# Solving the quadratic assignment problem by means of general purpose mixed integer linear programming solvers

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1 Solving the quadratic assignment problem by means of general purpose mixed integer linear programg solvers Huizhen Zhang Cesar Beltran-Royo Liang Ma 19/04/2010 Abstract The Quadratic Assignment Problem (QAP) can be solved by linearization, where one formulates the QAP as a mixed integer linear programg (MILP) problem. On the one hand, most of these linearization are tight, but hardly exploited within a reasonable computing time because of their size. On the other hand, Kaufman and Broeckx formulation [1] is the smallest of these linearizations, but very weak. In this paper, we analyze how Kaufman and Broeckx formulation can be tightened to obtain better QAP-MILP formulations. As we show in our numerical experiments, these tightened formulations remain small but computationally effective in order to solve the QAP by means of general purpose MILP solvers. Key words: quadratic assignment problem, mixed integer linear programg. 1. Introduction In 1957, the Quadratic Assignment Problem (QAP) was introduced by Koopmans and Beckmann [2] as a mathematical model for the location of a set of indivisible economic activities. In general, QAP can be described as a one-to-one assignment problem of n facilities to n locations, which imizes the sum of the total quadratic interaction cost, the flow between the facilities multiplied with their distances, and the total linear cost associated with allocating a facility to a certain location. Consider the set N = {1, 2,, n} and three n n matrices F = (f ij ), D = (d ij ) and C = (c ij ), the quadratic assignment problem with coefficient matrices F, D and C, shortly denoted by QAP, can be stated as follows: Statistics and Operations Research, Rey Juan Carlos University, Madrid, Spain. Corresponding author: Statistics and Operations Research, Rey Juan Carlos University, Madrid, Spain. School of Management,University of Shanghai for Science and Technology,Mail box 459, 516 Jungong Road, Shanghai ,China 1

2 f ik d jl x ij x kl + c ij x ij (1.1) where X = { x x ij = 1 i N (1.2) j=1 x ij = 1 j N (1.3) i=1 x ij {0, 1} i, j N }, (1.4) f ik denotes the amount of flow between facilities i and k, d jl denotes the distance between locations j and l, and c ij denotes the cost of locating facility i at location j. x ij = 1 if facility i is assigned to location j, otherwise, x ij = 0. In [3] a more general expression of (1.1) was introduced by using a four-dimensional array ˆq ijkl instead of the flow-distance products f ik d jl : ˆq ijkl x ij x kl + c ij x ij. (1.5) Obviously, the linear terms can be eliated by setting q ijij := ˆq ijij + c ij : q ijkl x ij x kl. (1.6) Without loss of generality, we can assume that q ijkl are nonnegative. If they are negative, we add a sufficiently large constant to all q ijkl, which does not change the optimal permutation and increase the objective function by n 2 times the added constant. Over the years, the QAP has drawn the researcher s attention worldwide and extensive research has been done. From the theoretical point of view, it is because of the high computational complexity: QAP is NP-hard, and even finding an ε-approximate solution is a hard problem [4]. Moreover, many well-known classical combinatorial optimization problems such as the traveling salesman problem, the graph partitioning problem, the maximum clique problem can be reformulated as special cases of the QAP, see [5] and [6] for details. From the practical point of view, it is because of the diversified applications of the QAP. The QAP has been applied in many fields such as backboard wiring [7], typewriter keyboards and control panels design [8], scheduling [9], numerical analysis [10], storage-and-retrieval [11], and many others. More advances in theoretical aspects, solution methods and applications of the QAP can be found in [5, 12, 13, 14, 15, 16, 17, 18]. One common approach to solve the QAP is to linearize it, that is, reformulate it as a pure or mixed integer linear programg problem. Lawler [3] replaced the quadratic terms x ij x kl in the objective function by n 4 variables y ijkl = x ij x kl. The main drawback of this approach is the huge number of variables. Kaufman and Broeckx [1] proposed a mixed integer linearization with n 2 binary variables and n 2 real variables (see Section 2.2.). Although this is the smallest QAP linearization, its LP relaxation is known to be usually weak. Recently, 2

