Weighted Acyclic Di-Graph Partitioning by Balanced Disjoint Paths

Transcription

1 Weighted Acyclic Di-Graph Partitioning by Balanced Disjoint Paths H. Murat AFSAR Olivier BRIANT G-SCOP Laboratory Grenoble Institute of Technology 1

2 Presentation Plan Introduction Problem Definition Solution Method Numerical Results Conclusions and Perspectives 2

3 Partitioning Problems - Disjoint Paths Covering all the nodes Each node appears on just one path Polynomially solvable - Flow 3

4 Problem Context Telecommunication networks Noninterfering messages Transportation Aircraft Rotation Problems 4

5 Problem Definition A directed acyclic graph G (V, A), Find K disjoint and weight balanced paths p 1, p 2,..., p K such that Each node V is covered by exactly one path. 5

6 Difficulty Objective : min K k=1 c pk Where cpk = g( w v )= 1 M (M w v ) 2 v p k v p k And M is the ideal path weight M = 1 K v V w v 6

7 Similar Problems Finding 2 node disjoint paths with bounded weight on a DAG NP-complete Vertex covering by minimum weight paths, on trees NP-Hard (even in unweighted case) Partitioning problem NP-hard 7

8 Partitoning Graph by Disjoint Paths 8

9 Partitoning Graph by Disjoint Paths

10 Partitoning Graph by Disjoint Paths

11 Problem Formulation min subject to (My (s,v) x (s,v) ) 2 v V \{s,t} y v,v (v,v ) δ (v ) (v,v ) δ (v ) x v,v v δ + (v) (s,v ) δ + (s) M y v,v =0 v V \{s, t} (v,v) δ + (v) (v,v) δ + (v ) x v,v = w v v V \{s, t} y v,v =1 v V \{s, t} y s,v = K x v,v W v y v,v (v, v ) A x v,v R (v, v ) Ay v,v {0, 1} (v, v ) A 9

12 Decomposition Dantzig-Wolfe Master Problem : Combining paths to partition the graph Slave Problem : Finding good paths 10

13 Master Problem min c p λ p p P subject to p P p P v p λ p = K λ p =1 v V λ p {0, 1} p P 11

14 Reduced Cost of a Path c p = g( v v p c p (u) = g( v v p w v ) w v ) v v p u v u 0 Where u v and u 0 are the dual variables associated to the node v and to the first constraint, respectively 12

15 Slave Problem Solution Methods Max Flow - Max Cost to cover all the nodes 2OPT on flow solution to minimize the minimum reduced cost : Finding a feasible solution with promissing paths. Optimal dynamic programming : A multi label setting dynamic model to find max u v for v s v each node v for each total weight v s v w v = W s v 13

16 Max Flow - Max Cost Flow 1 on graph G (V, A): cost(a) = p a p λ p a A Flow 2 on graph G (V, A ): With only a limited number of exiting arcs from all node v such that p a p λ p

17 2OPT For the paths p, p in the solution S, for every couple of nodes v p and v p compute c p new = g(w s v + W v t) (U s v + U v t) c p new = g(w s v + W v t) (U s v + U v t ) Where U s v is the sum of dual values of the nodes on the path between s and v If min{ c new p, c p new } < min{ c p, c p } then accept the swap 15

18 Dynamic programming f (v, W ) = u v + max, W w v ) v δ (v) And naturally f (s, 0) = 0 Where W is the total weight of the path between s and v 16

19 Dynamic programming f (v, W ) = u v + max, W w v ) v δ (v) And naturally f (s, 0) = 0 Where W is the total weight of the path between s and v min X p P(v) w v = W v p c p (u) = g(w ) f (v, W ) Where P(v) is the set of paths ending at the node v 16

20 Lagrangian Lower Bound c p (u) =g( w v ) v v p v v p u v 17

21 Lagrangian Lower Bound c p (u) =g( w v ) v v p v v p u v θ1(u) = K min c p(u)+ u v p P v V θ2 (u) = min c p(u)+ p P(v i ) i K v V u v Where P(v i ) is the set of paths ending with the node v i So each path ends at a different node Obviously θ 2 (u) θ 1 (u) 17

22 Instance Characteristics Table: General DAG with 700 nodes and K =80 A % w min =40 w max = 120 A % w min =40 w max = 120 A % w min =40 w max = 120 A % w min =40 w max = 120 A % w min =40 w max = 120 B % w min =1 w max =1 B % w min =1 w max =1 B % w min =1 w max =1 B % w min =1 w max =1 B % w min =1 w max =1 18

23 Instance Characteristics Table: Flight-like nodes with 700 nodes, w min = 40,w max = 120, K =80 F wait min =40 wait max =80 nb arcs : F wait min =40 wait max = 100 nb arcs : F wait min =40 wait max = 120 nb arcs : F wait min =40 wait max = 150 nb arcs : F wait min =40 wait max = 240 nb arcs :

24 RMP Linear Relaxation vs Lagrangean Relaxation Restricted Master Linear Relaxation 1 : λ p = K p P Restricted Master Linear Relaxation 2 : λ p K p P 19

25 RMP Linear Relaxation vs Lagrangean Relaxation 100 Lower and Upper Bounds objlp=k objlp<k GLB GUB

26 RMP Linear Relaxation vs Lagrangean Relaxation 900 Initial and Final Solutions

27 RMP Linear Relaxation vs Lagrangean Relaxation 45 Initial and Final Solution Costs

28 Numerical Results Instance iter ZLP 1 ZLP 2 GLB GUB Columns A ,141 0,070 0,141 10, A ,141 0,002 0,141 7, A ,141 0,002 0,141 3, A ,141 0,096 0,141 0, A ,141 0,106 0,141 0, B ,714 0,555 1,714 1, B ,714 0,940 1,714 1, B ,714 0,892 1,714 1, B ,714 0,937 1,714 1, B ,714 1,714 1,714 1, F ,945 2,945 2,945 2, F ,624 1,665 2,624 2, F ,689 2,689 2,689 2, F ,120 2,120 2,120 2, F ,946 1,946 1,946 1,

29 Total CPU Time (in sec.) per Method Instance Total Master Slave Flow 2OPT A A A A A B B B B B F F F F F

30 Conclusions Gap between lower and upper bounds when the graph is sparse Optimality proof of heuristic approach when the graph is dense 2-OPT and Master problem are dominating the resolution time 22

31 Perspectives More tests to come Improving the lower bound Branch & Price for the sparse graphs to show the optimality 23

32 Weighted Acyclic Di-Graph Partitioning by Balanced Disjoint Paths H. Murat AFSAR Olivier BRIANT G-SCOP Laboratory Grenoble Institute of Technology 24