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Rounding and Propagation Heuristics for Mixed Integer Programming

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Jul 11, 2011 - Mixed integer programming (MIP) is to solve the optimization problem ..... gramming. Bachelor's thesis, TU Berlin, 2011. [4] T. Koch, T.
Konrad-Zuse-Zentrum fur ¨ Informationstechnik Berlin

T OBIAS ACHTERBERG1 T IMO B ERTHOLD? G REGOR H ENDEL

Rounding and Propagation Heuristics for Mixed Integer Programming

1 ?

IBM Deutschland, Germany, [email protected] Supported by the DFG Research Center M ATHEON Mathematics for key technologies in Berlin.

ZIB-Report 11-29 (July 2011)

Takustraße 7 D-14195 Berlin-Dahlem Germany

Rounding and Propagation Heuristics for Mixed Integer Programming Tobias Achterberg∗, Timo Berthold†, Gregor Hendel‡ July 11, 2011

Abstract Primal heuristics are an important component of state-of-the-art codes for mixed integer programming. In this paper, we focus on primal heuristics that only employ computationally inexpensive procedures such as rounding and logical deductions (propagation). We give an overview of eight different approaches. To assess the impact of these primal heuristics on the ability to find feasible solutions, in particular early during search, we introduce a new performance measure, the primal integral. Computational experiments evaluate this and other measures on MIPLIB 2010 benchmark instances.

1

Introduction: primal heuristics for MIP

Mixed integer programming (MIP) is to solve the optimization problem x ˜opt = argmin{cT x | Ax ≤ b, l ≤ x ≤ u, xj ∈ Z for all j ∈ J},

(1)

ˆ n (with R ˆ := R ∪ {±∞}), and with A ∈ Rm×n , b ∈ Rm , c ∈ Rn , l, u ∈ R J ⊆ N = {1, . . . , n}. In state-of-the-art MIP solvers, primal heuristics play a major role in finding and improving integer feasible solutions at an early stage of the solution process. Knowing good solutions early during optimization helps to prune the search tree and to simplify the problem via dual reductions. Further, it proves the feasibility of a problem and a practitioner might be satisfied with a solution that is proven to be within a certain gap to optimality. This article gives an overview about rounding and propagation heuristics for MIP that are integrated into SCIP [1], which is a state-of-the-art noncommercial solver and framework for mixed integer programming. The ZI Round heuristic has been introduced by Wallace [5], more details on the other primal heuristics can be found in [2, 1, 3].

∗ IBM

Deutschland, Germany, [email protected] Institute Berlin, Takustr. 7, 14195 Berlin, Germany, [email protected], supported by the DFG Research Center Matheon Mathematics for key technologies in Berlin. ‡ Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany, [email protected] † Zuse

1

2

Rounding heuristics

The goal of rounding heuristics is to convert a fractional solution x ¯ of the system Ax ≤ b, l ≤ x ≤ u into an integral solution, i.e., xj ∈ Z ∀j ∈ J. All rounding heuristics described in this section use the notion of up- and down-locks. For a MIP (1), we call the number of positive coefficients ξj+ := |{i : aij > 0}| the up-locks of the variable xj ; the number of negative coefficients is called the down-locks ξj− of xj . Simple Rounding is a very cheap heuristic that iterates over the set of fractional variables of some LP-feasible point. It only performs roundings, which guarantee to keep all linear constraints satisfied. Consider an integer variable xj , j ∈ J, with fractional LP solution x ¯j . If ξj− = 0, we can safely set x ˜j := b¯ xj c without violating any linear constraint. Analogously, if ξj+ = 0, we can set x ˜j := d¯ xj e. If all integer variables with fractional x ¯j can be rounded that way, then x ˜ will be a feasible MIP solution. In contrast to Simple Rounding, Rounding also performs roundings which potentially lead to a violation of some linear constraints, trying to recover from this infeasibility by further roundings later on. The solutions that can be found by Rounding are a superset of the ones that can be found by Simple Rounding. Like Simple Rounding, the Rounding heuristic takes up- and down-locks of an integer variable with fractional LP value x ¯j into account. As long as no linear constraint is violated, the algorithm iterates over the fractional variables and applies a rounding into the direction of fewer locks, updating the activities Ai x ˜ of the LP rows after each step, Ai being the i-th row of A. If there is a violated linear constraint, hence Ai x ˜ > bi for some i, the heuristic will try to find a fractional variable that can be rounded in a direction such that the violation of the constraint is decreased, using the number of up- and down-locks as a tie breaker. If no rounding can decrease the violation of the constraint, the procedure is aborted. ZI Round[5] reduces the integer infeasibility of an LP solution step-by-step by shifting fractional values towards integrality, but not necessarily rounding them. For each integer variable xj with fractional solution value x ¯j , the heuristic calculates bounds for both possible rounding directions of x ¯j such that the obtained solution stays LP-feasible. x ¯j is shifted by the corresponding bound into the direction which reduces the fractionality min{¯ xj −b¯ xj c, d¯ xj e− x ¯j } most. The set of fractional variables might be processed several times by ZI Round. It either terminates with a MIP solution x ˜ or aborts if the integer infeasibility could not be decreased anymore or if a predefined iteration limit has been reached.

3

Propagation heuristics

The goal of propagation heuristics is to construct a feasible MIP solution x ˜ from scratch or from a start solution, while applying domain propagation to exclude variable values that would lead to an infeasibility or to a solution that is inferior to the incumbent. In contrast to diving or LNS heuristics (see, e.g., [2]), propagation heuristics do not solve any LP during search. The Shifting heuristic is similar to Rounding, but it tries to continue in the case that no rounding can decrease the violation of a linear constraint. In this case, the value of a continuous variable or an integer variable with integral

2

value will be shifted in order to decrease the violation of the constraint. To avoid cycling, the procedure terminates after a certain number of non-improving shifts. A shift is called non-improving, if it neither reduces the number of fractional variables nor the number of violated rows. Shift-and-Propagate tries to find a MIP solution by alternately fixing variables and propagating these fixings. Starting with an initial, typically infeasible, solution in which each variable is assumed to be at one of its bounds, it iterates over all integer variables xj , j ∈ J in nondecreasing order of their impact on the activity of the linear constraints. In each step, the heuristic fixes a variable xj to a value x ˜j such that the overall infeasibility gets maximally reduced. The fixing xj = x ˜j is then propagated to reduce further variable domains. In case that the domain propagation detects the infeasibility of the current partial solution, the domain of xj is reset to its previous state and the variable is postponed. The procedure is aborted when a predefined iteration limit is exceeded.