3 Xia and Yuan [19, 20, 21] tightened Kaufman-Broeckx formulation, basically, by introducing new constraints based on the Gilmore-Lawler constants (see Section 2.3.). In this paper we will concentrate on linerizations derived from the Kaufman-Broeckx formulation, which we will call the Kaufman-Broeckx (KB) family of formulations. The objective of this paper is to study the performance of the KB family of formulations when one uses a general purpose mixed integer linear programg solver to solve the QAP. This objective will be developed as follows. In section 2, we will review the Gilmore-Lawler bound, the Kaufman-Broeckx linearization and the Xia-Yuan linearization. In section 3, we will enhance the KB family of formulations with a new QAP linearization. In section 4, on the one hand we will show that the Kaufman-Broeckx formulation is the weakest possible QAP linearization and on the other hand, we will study the LP relaxation of the new linearization. In section 5 we will analyze the computational performance of the KB family of formulations. 2. The Kaufman-Broeckx family of linearizations Here we shortly review the Gilmore-Lawler bound, and the two current smallest QAP linearizations, namely, Kaufman-Broeckx linearization and Xia-Yuan linearization. As we already said, Xia and Yuan have derived his formulation from Kaufman-Broeckx formulation. In this paper we will study this type of formulations which we will call the Kaufman-Broeckx (KB) family of formulations Gilmore-Lawler bound The Gilmore-Lawler bound was derived by Gilmore [22] and Lawler [3]. Consider the Lawler QAP (1.5) with coefficient matrix Q = (q ijkl ). For each pair i, j N, solve the following linear assignment problem (LAP) and denote its optimal value as l ij : k=1 k i q ijkl x kl (2.7) l=1 l j x kl = 1, l N and l j (2.8) k i x kl = 1, k N and k i (2.9) l j x kl {0, 1}, k, l N and k i, j l (2.10) The Gilmore-Lawler bound for the QAP is given by the optimal value of the LAP of size n with cost matrix (l ij + q ijij ): GLB : (l ij + q ijij )x ij (2.11) 3

4 2.2. Kaufman-Broeckx linearization Kaufman and Broeckx [1, 23] introduced n 2 continuous variables z ij = x ij q ijkl x kl and n 2 + 2n constraints, to derive the following QAP linearization: KBL : s. t. z ij z ij (2.12) q ijkl x kl ã ij (1 x ij ) i, j N (2.13) z ij 0 i, j N (2.14) where ã ij = 2.3. Xia-Yuan linearization q ijkl. (2.15) Xia and Yuan [19, 20, 21] tightened Kaufman and Broeckx formulation, basically, by introducing new constraints based on the Gilmore-Lawler constants l ij : XY L : s. t. z ij ( z ij + q ijij x ij ) (2.16) k=1 k i q ijkl x kl ā ij (1 x ij ) i, j N (2.17) l=1 l j z ij l ij x ij i, j N (2.18) where ā ij = max{ k=1 k i q ijkl x kl : x kl satisfy (2.8) (2.10)} for i, j N (2.19) l=1 l j In [20] it is also proved that Xia-Yuan bound is stronger than the Gilmore-Lawler bound: Theorem 2.1 Let f RXY L be the optimal solution of the LP relaxation of (XY L) and f GLB the optimal solution of (GLB). Then: f RXY L f GLB. 4

