7200.937 57mbp-11-20a 14 opt 26.203 opt 0.64 14mbp-11-20b 16 opt 20.937 opt 0.203 16mbp-11-20c 17 opt 21.359 opt 0.062 17mbp-11-20d 17 opt 21.421 opt 0.125 17mbp-11-50a 33 opt 41.171 opt 7200.53 33mbp-11-50b 32 opt 35.656 opt 7202.421 32mbp-11-50c 34 opt 35.64 opt 7202.312 34mbp-11-50d 30 opt 35.844 opt 7202.61 30mbp-11-100a 55 opt 72.625 opt 7200.937 55mbp-11-100b 56 opt 60.671 opt 7200.953 56mbp-11-100c 59 opt 61.015 57 7201.078 59mbp-11-100d 52 opt 60.703 51 7201.031 52mbp-12-20a 17 opt 331.156 opt 0.265 17mbp-12-20b 17 opt 273.14 opt 0.515 17mbp-12-20c 18 opt 273.859 opt 0.11 18mbp-12-20d 17 opt 279.109 opt 0.328 17mbp-12-50a 34 opt 520.125 opt 7200.58 33mbp-12-50b 33 opt 443 32 7201.656 33mbp-12-50c 36 opt 443.922 opt 6401.687 36mbp-12-50d 34 opt 445.25 opt 7201.687 34mbp-12-100a 56 opt 861 opt 7201.015 56mbp-12-100b 57 opt 738.406 45 7201.094 56mbp-12-100c 55 opt 739.906 opt 7201 55mbp-12-100d 57 opt 743.687 opt 7201.11 57

Table 3Experimental results on larger instances.

Instance name opt enum CPLEX a ¼ 0:01 GA

sol t (seconds) sol t (seconds)

mbp-15-30a 26 24 7200.015 opt 14.328 25mbp-15-30b 27 22 7200.016 opt 0.781 27mbp-15-30c 26 25 7200.016 opt 0.875 26mbp-15-30d 25 24 7200.016 opt 9.125 25mbp-15-70a – 43 7200.015 46 7201.359 46mbp-15-70b – 40 7200.016 48 7201.39 49mbp-15-70c – 45 7200.016 49 7201.375 49mbp-15-70d – 43 7200.015 44 7201.359 46mbp-15-200a – 96 7200.015 100 7200.703 105mbp-15-200b – 98 7200.016 101 7200.797 103mbp-15-200c – 103 7200.016 99 7200.75 105mbp-15-200d – 98 7200.015 94 7200.765 105mbp-20-40a 37 24 7200.015 opt 0.781 36mbp-20-40b 37 28 7200.016 opt 4.562 36mbp-20-40c 34 22 7200.016 opt 110.953 33mbp-20-40d 34 22 7200.015 opt 144.219 32mbp-20-100a – 54 7200.015 60 7201.031 65mbp-20-100b – 60 7200.016 69 7200.969 66mbp-20-100c – 50 7200.016 67 7201.25 65mbp-20-100d – 48 7200.015 59 7201.047 66mbp-20-200a – 100 7200.015 114 7200.08 113mbp-20-200b – 98 7200.016 105 7200.828 116mbp-20-200c – 90 7200.016 109 7200.797 115mbp-20-200d – 84 7200.015 109 7200.844 116mbp-30-60a 55 30 7200.015 opt 2238.75 51mbp-30-60b – 33 7200.015 53 7201.36 51mbp-30-60c – 28 7200.015 54 7201.406 50mbp-30-60d – 33 7200.016 53 7201.468 49mbp-30-150a – 61 7200.015 104 7200.75 102mbp-30-150b – 67 7200.016 101 7200.735 98mbp-30-150c – 60 7200.015 103 7200.672 99mbp-30-150d – 68 7200.015 103 7200.75 103mbp-30-300a – 111 7200.015 173 7200.63 173mbp-30-300b – 126 7200.015 168 7200.672 167mbp-30-300c – 112 7200.015 174 7200.75 172mbp-30-300d – 114 7200.015 154 7200.641 186

526 A. Savic et al. / European Journal of Operational Research 206 (2010) 522–527

As can be seen from Table 2, total enumeration finished its workand produced optimal solutions for all instances up to n ¼ 12 ele-ments and arbitrary number of triples nt . With given time limit of7,200 seconds, total enumeration could not finish its work on lar-ger instances, for n > 12 (Table 3). Therefore, the produced resultsfor those instances are suboptimal.

On the other hand, CPLEX can handle instances with largernumber of elements. For example, in case of mbp� 30� 60a in-stance, CPLEX obtained optimal solutions in a reasonable short per-iod of time. Results from Tables 2 and 3 show that CPLEX has somedifficulties with a large number of triples. Altogether, it is clear thatCPLEX ða ¼ 0:01Þ produced significantly better results than totalenumeration, because it reached 30 out of 36 optimal solutionswhere total enumeration obtained the optimum. On larger in-stances where total enumeration did not reach the optimum, in34 out of 36 cases CPLEX gave better results in the same time limitand in 9 cases it even reached the optimum.

4. Conclusions

This paper is devoted to the maximum betweenness problem.We introduced its mixed integer linear programming formulation.Also, we proved the correctness of the corresponding formulation.Numbers of variables and constraints were relatively small com-pared to dimension of the problem.

We carried out numerical experiments using randomly gener-ated instances from the literature. Numerical results showed thatCPLEX solutions, based on this MILP formulation, are significantly

better than results of the total enumeration especially on the in-stances with large sets and a moderate number of triples.

This research can be extended in several ways. It would bedesirable to investigate the application of some exact methodusing the proposed MILP formulation. Further research also shouldbe directed at solving similar problems and testing on parallelcomputers.

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