4

Improvement heuristics

Improvement heuristics consider the incumbent solution as a starting point and try to construct an improved solution with better objective value. Oneopt is a straightforward improvement heuristic: given a feasible MIP solution x ˜, the value of an integer variable xj , j ∈ J, can be decreased for cj > 0 or increased for cj < 0 if the resulting solution is still feasible. If more than one variable can be shifted, they are sorted by non-decreasing impact |cj δj | on the objective and sequentially shifted until no more improvements can be obtained. Here, δj ∈ Z denotes how far the variable can be shifted into the desired direction without losing feasibility. Oneopt often succeeds in improving solutions which were found by the rounding heuristics described in Section 2, since their defensive approach to round into the direction of fewer locks tends to over-fulfill linear constraints, sacrificing solution quality. The Twoopt heuristic attempts to improve a feasible MIP solution x ˜ by altering the solution values of pairs of variables. Only variables which share a pre-defined ratio of LP rows are considered as pairs. Each step of the heuristic consists of improving the objective value by shifting one variable, and then compensating the resulting infeasibilities by shifting a second variable, without completely losing the objective improvement. Similarly to Oneopt, pairs are processed in non-decreasing order of their impact on the objective.

5

Computational experiments

In MIP solving, the running time to optimality and the number of branch-andbound nodes are typical measures for comparison. For primal heuristics, the time needed to find a first feasible solution, an optimal solution, or a solution within a certain gap to optimality are favorable measures that concentrate on the primal part of the solution process. Nevertheless, the trade-off between speed and solution quality is not well covered by any of them. We suggest a new performance measure that takes into account the overall solution process. The goal is to measure the progress of convergence towards the optimal solution over the entire solving time. Let x ˜ be a solution for a MIP,

3

and x ˜opt be an optimal (or best known) solution for that MIP. We define the primal gap γ ∈ [0, 1] of x ˜ as:   if |cT x ˜opt | = |cT x ˜| = 0, 0,  T T if c x ˜opt · c x ˜ < 0, (2) γ(˜ x) := 1, T T    |c Tx˜opt −c x˜T| , else. max{|c x ˜opt |, |c x ˜|} Let tmax ∈ R≥0 be a limit on the solution time of a MIP solver. Considering a log file of a MIP solver for a certain problem instance within a fixed computational environment, we define its primal gap function p : [0, tmax ] 7→ [0, 1]: ( 1, if no incumbent found until point t, (3) p(t) := γ(˜ x(t)), with x ˜(t) being the incumbent solution at point t. p(t) is a step function that changes whenever a new incumbent is found. It is zero from the point on at which the optimal solution is found. We define the primal integral P (T ) of a run until a point in time T ∈ [0, tmax ] as: ZT P (T ) :=

p(t) dt = t=0

I X

p(ti−1 ) · (ti − ti−1 ),

(4)

i=1

where ti ∈ [0, T ] for i ∈ 1, . . . , I − 1 are the points in time when a new incumbent solution is found, t0 = 0, tI = T . We suggest to use P (tmax ) for measuring the quality of primal heuristics. It favors finding good solutions early. The fraction P (tmax )/tmax can be seen as the average solution quality during the search process. Spoken differently, the smaller P (tmax ) is, the better is the expected quality of the incumbent solution if we stop the solver at an arbitrary point in time. We used the benchmark set of the MIPLIB 2010 [4] as test set for our experiments. Since we are interested in the primal part of the solution process, we excluded the four infeasible instances triptim1, enlight14, ns1766074, and ash608gpia-3col; further, mspp16 was excluded since it terminated for memory reasons during presolving for all our tests. Thus, 82 test instances remained. We performed four different runs: SCIP without any primal heuristics (Heur OFF), SCIP using only the rounding and propagation heuristics which are described in this paper (R&P), SCIP using all default heuristics except the ones described in this paper (NoR&P), and SCIP with default settings (DEF). All experiments were conducted with a time limit of one hour, a memory limit of 4 GB on a 3.00 GHz Intel® Core™2 Extreme CPU X9650 with 6144 KB Cache and 8 GB RAM. Figure 1 exemplarily shows the primal gap function p(t) for the four settings applied to the instance n3seq24. A square shows when a new primal solution is found and its quality. It can be seen that the two settings that use rounding and propagation heuristics find solutions earlier and hence have a smaller primal integral P (tmax ). For all four settings, SCIP found the optimal solution within an hour, but timed out without proving optimality. Interestingly, with disabled heuristics (Heur OFF), the optimal solution was found in the smallest amount of time. Nevertheless, the behavior of the default settings (DEF) seems favorable since primal solutions of reasonable quality are found much earlier. 4

NoR&P, P (tmax ) = 107282.2 p(t) in %

p(t) in %

DEF, P (tmax ) = 42166.8 75 50 25 0 0

1,000

2,000

75 50 25 0

3,000

0

1,000

t in sec.

75 50 25 0 1,000

2,000

3,000

Heur OFF, P (tmax ) = 105009.6 p(t) in %

p(t) in %

R&P, P (tmax ) = 79724.7

0

2,000 t in sec.

3,000

75 50 25 0 0

t in sec.

1,000

2,000

3,000

t in sec.

Figure 1: The solving process depicted for the instance n3seq24 DEF NoR&P R&P Heur OFF φ(P ) 0.44 0.47 0.61 1.0 φ(t1 ) 8.54 17.48 10.88 57.66 φ(topt ) 215.63 218.18 236.92 263.46 φ(tsolved ) 712.90 676.10 746.48 838.60 Table 1: Computed mean values for all four settings Table 1 shows aggregated results of our experiments. The first row shows the evaluation of the normalized primal integral over the entire testset. For each instance, the corresponding integral P (tmax ) is divided by the integral obtained with the setting Heur OFF. This reference value is then used to compute the geometric mean φ(P ) for each setting. The remainder of the table shows the geometric means of the time until the first (t1 ) and the best (topt ) primal solution were found and the mean solving time (tsolved ). The advantage of rounding and propagation heuristics can be best seen in the time to first solution t1 : they are valuable for finding start solutions. The impact on the time to the best solution and the overall solving time is much smaller, the latter even showing a degradation. The primal integral P (tmax ) implies the following ranking: To get good primal solutions, using cheap and expensive heuristics together (DEF) is slightly better than using only expensive heuristics (noR&P), which is considerably better than using only cheap ones (R&P), which is much better than using no heuristics (Heur OFF). This behavior is only partially expressed by the other measures. Detailed computational results including charts for the development of the primal gap function p(t) can be found in the Appendix.

