GRASP with evolutionary path-relinking for the antibandwidth problem

Transcription

1 GRASP with evolutionary path-relinking for the antibandwidth problem VIII Metaheuristics International Conference (MIC 009) Hamburg, Germany July 3-6, 009 Mauricio G. C. Resende AT&T Labs Research Florham Park, New Jersey Joint work with A. Duarte, R. Martí, & R. Silva

2 Summary Antibandwidth Integer programming formulation GRASP construction Local search GRASP with evolutionary path-relinking Experimental results Concluding remarks

3 Antibandwidth problem Given an undirected graph G = (V,E), where V is the set of nodes (n = V ) E is the set of edges (m = E ) A labeling f of V is a one-to-one mapping of {,,...,n} onto V. Each vertex v V has a unique label f(v) {,,...,n}

4 Antibandwidth problem Given an undirected graph G = (V,E), where V is the set of nodes (n = V ) E is the set of edges (m = E ) A labeling f of V is a one-to-one mapping of {,,...,n} onto V. Each vertex v V has a unique label f(v) {,,...,n} undirected graph G = (V,E)

5 Antibandwidth problem Given an undirected graph G = (V,E), where V is the set of nodes (n = V ) E is the set of edges (m = E ) A labeling f of V is a one-to-one mapping of {,,...,n} onto V. Each vertex v V has a unique label f(v) {,,...,n} undirected graph G = (V,E) with a labeling f

6 Antibandwidth problem Given an undirected graph G = (V,E), where V is the set of nodes (n = V ) E is the set of edges (m = E ) A labeling f of V is a one-to-one mapping of {,,...,n} onto V. Each vertex v V has a unique label f(v) {,,...,n} undirected graph G = (V,E) with another labeling f

7 Antibandwidth problem Given G and f, the antibandwidth ABf(v) of node v is smallest difference between f(v) and the labels of all of the nodes adjacent to v, i.e. ABf(v) = min { f(v) f(u) : u N(v) } where N(v) is the set of nodes adjacent to v undirected graph G = (V,E) with a labeling f

8 Antibandwidth problem Given G and f, the antibandwidth ABf(v) of node v is smallest difference between f(v) and the labels of all of the nodes adjacent to v, i.e. ABf(v) = min { f(v) f(u) : u N(v) } where N(v) is the set of nodes adjacent to v undirected graph G = (V,E) with a labeling f

9 Antibandwidth problem Given G and f, the antibandwidth ABf(v) of node v is smallest difference between f(v) and the labels of all of the nodes adjacent to v, i.e. ABf(v) = min { f(v) f(u) : u N(v) } where N(v) is the set of nodes adjacent to v =5 4 5 undirected graph G = (V,E) with a labeling f

10 Antibandwidth problem Given G and f, the antibandwidth ABf(v) of node v is smallest difference between f(v) and the labels of all of the nodes adjacent to v, i.e. ABf(v) = min { f(v) f(u) : u N(v) } where N(v) is the set of nodes adjacent to v =3 6-=5 undirected graph G = (V,E) with a labeling f

11 Antibandwidth problem Given G and f, the antibandwidth ABf(v) of node v is smallest difference between f(v) and the labels of all of the nodes adjacent to v, i.e. ABf(v) = min { f(v) f(u) : u N(v) } where N(v) is the set of nodes adjacent to v =3 6-=5 5-=4 undirected graph G = (V,E) with a labeling f

12 Antibandwidth problem Given G and f, the antibandwidth ABf(v) of node v is smallest difference between f(v) and the labels of all of the nodes adjacent to v, i.e. ABf(v) = min { f(v) f(u) : u N(v) } where N(v) is the set of nodes adjacent to v =3 6-=5 5-=4 -= undirected graph G = (V,E) with a labeling f

13 Antibandwidth problem Given G and f, the antibandwidth ABf(v) of node v is smallest difference between f(v) and the labels of all of the nodes adjacent to v, i.e. ABf(v) = min { f(v) f(u) : u N(v) } where N(v) is the set of nodes adjacent to v =3 6-=5 5-=4 -= ABf() = min {5,3,4,} = undirected graph G = (V,E) with a labeling f

14 Antibandwidth problem Given G and f, the antibandwidth ABf(G) of f is smallest antibandwidth over all nodes in V, i.e. ABf(G) = min { ABf(v) : v V } where N(v) is the set of nodes adjacent to v undirected graph G = (V,E) with a labeling f