5 3. A new linearization in Kaufman-Broeckx family Now we present another way to tighten Kaufman-Broeckx formulation by using the Gilmore- Lawler constants l ij. This formulation turns out to be similar to Xia-Yuan formulation, as we will see in Proposition 3.1. However, this new formulation may obtain slightly tighter LP bounds (see in Table 4, instances labeled by Bur ). This new QAP formulation, which we call Gilmore-Lawler Linearization (GLL), is as follows: GLL : s. t. z ij [z ij + (q ijij + l ij )x ij ] (3.20) q ijkl x kl a ij (1 x ij ) (l ij + q ijij )x ij i, j N (3.21) z ij 0 i, j N (3.22) where { a ij = max q ijkl x kl : } x X (3.23) Theorem 3.2 To solve GLL is equivalent to solve the QAP. Proof: Let us consider F QAP, f QAP (x) and fqap, the QAP feasible set, the QAP objective function and the QAP optimum, respectively. Analogously, we consider F GLL, f GLL (x, z) and fgll. Firstly, let us see that any given solution (x, z) F GLL can be projected to a solution x F QAP with the same objective value. We consider the following two cases: i). If x ij = 1, then z ij n k,l=1 q ijklx kl (l ij +q ijij )x ij by (3.21). Considering the definition of f GLL, we have z ij + (l ij + q ijij )x ij = q ijkl x kl = k,l=1 q ijkl x kl x ij. ii). If x ij = 0, then z ij + (l ij + q ijij )x ij = 0 by (3.21), (3.22), and the definition of f GLL. Therefore, we have z ij + (l ij + q ijij )x ij = q ijkl x kl x ij. Thus, we have seen that f GLL (x, z) = f QAP (x) as we wanted to see. This implies, f GLL f QAP. Secondly, let us see that any given solution x F QAP can be lifted to a solution (x, z) F GLL with the same objective value. Given a x F QAP, we define z ij = n k,l=1 q ijklx kl (l ij + q ijij ) if x ij = 1 and z ij = 0, otherwise. Obviously, (x, z) F GLL. Furthermore, for any i, j N, z ij + (l ij + q ijij )x ij = x ij q ijkl x kl = q ijkl x kl x ij, k,l=1 5 k,l=1 k,l=1 k,l=1

6 whether x ij = 1 or x ij = 0. Thus, we have seen that f QAP (x) = f GLL (x, z) as we wanted to see. This implies, f QAP f GLL and therefore f QAP = f GLL All in all, we have proved that if (x, z ) is optimal for GLL, then x is optimal for QAP. Proposition 3.1 GLL can be written as s. t. ẑ ij ẑ ij (3.24) q ijkl x kl a ij (1 x ij ) i, j N (3.25) ẑ ij (q ijij + l ij )x ij i, j N (3.26) Inequalities (3.21) and (3.22) are equivalent to the following inequalities, respecti- Proof: vely z ij + (q ijij + l ij )x ij q ijkl x kl a ij (1 x ij ) i, j N (3.27) z ij + (q ijij + l ij )x ij (q ijij + l ij )x ij i, j N (3.28) To finish the proof we only need to define ẑ ij = z ij + (q ijij + l ij )x ij, and replace z ij + (q ijij + l ij )x ij by ẑ ij in (3.20), (3.27) and (3.28). 4. Bounds based on the Kaufman-Broeckx family of formulations It is well known that the LP relaxation of the Kaufman-Broeckx linearization gives a poor QAP bound. In this section, we prove that it gives the worst possible QAP bound, that is, 0. We also prove that much better bounds can be obtained by combining Kaufman-Broeckx linearization with Gilmore-Lowler constants l ij, as in linerizations XY L and GLL. We name RGLL, RXY L and RKBL the LP relaxations of GLL, XY L and KBL, respectively. Theorem 4.3 Let frkbl linearization. Then be the optimal value of the LP relaxation of the Kaufman-Broeckx f RKBL = 0. Proof: Let F RKBL be the feasible set of RKBL. Considering that the objective value of RKBL is always positive, to prove this result, it is enough to see that there exists one solution (x 0, z 0 ) F RKBL such that f RKBL (x 0, z 0 ) = 0. As such point, we take x 0 ij = 1 n and z0 ij = 0 for all i, j N. Firstly, let us see that (x 0, z 0 ) F RKBL. Obviously, x 0 X and z ij 0 0. Furthermore, for each i, j N and for any 6