5

References [1] T. Achterberg. Constraint Integer Programming. PhD thesis, TU Berlin, 2007. [2] T. Berthold. Primal Heuristics for Mixed Integer Programs. Master’s thesis, TU Berlin, 2006. [3] G. Hendel. New rounding and propagation heuristics for mixed integer programming. Bachelor’s thesis, TU Berlin, 2011. [4] T. Koch, T. Achterberg, E. Andersen, O. Bastert, T. Berthold, R. E. Bixby, E. Danna, G. Gamrath, A. M. Gleixner, S. Heinz, A. Lodi, H. Mittelmann, T. Ralphs, D. Salvagnin, D. E. Steffy, and K. Wolter. MIPLIB 2010. Mathematical Programming C, 3(2):103–163, 2011. [5] C. Wallace. ZI round, a MIP rounding heuristic. J Heuristics, 16:715–722, 2009.

6

Appendix In Table 2, the results of the experiment are explicitly shown for each of the 82 instances in our test set. Different performance measures are evaluated for the four settings Heur OFF, R&P, NoR&P, and DEF, see above. For each instance, the first row shows the time until the first (t1 ), and the best (topt ) solution were found, the overall solving time (tsolved ) and the normalized integral value (P (tmax )). The second row shows the development of p(t) over time. Note that according to the definition of the normalized integral, P (tmax ) will always be 1.00 for the setting without primal heuristics. For the figures, a square at 3600 seconds indicates that the solver hit the time limit, because the incumbent solution was not proven to be optimal. Note that the light shaded areas, which indicate the period without an incumbent solution, are cut off at the top. DEF

533.64 3582.95 3600.0

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

1.03

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

226.27 226.27 226.27

1.26

Gap

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

536.83 536.83 536.84

2.98

0.8

0.8

0.6

0.6

0.6

0.4

0.2

0.2

0

0

0

100

200 300 Time(sec.)

400

500

0.33

60

60 Gap

80

20

0

0 500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

45.63

48.99

1.24

200 300 Time(sec.)

400

500

0.45

40

20 0

100

14.12 592.35 3516.76

80

40

0

41.34

100

200 300 Time(sec.)

400

2.10

80

60

60

40

56.39

58.69

1.27

35.13

0

38.71

42.33

1.06

33.05

6

6

6

0

10

20 30 Time(sec.)

40

0

50

1.34

Gap

6

Gap

8

4 2

0

10

20 30 Time(sec.)

137.01 3485.08 3485.09

40

0

50

0.60

10

20 30 Time(sec.)

3600.0 3600.0 3600.0

40

1.34

30

30

30

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

500.97 688.34 3600.0

0.72

Gap

30

Gap

40

0

20 10 0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

571.07 1905.29 3600.0

0.82

758.96

794.2 3600.0

1.11

6

6

0

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0.07 917.53 3600.0

0.05

Gap

6 Gap

6 Gap

8

2

4 2

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0.03 2690.61 3600.0

0.07

0.07

0.07 3600.0

0.21

15

15

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0

Gap

15 Gap

15 Gap

20

5

10 5

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

7

0

0

40

50

1.00

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0

1.00

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

3600.0 3600.0 3600.0

20

10

20 30 Time(sec.)

4

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

20

5

10

2 0

20

10

0

613.21 1980.27 3600.0

8

4

1.00

20

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

8

2

36.93

10 0

8

4

1.00

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

2685.16 2685.16 2685.17

40

10

500

4

0

50

40

20

400

2 0

40

20

0

36.82

8

2

200 300 Time(sec.)

40

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

8

4

100

20 0

8

4

0

32.64 630.78 3600.0

80

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

1.00

0.4

0

500

20 0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0.2 0

0.76 986.43 1886.23

Gap

0

Gap

0.4

Gap

0.8

0.6

Gap

0.8 Gap

1

0.2

0

180.29 180.29 180.29

1

0.4

1.00

20

0

1

Gap

Gap

0

1

0

Gap

Gap

Gap

40

20

0

P (tmax )

2680.8 3594.77 3600.0

40

20

0

3600.0 3600.0 3600.0

Gap

1.14

topt tsolved

40

20

10

beasleyC3

3600.0 3600.0 3600.0

t1

40

0

bab5

Heur OFF

topt tsolved P (tmax )

60

2

app1-2

0.67

t1

60

41.0

Gap

air04

R&P

topt tsolved P (tmax )

60

0.77 353.51 1274.95

Gap

aflow40b

0.45

t1

60

185.25 185.25 185.25

Gap

acc-tight5

NoR&P

topt tsolved P (tmax )

256.63 3287.83 3600.0

Gap

30n20b8

t1

Gap

Problem Name

1.00

10 5

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

DEF

4.52 386.06 419.45

0

0

0

200

400 Time(sec.)

95.03

600

800

169.6

0.46

2.25

400 Time(sec.)

600

0

800

30.69 223.84

0.44

Gap 0

200

400 Time(sec.)

600

40

0

800

0.97 120.42 125.46

0.42

4.05

20

20

10

10

0

0

0

40

60 80 Time(sec.)

100

120

0.49

1.32

0

20

40

60 80 Time(sec.)

100

105.5 131.83

120

0.44

10 5

0

20

40

60 80 Time(sec.)

100

0

120

0.77

20

40

60 80 Time(sec.)

100

50

100 150 Time(sec.)

0.79

0.26

0

120

0.82

3.15

Gap 50

100 150 Time(sec.)

20

40

60 80 Time(sec.)

100

709.23 709.23 709.24

0.47

1.01

0

50

100 150 Time(sec.)

1.38

0.8

0.6

0.6

8.43 219.18 219.19

0

2,000

0.10

7.21

500

145.7

1,000 1,500 Time(sec.)

145.7

0.08

200

400 Time(sec.)

0.01 234.81 3600.0

600

0.21

500

1,000 1,500 Time(sec.)

8.34 316.47 316.48

0.24

0.09

200

400 Time(sec.)

6.57 3600.0

0

600

0.10

0

0.0

200

400 Time(sec.)

7.36 3600.0

600

0.21

60

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

31.48 449.62 3189.21

0.60

Gap

60 Gap

60 Gap

60

0

40 20

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

10.14 374.06 2818.98

0.50

40.03 301.69 3544.72

1.17

20

20

10

10

10

0

0

0

20.74 3600.0

0.61

4.95 144.23 3600.0

20 Gap

10

0

0.49

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0.96

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

8

200

1.00

0

500

1,000 1,500 Time(sec.)

2,000

1.00

0

200

400 Time(sec.)

600

1.00

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

1.00

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

1.00

20

10

0 0

100 150 Time(sec.)

7.27 186.98 3600.0

20

10

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

10.43 102.04 3596.58

20

1.00

10

0

Gap

8.58

Gap

20

Gap

20

Gap

30

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

50

13.75 636.55 3600.0

30

0

120

40

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

30

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

100

20 0

30

0

60 80 Time(sec.)