15 Antibandwidth problem Given G and f, the antibandwidth ABf(G) of f is smallest antibandwidth over all nodes in V, i.e. ABf(G) = min { ABf(v) : v V } where N(v) is the set of nodes adjacent to v ABf(G) = min {,,,, } =

16 Antibandwidth problem Given G, the antibandwidth AB(G) of G is largest antibandwidth over all possible labelings, i.e. AB(G) = max { ABf(G) : f Πn } where Πn is the set of all permutations of {,,..., n} ABf(G) = min {,,,, } =

17 Antibandwidth problem Given G, the antibandwidth AB(G) of G is largest antibandwidth over all possible labelings, i.e. AB(G) = max { ABf(G) : f Πn } where Πn is the set of all permutations of {,,..., n} ABf(G) = min {,,,, 3 } = 4

18 Antibandwidth problem Given G, the antibandwidth AB(G) of G is largest antibandwidth over all possible labelings, i.e. AB(G) = max { ABf(G) : f Πn } where Πn is the set of all permutations of {,,..., n} AB(G) = 4

19 Antibandwidth problem NP-hard (Leung et al., 984) Special cases can be solved in polynomial time, e.g. complements of intervals, arborescent comparability, and on threshold graphs (Raspaud et al., 008)

20 Antibandwidth problem Yixum and Jinjiang (003) proposed the upper bound: min { floor( (n δ + )/), n }, where δ is the smallest degree over all v V is the largest degree over all v V

21 Antibandwidth problem Yixum and Jinjiang (003) proposed the upper bound: min { floor( (n δ + )/), n }, where δ is the smallest degree over all v V is the largest degree over all v V δ= =4 UB = min{, } = 3 3 deg = 4

22 Antibandwidth problem Yixum and Jinjiang (003) proposed the upper bound: min { floor( (n δ + )/), n }, where δ is the smallest degree over all v V is the largest degree over all v V δ= =4 UB = min{, } = ABf(G) = is opt AB(G) =

23 Antibandwidth problem: IP formulation Let xik be a binary variable that takes on the value if and only if f (i ) = k, i.e. node i takes label k. Define li = f (i ) {,,K, n} to be the label of node i. Finally, let b = AB f (G ) = min{ f (u ) f (v) : (u, v) E} be the antibandwidth of labeling f. In the antibandwidth problem we want to determine the labeling f * that maximizes b.

24 Antibandwidth problem: IP formulation Objective: maximize b Constraints: One label is assigned to each node: n i= xik =, k =,K, n

25 Antibandwidth problem: IP formulation Objective: maximize b Constraints: One node is assigned to each label: n k= xik =, i =,K, n

26 Antibandwidth problem: IP formulation Objective: maximize b Constraints: Each label li is a function of the binary variables xik: n k= k xik = li, i =,K, n

27 Antibandwidth problem: IP formulation Objective: maximize b Constraints: Require that b = min{ li l j : (i, j ) E} : b li l j,(i, j ) E

28 Antibandwidth problem: IP formulation Objective: maximize b Constraints: Binary variables xik can only take values 0 or : xik {0,}, i, k =,K, n

29 Antibandwidth problem: IP formulation Objective: maximize b Constraints: Labels li can only take on values {,..., n} : li {,,K, n}, i =,K, n

30 Antibandwidth problem: IP formulation Objective: maximize b Constraints b li l j, (i, j ) E are nonlinear: If li l j then b li l j Otherwise, b (li l j ) Introduce two binary variables to indicate case: If li l j then yij = 0 and zij = Otherwise, yij = and zij = 0

31 Antibandwidth problem: IP formulation Objective: maximize b Constraints b li l j, (i, j ) E become: b (li l j ) yij (n ), (i, j ) E b + (li l j ) zij (n ), (i, j ) E li l j then li < l j then yij + zij =, (i, j ) E b yij = 0 zij = yij = zij = 0

32 Antibandwidth problem: IP formulation max b n xik =, k =,K, n xik =, i =,K, n i= n k= n k= k xik = li, i =,K, n b (li l j ) yij (n ), (i, j ) E b + (li l j ) zij (n ), (i, j ) E yij + zij =, (i, j ) E b xik {0,}, (i, k ) E li {,,K n}, i =,,K n yik {0,}, (i, k ) E zik {0,}, (i, k ) E