7 n > 2, we have = 1 n ( q ijkl 1 n ã ij(1 1 n ) q ijkl ã ij (n 1)) = 1 n (ã ij ã ij (n 1)) = 2 n n < 0. ã ij Therefore, (x 0, z 0 ) also satisfies (2.13) and thus (x 0, z 0 ) F RKBL. On the other hand, it is clear that f RKBL (x 0, z 0 ) = 0. Theorem 4.4 Let frgll, f GLB and f RKBL and RKBL, respectively. Then be the optimal objective values of RGLL, GLB frgll fglb frkbl (4.29) Proof: We denote by RGLB the LP relaxation of GLB. Considering that GLB is a linear assignment problem we have that frglb = f GLB. One can write RGLB as RGLB : z ij (4.30) s. t. (1.2) (1.3) z ij (l ij + q ijij )x ij i, j N (4.31) 0 x ij 0 i, j N (4.32) Thus, to prove this theorem, we only need to prove f RGLL f RGLB and f RGLB f RKBL. i). Firstly, let us see frgll f RGLB. By Proposition 3.1 it is clear that RGLB is a relaxation of RGLL. Therefore frgll f RGLB. ii). Secondly, let us see frglb f RKBL. By constraints (4.31) we have z ij 0 for i, j N. Therefore, frglb 0 and by Theorem 4.3 we can conclude that f RGLB f RKBL. Lemma 4.1 For a given QAP instance, if f ii = 0 and d ii = 0 for all i N, then Proof: Trivial. c ij + k=1 k i q ijkl = l=1 l j q ijkl i, j N (4.33) c ij + ā ij = a ij i, j N (4.34) 7

8 Proposition 4.2 Given a QAP instance defined by flow matrix F and distance matrix D, if f ii = 0 and d ii = 0 for all i N, then XY L can be written as s. t. ẑ ij ẑ ij (4.35) q ijkl x kl a ij (1 x ij ) + c ij (1 x ij ) i, j N (4.36) ẑ ij (q ijij + l ij )x ij i, j N (4.37) Proof: If f ii = 0 and d ii = 0 for all i N, we have q ijij = c ij and n k=1 q ijkj + n l=1 q ijil = k i l j 0. Furthermore, constraint (2.17) can be rewritten as follows (2.17) z ij + c ij x ij k=1 k i q ijkl x kl ā ij (1 x ij ) + c ij x ij l=1 l j (By Lemma 4.1) (4.38) z ij + q ijij x ij q ijkl x kl (a ij c ij )(1 x ij ) (4.39) z ij + q ijij x ij Similarly, constraint (2.18) can be rewritten as follows q ijkl x kl a ij (1 x ij ) + c ij (1 x ij ) (4.40) (2.18) z ij + c ij x ij l ij x ij + c ij x ij z ij + q ijij x ij (l ij + q ijij )x ij (4.41) Now, we can replace z ij + q ijij x ij by ẑ ij in (2.16), (4.40) and (4.41), to obtain the following formulation equivalent to XY L: s. t. ẑ ij ẑ ij (4.42) q ijkl x kl a ij (1 x ij ) + c ij (1 x ij ) i, j N (4.43) ẑ ij (l ij + q ijij )x ij i, j N (4.44) Theorem 4.5 Given a QAP instance defined by flow matrix F and distance matrix D, if f ii = 0 and d ii = 0 for all i N, then f RXY L f RGLL, where f RXY L and f RGLL are the optimal objective values of RXY L and RGLL, respectively. Proof: From Propositions 3.1 and 4.2 it is clear that the feasible set of formulation GLL contains the feasible set of formulation XY L. Furthermore, the two formulations have the same objective function. 8