7.28 3600.0 80

20

0

6.91

80

40

40

20

0

80

20

20

40

80

40

0

95.84 778.98 778.99

20

0

1.00

0.4

0

2,000

Gap

Gap

Gap

0 0

120

0.2 0

40

20

20 0

0

2,000

40

40

0.4 0.2

0

100

Gap

1,000 1,500 Time(sec.)

Gap

0.8

0.6

Gap

0.8

0.6

Gap

0.8

500

60 80 Time(sec.)

1495.95 1495.95 1495.96 1

0

40

10

0

200

1

0

20

20

2067.97 2067.97 2067.97

0.2

0

243.42 244.21 244.22

1

0.4

1.00

10

1

0.4

800

98.17 171.45

0

120

10

200

600

5 0

245.34 245.34 245.34

0 0

400 Time(sec.)

15

20

10

0

200

100

10

0

120

184.61 212.53 212.55

Gap 0

60 80 Time(sec.)

5 0

20

10

40

15 Gap

10

20

2.55 136.23 179.03

15

200

10

0

Gap

20

Gap

0

Gap

20

Gap

20 10

0

80.44 149.69 30

Gap

Gap Gap Gap

200

30

393.34 393.34 393.34

Gap

0

1.00

20

20

30

181.68 206.85 206.85

Gap

40

30

1.36 102.26 129.01

Gap

Gap

Gap

60

20

P (tmax )

635.66 745.69 944.05

60

40

20

0.2

danoint

0.69

60

40

0

csched010

838.8 897.29

topt tsolved

60

20

cov1075

2.5

t1

80

0

core2536-691

topt tsolved P (tmax )

80

5

bnatt350

0.02

Heur OFF

t1

80

15

bley xl1

R&P

topt tsolved P (tmax )

Gap

binkar10 1

0.07

t1

80

1.1

Gap

bienst2

NoR&P

topt tsolved P (tmax )

2.51 619.04 701.01

Gap

biella1

t1

Gap

Problem Name

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

10

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

DEF

19.44 104.69

0.39

0.01

0

5

10

82.59

15 20 Time(sec.)

25

94.24

0

30

0.99

19.53

Gap

Gap 5

10

15 20 Time(sec.)

57.52

25

80.3

0

30

1.02

13.48

40 20

0

5

10

15 20 Time(sec.)

37.05

25

76.56

0

30

0.87

18.71 80

60

60

40 20

20

40 Time(sec.)

60

241.0

0

80

0.70

20 0

20

40 Time(sec.)

60

23.55 128.95 158.15

0

80

1.15

0 0

50

100 150 Time(sec.)

200

10.62 1938.46

0.05

Gap

Gap

20

0

20

40 Time(sec.)

60

0.71

50

100 150 Time(sec.)

315.02 315.02 452.68

1.62

66.63

0 50

100 150 Time(sec.)

66.63 3600.0

200

0.34

0.6

0.6

0.6

0.6

0.4

0.2

0.2

0

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

1.00

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

131.96 131.96 131.96

1.01

Gap

1 0.8

Gap

1 0.8

0.4

130.36 130.36 130.36

0.99

0.8

0.8

0.6

0.6

0.6

40

60 80 Time(sec.)

100

0.58 1679.56 3600.0

0

120

0.95

0.2 0

20

40

60 80 Time(sec.)

100

0.43 2429.37 3600.0

0

120

0.96

Gap

Gap

20

0

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0.0 2004.04 3600.0

0.14

40

60 80 Time(sec.)

100

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0.36 2908.79 3193.88

0.18

0.87

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0.15

60

0.0

0 500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

28.62 1988.89

0.19

0.12

Gap

60

Gap

60

Gap

60

0

40

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

27.81 1863.8

0.42

0.0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

27.13 1545.83

0.22

23.28

40

40

40

40

0

5

10

15 20 Time(sec.)

0.01 127.69 3600.0

25

0.33

20

0

0

0.3

0

5

10

15 20 Time(sec.)

65.23 3600.0

25

0.39

0.0

Gap

60

Gap

60

20

5

10

15 20 Time(sec.)

94.46 3600.0

0

25

0.40

27.08

60

60

60

60

20

0

0 0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

Gap

80

Gap

80

40

40

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

9

0

40

60 80 Time(sec.)

100

120

1.00

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

1.00

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

1.00

0

5

10

15 20 Time(sec.)

25

1.00

40 20

20 0

20

43.43 3600.0

80

20

0

20 0

80

40

1.00

26.77 1478.99

60

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

40

0 0

60

20

0

20

20 0

1.00

3.22 1832.67 3276.45 80

0

200

20 0

0.0 650.65 3600.0

20

100 150 Time(sec.)

40

80

40

50

60

80

20

0

0.94 2866.82 2964.95

80

40

1.00

0.4

0

120

40

20 0

20

60

40

20

80

0.2 0

0.38 2392.02 2489.41

60

40

0.4

Gap

20

Gap

0.8

0.6

Gap

0.8 Gap

1

0

60

131.23 131.23 131.23

1

0.2

40 Time(sec.)

0.4

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

1

0.4

20

0.2 0

1

0.4

0

193.77 193.77 2564.5

1 0.8

0

1.00

20 0

1

0.2

81.94

30

40

0.8

0.4

25

60

40

0

200

15 20 Time(sec.)

15.26 201.53 248.37

20 0

10

40

0

80

60

40

20

5

20 0

2.58 205.23 235.44

60

40

40

Gap

0

Gap

80

60

Gap

80

60 40

0

50.11

80

Gap

Gap Gap Gap

0

100

131.17 131.17 131.17

Gap

20

100

10.62

Gap

40

100

2.47 220.59

Gap

1.00

100

Gap

Gap

13.51

Gap

31.43 125.07

60

40 20

60

iis-bupa-cov

P (tmax )

1.14

60

0

iis-100-0-cov

0.55

60

40

0.2

gmu-35-40

98.04

60

60

glass4

18.47

topt tsolved

80

0

ex9

0.02

t1

80

20

enlight13

0.45

Heur OFF

topt tsolved P (tmax )

80

0

eilB101

24.89 108.87

t1

80

20

eil33-2

R&P

topt tsolved P (tmax )

Gap

0.02

t1

Gap

dfn-gwin-UUM

NoR&P

topt tsolved P (tmax )

Gap

t1

Gap

Problem Name

0 0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

DEF

0.35 224.63 657.33

20

0

0 0

100

79.06

200 Time(sec.)

300

400

79.07

0.31

14.97

300

0

400

83.63

0.32

15.55

Gap

40

0 0

100

93.52

200 Time(sec.)

300

400

93.53

0.36

40

40

20

20

20

0

0

0

0

50

100 150 Time(sec.)