33 Antibandwidth problem: IP formulation max b n xik =, k =,K, n xik =, i =,K, n i= n k= n k= k xik = li, i =,K, n b (li l j ) yij (n ), (i, j ) E b + (li l j ) zij (n ), (i, j ) E yij + zij =, (i, j ) E b xik {0,}, (i, k ) E li {,,K n}, i =,,K n yik {0,}, (i, k ) E zik {0,}, (i, k ) E IP has O(n ) variables and O(n) constraints

34 GRASP with evolutionary path-relinking Repeat outer loop Repeat inner loop ) Construct greedy randomized ) Local search 3) Mixed path-relinking 4) Update pool Evolutionary-PR

35 GRASP construction procedure We use the sampled greedy construction scheme of R. & Werneck (004) Select first node at random & label it n/ Select a small set C of unlabeled nodes Repeat while there are unlabeled nodes Evaluate incremental For each value of node c node c in C: (determine best label for c) Select the node in C with the best incremental value and label it with its best label

36 GRASP construction procedure: Selecting a small set C of unlabeled nodes The set CL of candidate nodes is made up of nodes adjacent to labeled nodes The small set C of candidate nodes is a set of α CL randomly sampled nodes from CL, where α is a random real number [0,] The value of α does not change during construction Values of α makes sampled greedy resemble a greedy construction, while values of α 0 makes it behave like a random construction

37 GRASP construction procedure: Determine the best label for a candidate node c ( lc ) lc Let and be, respectively, the smallest and largest assigned labels to the the nodes adjacent to c The best label for c is ) ( l = argmax{min( l lc, l lc ): l =,K, n} * The closest available label to l * is assigned to c

38 GRASP construction procedure: Determine the best label for a candidate node ( lc ) lc Let and be, respectively, the smallest and largest assigned labels to the the nodes adjacent to c The best label for c is ) ( l = argmax{min( l lu, l lu ): l =,K, n} * The closest available label to l * is assigned to c 3 Choose first node at random and label it n/ = 6/ = 3

39 GRASP construction procedure 3 Best label for both candidates is 6. Label one of the nodes with a 6 Candidate node

40 GRASP construction procedure 3 6 Best label for both candidates on left is and on right is 6. Label node on right with a 5 (closest available label to 6)

41 GRASP construction procedure 3 6 Best label for both candidates on left is and on right is 6. Label node on right with a 5 (closest available label to 6) Candidate node

42 GRASP construction procedure Best label for both candidates on left is and on right is 6. Label node on right with a 5 (closest available label to 6)

43 GRASP construction procedure Best label for both candidates is. Label node on top with a. Candidate node

44 GRASP construction procedure Best label for both candidates is. Label node on top with a.

45 GRASP construction procedure Best label for node on left is 6 and for node on bottom is 3. Label node on bottom with a. Candidate node

46 GRASP construction procedure Best label for node on left is 6 and for node on bottom is or 3. Label node on bottom with a. Candidate node

47 GRASP construction procedure Remaining node must be labeled with a 4.

48 GRASP construction procedure Remaining node must be labeled with a 4. ABf(G) =

49 Local Search

50 GRASP local search procedure Antibandwidth problem has a flat landscape: many solutions have same cost For a given labeling f, there may be multiple nodes u such that ABf(u) = ABf(G) Therefore, in local search, a move (swap of labels of a pair of nodes) that improves ABf(u) does not necessarily change the value of the solution ABf(G)

51 GRASP local search procedure Nodes u with unequal ABf(u) values but that are close to ABf(G) can be crucial in future iterations (swaps) of the local search, even though they cannot affect the value of the current labeling Define the set of crucial vertices of a labeling f to be C ( f ) = {u V : AB f (u ) β AB f (G )} ( β )

52 GRASP local search procedure Given a labeling f, operator move(u,v) assigns the label f (u) to node v and the label f (v) to node u, resulting in a new labeling f ' Local search scans nodes u in C(f ), changing their labels to increase their antibandwidths ( ) Let lu and lu be, respectively, the smallest and largest assigned labels to the the nodes adjacent to u The best label for u is ) ( lu = argmax{min( l lu, l lu ): l =,K, n} *