9 Corollary 4.1 Given a QAP instance defined by flow matrix F and distance matrix D, if f ii = 0, d ii = 0 and c ii = 0 for all i N, then Proof: From Propositions 3.1 and 4.2. f RXY L = f RGLL. 5. Numerical experiments The objective of this section is to assess the effectiveness of the small QAP linearizations presented in this paper. We present the experimental results obtained with formulations GLL, KBL and XY L used to solve a representative set of instances from QAPLIB [24]. Furthermore, we compare these results with formulation IP QAP R III presented in [25]. To the best of our knowledge, formulation IP QAP R III is the best one in order to solve the QAP by a general purpose MILP solver. The CPU time limit was set to seconds. The experiments were conducted on a laptop with a processor Intel Core Duo 2.80GHz and with 3.95 GB of RAM. Cplex 11.2 (default parameters) interfaced with Matlab 2008b [26] was used to solve the QAP instances. We have to take into account that, as reported in [25], the IP QAP R III results were obtained by using a laptop with a processor Intel Pentium M 1.70GHz and with 1.23 GB of RAM. Cplex 9.0 (default parameters) interfaced with Matlab 7.0 was used to solve the QAP instances. In table 1 we describe the instances used in our test, namely, the size of the instances, the number of variables x ij, y ij and z ij, and the number of constraints. In these experiments, we studied two groups of instances: sparse instances (Chr18a-Scr20) and dense ones (Bur26a- Nug30). In table 1, we also report the density of the flow matrix F = (f ij ) (DFM), which is defined as the proportion of non-zero elements in the matrix (in %). In table 2 we report the quality of the MILP solution computed by Cplex. We present its objective function (Cost), its brand and bound gap as computed by Cplex B&B Gap = Cost B&BBound Cost Opt. Cost 100% and its optimality gap Opt. Gap = Cost Opt. Cost 100%, where Opt. Cost denotes the optimal or best known cost reported in literature. For the 31 tested instances, there are 17, 13, 20 and 9 near optimal solutions (Opt. Gap 1% ) computed by GLL, KBL, XY L and IP QAP R III, respectively. By using the same four formulations, for 6, 1, 6 and 6 instances, respectively, optimality was proved (B&B. Gap = 0%). Therefore, these results also imply that GLL and XY L are the improvements of the KBL. We observe that, regarding the solution quality, formulation XY L outperforms the other three formulations, closely followed by formulation GLL. In table 3 we report the CPU time spent by Cplex to solve the corresponding MILP and the number of branch and bound nodes. In the first group of instances (sparse instances) we observe that, regarding the CPU time, formulation IP QAP R III can compete with formulations GLL and XY L (IP QAP R III was designed to exploit the sparsity of the QAP cost matrix). We also observe that in formulation GLL, and especially in formulation KBL, Cplex stops because an out of memory. Those results have been obtained by using Cplex default parameters (the results obtained by tuning Cplex memory management parameters have been similar). Therefore, regarding the CPU time and memory requirements, formulation XY L outperforms the other three formulations. 9

10 In table 4 we present the quality of the LP bound and the CPU time required to obtain Opt. Cost LP cost it. Furthermore, we compute the LP gap = Opt. Cost 100%, where Opt. Cost is the optimal or best known objective value. In this respect, formulations GLL and XY L clearly outperform the other two formulations. As proved in Theorem 4.3, KBL always gives the worst possible gap and formulation IP QAP R III shows much longer CPU times for dense instances. Finally, we observe that formulation GLL may obtain slightly tighter bounds than formulation XY L, as it is the case for the Bur group of instances. 6. Concluding Remarks The main objective of this paper is to analyze the possibility of solving the Quadratic Assignment Problem (QAP) by means of general purpose mixed integer linear programg solvers, as for example CPLEX. The main conclusion is that for this purpose the Kaufman-Broeckx family of formulations, especially Xia-Yuan formulation, is the most effective. We have considered three members in this family. The first member of this family is the Kaufman-Broeckx linearization (KBL). As it is known, this formulation is the smallest one but very weak in general. The second member is the Xia-Yuan linearization (XYL), which tightens KBL by introducing new constraints based on the Gilmore-Lowler constants l ij. We have introduced the third member of the Kaufman-Broeckx family which we have called the Gilmore-Lowler linearization (GLL). From a theoretical point of view first, we have proved that the KBL LP bound is always 0, the worse possible bound obtained by linearization. Second, we have proved that the bound given by GLL, the new QAP formulation, is stronger than the Gilmore-Lowler bound. Third, we have proved, that under particular conditions, XYL is stronger than GLL. However, in general, we cannot say that XYL is stronger than GLL: in Table 2 we have observed that for the Bur instances the GLL LP bound is stronger than the XYL one. From a numerical point of view, we have compared Kaufman-Broeckx family of formulations to formulation IP QAP R III, a state of the art formulation used to solve the QAP with a general purpose MILP. in Table 1 we have observed that the Kaufman-Broeckx family of formulations has a very reduced size. In Table 2, we have observed that GLL and specially XYL, obtain very good feasible solutions (the optimality error is under 1% in 70% of the cases). Compared to other linearizations, the great advantage of GLL and XYL, is that they have a moderate LP gap (average around 20%) and that the LP solving time is very small (around 2 seconds) as shown in Table 4. The main drawback of these two formulations, specially in the case of GLL, is the fast saturation of the branch-and-bound tree (Table 3). Acknowledgements Thanks are due to the support of the National Natural Science Foundation of China (Grant No ), and to the support from MAEC-AECID fellowship from the Spanish government. We wish to thank the Faculty of Sciences of the University of Lisbon for providing the software Cplex We also thank the support of the grant S2009/esp-1594 from the Comunidad de Madrid (Spain) and the grant MTM C06-03 from the Spanish government. 10

11 Table 1. Instances description: DFM stands for Density of the flow matrix Ins. n DFM Nb of x Nb of z Nb of y Nb of constraints (%) GLL/KLB/XYL R-III a GLL/KLB XYL R-III Chr18a Chr20a Chr22a Chr25a P. ave. b Esc16a Esc16b Esc16c Esc32a Esc32b Esc32c P. ave Kra30a Kra30b Kra P. ave Scr Scr Scr P. ave Bur26a Bur26b Bur26c Bur26d Bur26e Bur26f Bur26g Bur26h P. ave Nug Nug Nug Nug Nug Nug Nug P. ave T. ave. c a In these tables, R-III signify IPQAPR-III. b In this paper, P. ave. is the abbreviation for partial average. c T. ave. is the abbreviation for total average. 11

12 Table 2. Quality of the mixed integer linear programg solution (Mixed) ILP Ins. Opt. or B. known Cost Cost GLL KBL XYL R-III B&B/Opt. B&B/Opt. B&B/Opt. B&B/Opt. Cost Cost Cost Gap(%) Gap(%) Gap(%) Gap(%) Chr18a / / / /0 Chr20a / / / /0 Chr22a / / / /0 Chr25a / / / /0 P. ave / / / /0 Esc16a / /0 68 0/ /0 Esc16b / / / /0 Esc16c / / / /1 Esc32a 130(B) a / / / /88 Esc32b 168(B) / / / /138 Esc32c 642(B) / /ɛ b / /12 P. ave / / / /40 Kra30a / / / /30 Kra30b / / / /23 Kra / / / /35 P. ave / / / /29 Scr / / / /0 Scr / / / /0 Scr / / / /8 P. ave / / / /3 Bur26a /ɛ /ɛ /ɛ 0 100/100 Bur26b /ɛ /ɛ /ɛ /9 Bur26c /ɛ /ɛ /ɛ /7 Bur26d /ɛ /ɛ /ɛ 0 100/100 Bur26e /ɛ /ɛ /ɛ 0 100/100 Bur26f /ɛ /ɛ /ɛ 0 100/100 Bur26g /ɛ /ɛ /ɛ 0 100/100 Bur26h /ɛ /ɛ /ɛ /2 P. ave /ɛ /ɛ /ɛ /65 Nug / / / /18 Nug / / / /13 Nug / / / /28 Nug / / /ɛ /25 Nug / / / /37 Nug / / / /40 Nug / / / /31 P. ave / / / /27 T. ave / / / /34 a B means the value is the best known solution. b ɛ means that the corresponding value is less than

13 Table 3. Cplex CPU time and number of branch and bound nodes Ins. Cplex Time (Sec.) Nb of Nodes ( 10 3 ) GLL KBL XYL R-III GLL KBL XYL R-III Chr18a (M) a Chr20a (M) Chr22a (M) Chr25a 1586(M) 2790(M) 5869(M) P. ave Esc16a (M) (*) b Esc16b 8033(M) 6879(M) 13197(M) 14400(*) Esc16c 3577(M) 3092(M) 5086(M) 14400(*) Esc32a 14400(*) 9976(M) 14400(*) 14400(*) Esc32b 11437(M) 9395(M) 14400(*) 14400(*) Esc32c 14400(*) 7025(M) 14400(*) 14400(*) P. ave (*) Kra30a 5918(M) 9632(M) 9429(M) 14400(*) Kra30b 6482(M) 10053(M) 14400(*) 14400(*) Kra (M) 10913(M) 14400(*) 14400(*) P. ave (*) Scr Scr (M) Scr (*) P. ave Bur26a 14400(*) 14400(*) 14400(*) 14400(*) Bur26b 14400(*) 14400(*) 14400(*) 14400(*) Bur26c 14400(*) 14400(*) 14400(*) 14400(*) Bur26d 14400(*) 14400(*) 14400(*) 14400(*) Bur26e 14400(*) 14400(*) 14400(*) 14400(*) Bur26f 14400(*) 14400(*) 14400(*) 14400(*) Bur26g 14400(*) 14400(*) 14400(*) 14400(*) Bur26h 14400(*) 14400(*) 14400(*) 14400(*) P. ave (*) 14400(*) 14400(*) 14400(*) Nug (M) 4622(M) 5816(M) 14400(*) Nug (M) 4937(M) 14400(*) 14400(*) Nug (M) 5648(M) 14400(*) 14400(*) Nug (M) 6527(M) 14400(*) 14400(*) Nug (M) 8190(M) 14400(*) 14400(*) Nug (M) 9262(M) 14400(*) 14400(*) Nug (M) 11529(M) 14400(*) 14400(*) P. ave (*) T. ave a Because of out of memory, Cplex stopped before the CPU time limit was reached. b The CPU time limit, seconds, was reached. 13

14 Table 4. Quality of the LP bound Linear relaxation Ins. LPcost GLL KBL XYL R-III LP CPU Time (Sec.) LPgap LPcost LPgap LPcost LPgap LPcost LPgap GLL KBL XYL R-III (%) (%) (%) (%) Chr18a ɛ a 1 1 ɛ Chr20a ɛ ɛ 1 ɛ Chr22a ɛ ɛ 1 1 Chr25a ɛ ɛ 1 1 P. ave ɛ ɛ 1 1 Esc16a ɛ 1 Esc16b ɛ 6 Esc16c ɛ 2 Esc32a ɛ 94 Esc32b Esc32c P. ave ɛ 132 Kra30a Kra30b Kra ɛ P. ave Scr ɛ ɛ ɛ ɛ Scr ɛ ɛ ɛ 2 Scr ɛ ɛ 7 P. ave ɛ ɛ ɛ 3 Bur26a ɛ Bur26b ɛ Bur26c ɛ Bur26d ɛ Bur26e ɛ Bur26f ɛ Bur26g Bur26h P. ave ɛ Nug ɛ Nug ɛ Nug ɛ Nug ɛ Nug ɛ Nug ɛ Nug P. ave ɛ T. ave ɛ a ɛ means that the corresponding value is less than

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