200

18.4 3600.0

1.00

0

0.03

50

100 150 Time(sec.)

200

12.3 3600.0

1.00

Gap

60

Gap

60

Gap

60

40

0.01

50

100 150 Time(sec.)

40.73 3600.0

1.01

80

80

60

60

60

40

0

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

205.4 3600.0

0.22

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

10.32 1588.24 3600.0

0.14

40 20 0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0.22

2.08

40

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

54.51 424.76 500.01

0.46

0

4.33 354.17 376.22

0.33

80

80

60

60

60

100

200 300 Time(sec.)

0

400

0.10

Gap

80

60

Gap

80 Gap

100

0

40 20

0

100

200 300 Time(sec.)

0

400

52.88 372.95 472.83

0.23

4.56

100

200 300 Time(sec.)

316.8 405.42

0.16

80

80

60

60

60

40 20

0

0

0

50

100

150 200 250 Time(sec.)

300

0.11 157.72 159.86

350

0.21

0.38

50

100

150 200 250 Time(sec.)

300

35.74 184.27

350

0.13

0.12

40

20

20 0

0

0.0

20

40

60 80 100 120 140 Time(sec.)

0.13 131.97

0.01

0

0.01

20

40

50

100

150 200 250 Time(sec.)

300

37.37 148.26

0.11 130.49

0.01

0.21

6.5

0.0

20

40

60 80 100 120 140 Time(sec.)

5.16 122.18

0.01

60

60

60

0.01

0

100

44.99

200 300 Time(sec.)

0

400

46.64

0.69

36.8

Gap

60

Gap

80

Gap

100

80

0

40 20

0

100

50.29

200 300 Time(sec.)

0

400

53.43

0.95

0.01

100

45.64

200 300 Time(sec.)

46.78

0.69

39.03

60

60

60

40

20

20

0

0

0

10

20 30 Time(sec.)

40

50

0

10

20 30 Time(sec.)

10

40

50

Gap

60

Gap

80

40

0

50

100

0

20

40

0

100

50.58

80

0

1.00

150 200 250 Time(sec.)

300

350

1.00

60 80 100 120 140 Time(sec.)

1.00

40

0

400

80

20

400

20 0

80

40

200 300 Time(sec.)

1.01 432.24 3193.47

100

80

20

100

20 0

0

100

40

0

40

80

20

1.00

29.95 178.24

100

40

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

40

0

350

20 0

60 80 100 120 140 Time(sec.)

0

20 0

40 Gap

Gap

40

0

0

Gap

0

Gap

80

60

Gap

80 Gap

100

20

1.00

289.64 369.28 403.33

100

40

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

40

0

400

100

20

0

20 0

100

40

1.00

196.31 265.93 318.38

100

20

200

40

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

100

40

100 150 Time(sec.)

20 0

100

40

50

60

20 0

1.00

35.91 3049.12 3600.0

60 Gap

40

239.2

40

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0.0 3403.14 3600.0

60

400

20 0

Gap

0

Gap

80

60

Gap

80

20

300

2.88 3600.0 100

20

0

0.68

100

40

200 Time(sec.)

40

0

200

100

20

100

20 0

100

40

0

101.17 239.19

60

Gap

Gap Gap Gap

83.62

200 Time(sec.)

80

4.62 316.41 419.39

Gap

100

80

4.4 246.77 317.26

Gap

0

1.00

20

20

80

0.01

Gap

40

80

0

mine-166-5

Gap

60

Gap

60

40

20

P (tmax )

42.08 406.79 666.64

60

40

20

mik-250-1-100-1

0.19

topt tsolved

60

0

mcsched

0.02 177.67 583.23

t1

80

20

map20

Heur OFF

topt tsolved P (tmax )

80

60

map18

0.37

t1

Gap

macrophage

R&P

topt tsolved P (tmax )

80

0.0

Gap

m100n500k4r1

0.23

t1

80

14.8

Gap

lectsched-4-obj

NoR&P

topt tsolved P (tmax )

0.01 245.66 641.41

Gap

iis-pima-cov

t1

Gap

Problem Name

200 300 Time(sec.)

400

53.46

1.00

40 20 0

0

10

20 30 Time(sec.)

40

50

0

10

20 30 Time(sec.)

40

50

DEF

0.84

t1

24.65 438.38 603.12

60

0

0 0

200 400 600 800 1,000 1,200 1,400 Time(sec.)

394.86 3481.54 3600.0

0.19

Gap

0

0.93

0.05

0

0.99

111.95 80

60

60

40 20

0

0

0

0

50

100 150 Time(sec.)

200

0.41

0

50

100 150 Time(sec.)

200

110.48 2962.52 3600.0

1.05

Gap

80

60

Gap

80

60

Gap

80

20

50

100 150 Time(sec.)

2.42 818.07 3600.0

0.31

60

60

60

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0.40

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

994.89 1697.2 3600.0

1.02

Gap

Gap

20

0

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0.12

0.76

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0.09 109.28 1032.29

0.08

0.06

0 500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

34.61 523.35

0.09

77.12

60

40 20

0

0

7.91

0

20

40

60 80 100 Time(sec.)

120

15.99 235.22

140

1.06

1.62

0

20

40

60 80 100 Time(sec.)

120

15.11 163.08

140

0.42

Gap

60

Gap

60

Gap

60

20

7.86

40

60 80 100 Time(sec.)

120

11.17 279.51

140

1.07

7.74

20

20

6 8 10 Time(sec.)

12

26.84 2372.34 3600.0

14

0

16

1.02

25.18

Gap

20

Gap

20

Gap

30

4

10

0

2

4

6 8 10 Time(sec.)

12

39.38 3600.0

14

0

16

0.96

2

4

6 8 10 Time(sec.)

12

24.28 181.57 3600.0

14

0

16

0.93

26.23 1 0.8

0.6

0.6

0.6

0.6

0

31.86

0.2 0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

47.19 150.31

1.17

Gap

1 0.8

Gap

1 0.8

0.4

0.4 0.2

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

28.59 166.45 392.23

0.98

36.45

92.39 260.96

1.60

60

60

60

0

0

50

100 150 200 Time(sec.)

250

0

Gap

60 Gap

80

20

40 20

0

50

100 150 200 Time(sec.)

11

250

0

120

140

1.00

0

2

4

6 8 10 Time(sec.)

12

0

14

16

1.00

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

24.99 282.79 447.85

80

40

60 80 100 Time(sec.)

0.4

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

80

20

40

0.2 0

80

40

20

29.55 3600.0

1

0.2

0

10

0

0.8

0.4

1.00

10.52 304.19

30

2

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

40

0 20

30

0

0

20 0

30

10

1.00

89.77 799.61 80

40

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

20 0

80

0

0

40

80

20

1.00

1030.03 1278.57 3600.0

80

40

200

60

40

20 0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

60

40

20

100 150 Time(sec.)

40

0 0

31.25 3590.62 3600.0

60

40

50

20

20

Gap

0 0

Gap

60

Gap

80

40

0

106.9 2202.45 3600.0

80

20

1.00

40

0

200

80

40

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

20 0

80

40

0

205.9 277.44 100

40

1.00

10 0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

234.6 285.85

20

200 400 600 800 1,000 1,200 1,400 Time(sec.)

3600.0 3600.0 3600.0

100

40

0

20

10

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

103.0 203.92 220.85

Gap

Gap Gap Gap

1.00

100

0.07 148.95 631.79

Gap

3600.0 3600.0 3600.0

Gap

Gap Gap

0.96

200 400 600 800 1,000 1,200 1,400 Time(sec.)

20

0

40

0 0

100

31.46 1297.75 3600.0

Gap

0.24

10 0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

2.41 818.48 3600.0

Gap

0

200 400 600 800 1,000 1,200 1,400 Time(sec.)

840.17 1064.75 3600.0

Gap 0

0.04 141.49 288.18

0

neos-1396125

0

1.00

20

20

20

10

40

Gap

20

10

neos-1337307

40

20

Gap

60

Gap

60

40

P (tmax )

28.06 1525.87 1647.92

60

60

neos-1109824

1.02

topt tsolved

80

0

n4-3

0.01 1265.59 1514.87

t1

80

20

n3seq24

Heur OFF

topt tsolved P (tmax )

80

0

n3div36

0.71

t1

80

20

mzzv11

R&P

topt tsolved P (tmax )

Gap

msc98-ip

NoR&P

topt tsolved P (tmax )

0.01 1657.66 1911.35

Gap

mine-90-10

t1

Gap

Problem Name

1.00

40 20

0

50

100 150 200 Time(sec.)

250

0

0

50

100 150 200 Time(sec.)

250

DEF

0.42

t1

2.55 202.86 238.96

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

3.38

237.45 2946.48 3600.0

Gap

1419.96 2345.89 3600.0

2.74

80

60

60

40 20

37.64 149.83

0.92

0.37

20 0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

13.56

87.4

0.31

0.16

60 40

20

20

0

0

0

10

20 Time(sec.)

30

0.51

10

20 Time(sec.)

223.77 403.65 428.65

0.70

160.06

100

200 300 Time(sec.)

56.85

65.37

400

1.11

1

0

500

0

18.08

100

200 300 Time(sec.)

80.7

400

93.09

1.43

9.95

10

20 Time(sec.)

0

30

0.51

0

100

200 300 Time(sec.)

45.51

400

55.88

0.75

14.7

0

20

40 Time(sec.)

60

80

0.81

0

20

40 Time(sec.)

1396.76 1396.76 1396.76

60

80

1.66

Gap

40

Gap

40

Gap

40

0

20

40 Time(sec.)

801.06 801.06 801.07

60

0

80

0.95

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.2

0.2

0

0

0

200

400

600 800 1,000 1,200 Time(sec.)

43.97 132.83 563.83

0.19

200

400

600 800 1,000 1,200 Time(sec.)

33.92 115.29 528.25

0.15

Gap

Gap

20

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0.11 823.18 823.19

0.01

200

400

0.70

13.86 631.21 631.21

0.02

0.11 389.54 389.54

0.05

60

60

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

158.93 252.71 3600.0

0.57

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

2.04

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

Gap

Gap 0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

2.15

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

12

80

1.00

200

400

600 800 1,000 1,200 Time(sec.)

1.00

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

1.00

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

1.00

20

10 0

0

298.99 354.52 3045.95

20

10

60

40

0 0

504.37 1331.99 3004.87

20

10 0

0

40 Time(sec.)

20

20

127.13 3241.0 3600.0

20

40

Gap

0

0

Gap

60

Gap

60

Gap

80

20

20

3600.0 3600.0 3600.0

80

40

0

20

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

80

20

1.00

10 0

80

40

500

30

20

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

400

66.26

127.81 2527.55 3600.0

10 0

200 300 Time(sec.)

0.4

0

600 800 1,000 1,200 Time(sec.)

30

10 0

1.00

0.2 0

92.69 2179.78 3600.0

30

20

0

0

Gap

0

Gap

0.8

Gap

1

Gap

1

0.4

100

842.15 842.15 842.16

1

0.2

30

20

0

1

0.4

0

54.83

40

20

20 Time(sec.)

1

0

500

60

0

10

2

60

20

0

319.77 414.48 456.84

60

0

1.00

40

60

20

101.2

20 0

1

11.12

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

60

511.8 556.23

0

500

0

8.86

2 Gap

Gap 0

0.98

40

0

30

2

1

32.1 124.88

20 0

1.00

40

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

60 Gap

40

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

20 0

Gap

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

40

Gap

0

Gap

80

60

Gap

80

40

0

797.54 797.54 797.55

60

Gap

Gap Gap Gap

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

80

684.53 684.53 684.53

Gap

2.73

1.00

20 0

100

11.88

Gap

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

100

162.55 403.84 421.71

Gap

0

100

0.16

Gap

20

100

10

net12

Gap

Gap

40

0

P (tmax )

22.12 1538.69 3600.0

40

20

30

neos-934278

0.53

40

20

0

neos-916792

98.44 124.33

topt tsolved

40

2

neos-849702

2.1

t1

60

60

neos-686190

topt tsolved P (tmax )

60

0

neos-476283

0.22

Heur OFF

t1

60

20

neos18

R&P

topt tsolved P (tmax )

60

246.59 3517.45 3517.46

Gap

neos-1601936

NoR&P

topt tsolved P (tmax )

2.17 193.92 206.58

Gap

neos13

t1

Gap

Problem Name

10 0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

DEF

NoR&P

360.34 360.34 431.38

1.51

t1

422.65 422.65 554.26

100

200 300 Time(sec.)

1.76

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

15.41 381.75

0.02

0

0.09

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

2.14 302.55

0.83

0.01

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

75.28 632.39

0.00

40

40

1.5

0

2

7.76

Gap

40

Gap

40

Gap

60

1 Time(sec.)

20

0

0.5

1 Time(sec.)

1191.82 1191.82 1903.62

1.5

0

2

2.57

0.5

1 Time(sec.)

695.84 695.84 695.85

1.5

0

2

1.50

463.8 1 0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.2

0.2

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0.55

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

81.71 629.33 629.34

0.53

0

1.00

10

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

1.16

0

3600.0 3600.0 3600.0

1.00

0

3600.0 3600.0 3600.0

1.00

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.2

0.2

0

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

65.03 183.69 682.52

0.85

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

22.07 416.86 746.31

0.75

71.14

20

0

100

200 300 Time(sec.)

0.03 436.38 442.66

0.19

20

0

400

0

100

200 300 Time(sec.)

470.8 874.02

1.14

7.29 740.62 743.26

0.31

0

100

200 300 Time(sec.)

0.03 380.38 382.35

0.20

60

40 20

200

400

0

600 800 1,000 1,200 1,400 Time(sec.)

0.01 1313.26 1417.1

0.10

Gap

80

60

Gap

80

60

0

40 20

0

200

400

0

600 800 1,000 1,200 1,400 Time(sec.)

3.6 1123.98 1272.91

0.21

200

400

0.0 1824.41 1909.51

0.26

80

80

80

60

60

60

60

0

0

500

1,000 Time(sec.)

1,500

0

Gap

80 Gap

100

20

40 20

0

500

1,000 Time(sec.)

13

1,500

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

1.00

0

100

200 300 Time(sec.)

400

1.00

0

200

400

600 800 1,000 1,200 1,400 Time(sec.)

17.46 1250.58 1473.57

100

40

1.00

40

0

600 800 1,000 1,200 1,400 Time(sec.)

100

20

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

20 0

100

40

0

148.18 1502.84 1504.68

80

0

1.00

20

0

400

60

20

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

46.61 433.98 784.68

80

40

0

40

20

0

400

1.00

0.4

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

40 Gap

Gap

40

2

0.2 0

Gap

0

Gap

0.8

Gap

1

Gap

1

0.4

1.5

3600.0 3600.0 3600.0

1

0.2

1 Time(sec.)

10

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

1

0.4

0.5

20

10

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0

30.58 2548.98 3600.0

20 Gap

Gap

10

1.00

0.4

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

334.67 1552.74 1552.82

20

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0.2 0

Gap

0

Gap

1 0.8 Gap

1 0.8

0

1.00

463.8 585.43

1

0.2

400

20

0

0.8

0.4

200 300 Time(sec.)

0.84 219.48

60

0.5

0

0.12

60

0

100

10

0

60

20

0

6.88 714.09 3600.0

Gap

Gap

Gap

0

Gap

Gap Gap Gap

1.68

20

10

0

3600.0 3600.0 3600.0

Gap

400

20

10

155.61 541.74 541.75

Gap

200 300 Time(sec.)

20

0

3600.0 3600.0 3600.0

Gap

100

1.56 1437.89 3600.0

10

0

pg5 34

10 0

0

20

40

opm2-z7-s2

0.65

10 0

400

1.00

20

30

0

ns1830653

200 300 Time(sec.)

P (tmax )

237.76 293.44 328.04

30

20

ns1758913

100

topt tsolved

30

0

ns1688347

0

3.57 1099.34 3459.53

20

ns1208400

1.20

20

10 0

400

338.6 401.51

t1

30

0.01

Gap

noswot

0

Heur OFF

topt tsolved P (tmax )

285.64

Gap

10

1.78 3598.52 3600.0

Gap

newdano

1.77

t1

20 Gap

Gap

20

0

R&P

topt tsolved P (tmax )

Gap

topt tsolved P (tmax )

Gap

netdiversion

t1

Gap

Problem Name

1.00

40 20

0

500

1,000 Time(sec.)

1,500

0

0

500

1,000 Time(sec.)

1,500

DEF t1

pigeon-10

0.0 152.42 3600.0

NoR&P

topt tsolved P (tmax ) 4.84

t1

60

60

60

40

20 0

1.22

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

26.38 108.41 3600.0

1.08

28.73

20

10

10

0

0

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

6.09

61.76

0.84

0.04

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

11.06

81.57

1.04

0.06

Gap

20

Gap

20

10

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

10.52

54.15

0.87

80

80

60

60

60

40 20

0

0

0

0

2

4

6 Time(sec.)

8

10

207.1 303.18

0.05

0

2

4

6 Time(sec.)

8

10

12.31 260.18 361.65

0.05

Gap

80

60

Gap

80 Gap

100

20

7.38

2

4

6 Time(sec.)

8

611.5 679.32

0.23

60

60

100

200

300 400 Time(sec.)

500

37.42 213.09

600

0.11

0.0

0

100

200

300 400 Time(sec.)

500

37.12 251.08

0

600

0.08

0.02

10

5

0

0

0

20

40 60 Time(sec.)

80

0.43

11.19

200

300 400 Time(sec.)

500

600

0.13

0

20

40 60 Time(sec.)

80

65.31 283.58

0.74

0.01

4.63

20

40 60 Time(sec.)

0

80

71.61 209.67

0.48

16.27

60

1.2

0.08

10

20 Time(sec.)

30

76.76 115.69

40

0.09

0

10

20 Time(sec.)

30

0.78

8.05 101.26 113.47

0.13

20

40 60 Time(sec.)

0.01 1893.13 2808.4

80

0.07

20 0

100

0

20

40 60 Time(sec.)

5.87 1182.11 1925.61

80

0.13

Gap

40

36.61

10

20 Time(sec.)

30

0

0

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

14

500

600

1.00

20

40 60 Time(sec.)

80

1.00

0

20

40

60 80 100 120 140 160 Time(sec.)

67.8

1.00

20

92.94

40

0.19

0

10

20 Time(sec.)

30

87.99 102.99 113.03

40

1.00

40

0

20

40 60 Time(sec.)

80

0.08

20 0

100

0

20

40 60 Time(sec.)

33.42 2738.63 3600.0

80

100

1.00

60

40 20

0

41.35

0 0

60

40 20

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0.26

20

20 0

56.33

0.0 1942.44 3600.0

60

60

300 400 Time(sec.)

40

50.96

0

100

200

40

0

60 80 100 120 140 160 Time(sec.)

40 Gap

Gap 0

40

20

1.06

40

20

20

38.9

0

40

100

20 0

40

20 0

0

40

0

66.71

Gap

Gap 20

1.06

35.66

0

60 80 100 120 140 160 Time(sec.)

40

40

0

40

Gap

0.08

20

Gap

59.2

0

Gap

24.98

0

60 80 100 120 140 160 Time(sec.)