53 GRASP local search procedure Once we determine the best label l* for u, we determine the node v with this label to evaluate move(u,v) We know that label l* is good for u, but we need to determine whether label f(u) is good for node v We extend the search for a good label for u not only to node v with label l*, but also to nodes with labels close to l* The set N'(u) of suitable swapping nodes for u depends on ) ( the relationship between l*, lu, and lu

54 GRASP local search procedure ( ( * If l < lu then N '(u ) = {v V : lu f (v) lu AB f (G )} * u ) ) If l > lu then N '(u ) = {v V : lu + AB f (G ) f (v) lu*} * u ( * ) If lu lu lu then ) ) N '(u ) = {v V : lu + AB f (G ) f (v) lu AB f (G )}

55 GRASP local search procedure ( ( * If l < lu then N '(u ) = {v V : lu f (v) lu AB f (G )} * u ) ) If l > lu then N '(u ) = {v V : lu + AB f (G ) f (v) lu*} * u ( * ) If lu lu lu then ( ) N '(u ) = {v V : lu + AB f (G ) f (v) lu AB f (G )} If N'(u) =, then ABf(u) cannot be increased in a single step by changing the current label of u.

56 GRASP local search procedure a b 6 4 c d 5 e 3 f ABf(G) = v ABf(v) a b c d e f

57 GRASP local search procedure a b 6 4 c d 5 e ABf(G) = v 3 f a b c d e f ABf(v) crucial crucial crucial crucial

58 GRASP local search procedure ) lb = 6 ( lb = a b 6 4 c d 5 ABf(G) = e 3 f l ) l lu ( l lu

59 GRASP local search procedure ) lb = 6 ( lb = a b 6 4 c d 5 ABf(G) = e 3 f ) ( lu = argmax{min( l lu, l lu ): l =,K, n} = 3 * l ) l lu ( l lu

60 GRASP local search procedure ) lb = 6 ( lb = a b 6 4 d 5 ABf(G) = e l 3 f 3 ) ( lu * = argmax{min( l lu, l lu ): l =,K, n} = ( ) * Since = lu lu lu = 6 then c ) l lu ( ) N '(u ) = {v V : lu + AB f (G ) f (v ) lu AB f (G )} = {d, e, f } ( l lu

61 GRASP local search procedure ) lb = 6 ( lb = a b 6 4 d 5 swap ABf(G) = e l 3 f 3 ) ( lu * = argmax{min( l lu, l lu ): l =,K, n} = ( ) * Since = lu lu lu = 6 then c ) l lu ( ) N '(u ) = {v V : lu + AB f (G ) f (v ) lu AB f (G )} = {d, e, f } ( l lu

62 GRASP local search procedure a b 6 3 c d 5 ABf(G) = e optimal! 4 f

63 GRASP local search procedure Value of a move: Common practice is to define it as change in objective function value In antibandwidth, change in objective function provides little information Given node u and node v C(u), we define value of move(u,v) to be the difference in the antibandwidth of u.

64 GRASP local search procedure If f is the original labeling and f ' is the resulting labeling after move(u,v), then movevalue(u,v) = ABf'(u) ABf(u) Perform move(u,v) only if movevalue(u,v) > 0 and ABf'(v) ABf(G) Computation of ABf(G) is expensive: requires examination of all vertices in graph ABf(G) is not updated after each move, only when C(f) is computed (a la Glover & Laguna (997))

65 GRASP local search procedure Define the crucial vertices set C(f) from labeling f: compute ABf(G) Randomly select and remove u from C(f) Find the best label l(u) for u Find the vertex v with f(v) = l(u) while AB(G) is improving while C(f) is not empty Compute neighborhood N'(u) Select the best vertex v in N'(u) while N'(u) is not empty and Remove v from N'(u) there is no improvement If OK, swap labels f(u) and f(v) Update labeling f

66 GRASP with evolutionary path-relinking

67 GRASP with evolutionary path-relinking Initialize pool as empty set Repeat outer loop Repeat inner loop ) f construct greedy randomized ) f local search(f) 3) If pool not empty: select f' from pool 3) f mixed path-relinking (f, f') 4) Attempt to update pool with f pool evolutionary-pr(pool)

68 Mixed path-relinking Variants: trade-offs between computation time and solution quality Mixed path-relinking (Glover, 997; Rosseti, 003) G I

69 Mixed path-relinking Variants: trade-offs between computation time and solution quality Mixed path-relinking (Glover, 997; Rosseti, 003)

70 Mixed path-relinking Variants: trade-offs between computation time and solution quality Mixed path-relinking (Glover, 997; Rosseti, 003) G I

71 Mixed path-relinking Variants: trade-offs between computation time and solution quality Mixed path-relinking (Glover, 997; Rosseti, 003)

72 Mixed path-relinking Variants: trade-offs between computation time and solution quality Mixed path-relinking (Glover, 997; Rosseti, 003) I G

73 Mixed path-relinking Variants: trade-offs between computation time and solution quality Mixed path-relinking (Glover, 997; Rosseti, 003)

74 Mixed path-relinking Variants: trade-offs between computation time and solution quality Mixed path-relinking (Glover, 997; Rosseti, 003) I G

75 Mixed path-relinking Variants: trade-offs between computation time and solution quality Mixed path-relinking (Glover, 997; Rosseti, 003)

76 Mixed path-relinking Variants: trade-offs between computation time and solution quality Mixed path-relinking (Glover, 997; Rosseti, 003) I G

77 Mixed path-relinking Variants: trade-offs between computation time and solution quality Mixed path-relinking (Glover, 997; Rosseti, 003)

78 Mixed path-relinking Variants: trade-offs between computation time and solution quality Mixed path-relinking (Glover, 997; Rosseti, 003) G I

79 Mixed path-relinking Variants: trade-offs between computation time and solution quality Mixed path-relinking (Glover, 997; Rosseti, 003)

80 Mixed path-relinking Variants: trade-offs between computation time and solution quality Mixed path-relinking (Glover, 997; Rosseti, 003) Advantage: explore around neighborhoods of both input solutions.

81 Pool management Pool has at most p (e.g. p = 0 elements) ordered from best {f()} to worst {f(p)}. Let ABf()(G) be the antibandwidth of the best labeling {f()} in the pool Labeling f is accepted to the pool if ABf(G) > ABf()(G) or if ABf(G) > ABf(p)(G) and (f, pool) > δ, where n ( f, pool) = min{ f (k ) f (k ) : i pool} i k= If the pool is full and f is accepted into the pool: among all labelings f' such that ABf'(G) < ABf(G) we remove from the pool the labeling closest to f.

82 Evolutionary path-relinking ( Resende & Werneck, 004, 006 ) Evolutionary path-relinking evolves the pool, i.e. transforms it into a pool of diverse elements whose solution values are better than those of the original pool. Evolutionary path-relinking can be used as an intensification procedure at certain points of the solution process; as a post-optimization procedure at the end of the solution process.

83 Evolutionary path-relinking (EvPR) We use a variant of EvPR introduced in Resende, Martí, Gallego, & Duarte (008) Start with the pool of elite solutions

84 Evolutionary path-relinking (EvPR) While there exists a pair of pool solutions that have not yet been relinked:

85 Evolutionary path-relinking (EvPR) While there exists a pair of pool solutions that have not yet been relinked: Apply mixed path-relinking between pair

86 Evolutionary path-relinking (EvPR) While there exists a pair of pool solutions that have not yet been relinked: Apply mixed path-relinking between pair Solution of path-relinking is candidate to enter the pool: if accepted, it replaces closest solution with smaller antibandwidth

87 Evolutionary path-relinking (EvPR) While there exists a pair of pool solutions that have not yet been relinked: Apply mixed path-relinking between pair Solution of path-relinking is candidate to enter the pool: if accepted, it replaces closest solution with smaller antibandwidth

88 Evolutionary path-relinking (EvPR) EvPR ends when all pairs of pool solutions have been relinked and resulting labelings are not accepted to enter the pool.

89 Preliminary experimental results

90 Experiments Heuristics were coded in C and testing was done on a 3.0 GHz Pentium 4 PC with 3 Gb of memory CPLEX. was used to solve the integer program on a.6 GHz Itanium computer with 56 Gb of memory Four sets of test problems serve as our benchmark

91 Experiments Test problems derived from the Harwell-Boeing Sparse Matrix Collection small instances (having between 30 and 00 vertices) large instances (having between 400 and 900 vertices) -dim meshes with optimal solutions known by construction (Raspaud et al., 008) small instances (having between 90 and 0 vertices) large instances (having between 900 and 00 vertices)

92 Experiments All instances are available at Test problems from the Harwell-Boeing Sparse Matrix Collection small instances (having between 30 and 00 vertices) large instances (having between 400 and 900 vertices) -dim meshes with optimal solutions known by construction (Raspaud et al., 008) small instances (having between 90 and 0 vertices) large instances (having between 900 and 00 vertices)