40 20

Gap

0.77

40

Gap

60

Gap

60

Gap

60

20

0

35.81 206.25 80

0

1.00

5 0

80

20

10

10

80

40

8

92.13 223.25

80

40

6 Time(sec.)

15

10

5 0

100

31.31 210.66

5

4

40

0 0

15 Gap

15

10

2

20

20

Gap

0 0

Gap

60

Gap

60

Gap

80

40

0

278.95 380.74 453.91

80

20

1.00

40

0

10

80

40

63.14

20 0

80

40

1.00

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

7.9

100

40

0

7.13

100

20

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

10

0

100

40

0

52.25 3600.0

20

Gap

Gap

1.20

40 20

0

30

0.0 160.41 245.61

Gap

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

30

0.01

Gap

0

30

7.43

Gap

20

26.69 138.89 3600.0

Gap

Gap Gap

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

40

30

0.06

Gap

40

Gap

60

Gap

80

0

rmine6

1.00

100

20

rmatr100-p5

31.44 3600.0

80

15

rmatr100-p10

P (tmax )

31.44

100

0

reblock67

3.60

topt tsolved

80

20

ran16x16

0.0 113.37 3600.0

t1

100

23.65 171.43 3600.0

Gap

rail507

Heur OFF

topt tsolved P (tmax )

80

0

qiu

6.94

t1

100

20

pw-myciel4

R&P

topt tsolved P (tmax )

218.41 218.41 3600.0

Gap

Gap

Problem Name

40 20 0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

DEF

NoR&P

22.77 117.16 139.87

0.55

t1 28.92

20

0

0

0

50

100 150 Time(sec.)

200

485.0 1626.56

250

0.24

200

0

250

20

0.20

0.1

0 0

50

100 150 Time(sec.)

200

313.1 1415.92

250

0.34

100

200

300 400 Time(sec.)

500

600

0.48

0

100

200

300 400 Time(sec.)

500

3.61 245.47 3600.0

0

600

0.74

0.44

100

200

300 400 Time(sec.)

500

600

2.34

Gap

10

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

285.96 625.36 1244.68

0.66

277.48 1086.96 1660.38

0.88

60

60

60

60

40

20

20

0

0

0

200

400 600 Time(sec.)

800

1,000

0.46

0

200

400 600 Time(sec.)

800

10.69 2626.06 3600.0

1,000

0.29

Gap

80

Gap

100

80

Gap

100

80

40

200

400 600 Time(sec.)

800

2.02 2871.37 3600.0

0.56

60

60

40

20

20

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

7.2 231.91 286.09

0.52

7.18

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

60.19 163.48

0.47

0

0.16

50

100 150 Time(sec.)

0

200

1.28 3600.0

0.44

0

30.85 108.43

0.61

16.79

50

100 150 Time(sec.)

0

200

276.61 702.24 1596.85

0.08

50

100 150 Time(sec.)

0

200

0.16 3600.0 3600.0

0.44

60

60

0

0.01

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

20.01

48.35

0.79

Gap

60 Gap

60 Gap

80

20

40

3.24

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

3.24

3.77

0.59

0.02

0

13.02

43.76

0.79

5.42

60

60

60

60

20

0

0 0

5

10 Time(sec.)

15

1.16 212.69 591.87

20

0.44

Gap

80

Gap

80

40

40

5

10 Time(sec.)

15

1.26 102.86 529.93

0

20

0.37

5

10 Time(sec.)

15

11.09 194.58 515.85

20

1.01

14.09

30

30

30

30

20

10

10

10

0

0

0

50

100 150 Time(sec.)

200

0

50

100 150 Time(sec.)

15

200

Gap

40

Gap

40

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

140.4

1.00

0

50

100 150 Time(sec.)

200

1.00

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

12.16

1.00

0

5

10 Time(sec.)

15

26.16 439.05

40

20

1.00

40

0 0

40

20

1,000

20

20 0

0

11.74

80

20

800

40

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

80

40

400 600 Time(sec.)

20

20 0

200

3600.0 3600.0 3600.0

80

40

1.00

10

0

80

20

0

40.38

80

40

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

20

10

0

0

40

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

Gap 10

1.00

20 0

20

Gap

20

Gap 0

10.09

600

Gap

0

Gap

60 Gap

60

Gap

80

40

500

209.69 2312.39 3600.0

80

0

300 400 Time(sec.)

40

0

1,000

80

20

200

20 0

80

40

100

267.51 950.24 2374.85

100

0

0

10

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

80

20

1.00

5 0

100

40

981.9

250

15

5 0

200

12.64 484.77 3600.0

15

10

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

100 150 Time(sec.)

40

0 0

16.59 572.78 3600.0

5 0

50

20

20

15

10

40

Gap

0 0

Gap

Gap

40

0

13.84 384.62

60

Gap

Gap Gap Gap

100 150 Time(sec.)

60

2.04 1153.66 3600.0

Gap

50

20

303.35 1417.32 1419.37

Gap

0

40

60

40

3.63 479.02 3600.0

Gap

40

60

10

timtab1

1.00

60

80

20

tanglegram2

255.2 259.82

80

0

tanglegram1

P (tmax )

30.15

80

5

sp98ir

0.67

topt tsolved

20

3.37 646.23 1704.96

15

sp98ic

29.43 132.75 140.52

t1

80

0

satellites1-25

Heur OFF

topt tsolved P (tmax )

60

40

20

20

roll3000

0.57

Gap

Gap

40

0.09

Gap

rococoC10-001000

96.07 108.68

t1

60

Gap

Gap

60

R&P

topt tsolved P (tmax )

Gap

topt tsolved P (tmax )

Gap

rocII-4-11

t1

Gap

Problem Name

20

1.00

20 10

0

50

100 150 Time(sec.)

200

0

0

50

100 150 Time(sec.)

200

DEF

NoR&P

topt tsolved P (tmax )

172.12 825.49 1695.51

0.72

t1

166.6 805.37 1051.57

0.96

topt tsolved

242.79 1159.38 1438.29

3

3

3

2

2

2

1 0

200

400 600 Time(sec.)

0

800

0.84

Gap

3

Gap

4

1 0

200

400 600 Time(sec.)

609.93 2635.11 3600.0

0

800

0.84

200

400 600 Time(sec.)

3600.0 3600.0 3600.0

1.00

60

60

60

0.34

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

49.47 3600.0

0.42

0.1

Gap

20

0

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

43.81 3600.0

0.24

0.33

40

40

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

0

0.45

Table 2: The results for every instance

16

800

1.00

15.05

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

54.43 3600.0

1.00

40

20

0

400 600 Time(sec.)

40

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

62.66 3600.0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

200

20 0

40

20

0

40 20

20

Gap

0

Gap

60

Gap

80

40

0

3600.0 3600.0 3600.0

80

20

1.00

2

0

800

80

40

P (tmax )

1 0

80 Gap

Gap

231.51 935.35 1333.29

t1

4

698.86 2842.75 3600.0

Gap

Heur OFF

topt tsolved P (tmax )

4

0

zib54-UUE

0.69

t1

4

1

vpphard

R&P

topt tsolved P (tmax )

Gap

Gap

unitcal 7

t1

Gap

Problem Name

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)

20

0

0

500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)