93 Integer programming: small Harwell-Boeing instances name n m nz Iters million B&B million Time (secs) Soln UB bcspwr bcspwr >4h ibm pores >4h 6 8 curtis >4h 0 3 will >4h 4 bcsstk >4h 6 dwt >4h 3 58 ash >4h 7 bcspwr >4h 57 impcol.b >4h 5 nos >4h 0 48

94 Integer programming: large Harwell-Boeing instances name n m nz thous Iters million B&B nodes Time (secs) Soln UB 494bus >4h 47 66bus >4h bus >4h 3 34 bcsstk >4h 0 bcsstk >4h 0 can >4h can >4h 357 dwt >4h 50 dwt >4h 95 impcold >4h nos >4h sherman >4h 5 7

95 Experiments with GRASP For each of the 48 instances, we apply G+evPR and G+PR 30 times G+evPR: 5 iterations of inner loop and 4 iterations of the outer loop (total of 00 GRASP iterations) G+PR: 50 iterations Size of elite set is 0

96 Deviation w.r.t. best or optimum Small grids Large grids Small H-B Large H-B minimum maximum average G+PR.9 % 5.9 % 3.8 % G+evPR. % 4.8 % 3.4 % G+PR.4 % 3.8 % 3.3 % G+evPR. % 3.6 % 3.0 % G+PR 0.6 % 5.9 % 3.8 % G+evPR 0.0 % 5.9 % 3. % G+PR.0 % 3.9 %.7 % G+evPR 0.0 % 3.4 %. %

97 CPU time (seconds) Small grids Large grids Small H-B Large H-B minimum maximum average G+PR G+evPR G+PR G+evPR G+PR.0.. G+evPR G+PR G+evPR

98 Number of best (optimal) solutions Small grids Large grids Small H-B Large H-B minimum maximum % best G+PR % G+evPR % G+PR 0 0 0% G+evPR 0 0 0% G+PR % G+evPR 30 57% G+PR 0 30 % G+evPR 30 7%

99 Number of best (optimal) solutions Small grids Large grids Small H-B Large H-B minimum maximum % best G+PR % G+evPR % G+PR 0 0 0% G+evPR 0 0 0% G+PR % G+evPR 30 57% G+PR 0 30 % G+evPR 30 7% A minimum of 0 implies at least one instance (of the ) for which all 30 runs failed to find the best/opt

100 Number of best (optimal) solutions Small grids Large grids Small H-B Large H-B minimum maximum % best G+PR % G+evPR % G+PR 0 0 0% G+evPR 0 0 0% G+PR % G+evPR 30 57% G+PR 0 30 % G+evPR 30 7% A maximum of 30 implies at least one instance (of the ) for which all 30 runs found the best/opt

101 Number of best (optimal) solutions Small grids Large grids Small H-B Large H-B minimum maximum % best G+PR % G+evPR % G+PR 0 0 0% G+evPR 0 0 0% G+PR % G+evPR 30 57% G+PR 0 30 % G+evPR 30 7% %best = total number of runs that found best/opt / ( 30)

102 Small Harwell-Boeing instances (solution values) G+PR G+evPR name IP CPLEX max min avg max min avg bcspwr bcspwr ibm pores curtis54 0 will bcsstk dwt ash bcspwr impcol.b nos

103 Large Harwell-Boeing instances (solution values) G+PR G+evPR name IP CPLEX max min avg max min avg 494bus bus bus bcsstk bcsstk can can dwr dwr impcol.d nos sherman

104 Concluding remarks We described a GRASP with evolutionary path-relinking for the antibandwidth problem. The antibandwidth problem has an important application in frequency assignment in cellular telephony. To complete the experiments, we will derive run time distributions for the heuristics. Preliminary results indicate that G+PR and G+evPR have similar run time distributions. We will also conclude the CPLEX runs on the mesh instances. Preliminary results indicate that CPLEX cannot solve optimally even the smallest of the mesh instances. Our current G+evPR implementation can be made more efficient, resulting in a reduction in the number of path-relinking operations in the evolutionary path-relinking procedure.

105 Coauthors Ricardo M. A. Silva Rafael Martí & Abraham Duarte

106 The End These slides and a technical report can be downloaded from my homepage: