Friday, September 21, Flows

Transcription

1 Flows

2 Building evacuation plan people to evacuate from the offices corridors and stairways capacity For each person determine the path to follow to reach the meeting point Do not violate the capacity

3 Maximum flow problem Given a directed graph G =(N,A) and two nodes s, t N (integer) capacities u ij 0, (i, j) A Send as much flow as possible from s constraints arc capacities to t flow conservation at intermediate nodes

4 Example s 6 t 4 4 6

5 Building evacuation plan people to evacuate from the offices t corridors and stairways capacity s 50

6 Scheduling problem m identical machines set of jobs J = {1,...,n} processing time (p j,r j,d j ) j J release time due date jobs executed on a single machine they can be interrupted and resumed without loss of time Determine an assignment job-machine timing is satisfied

7 Alternative viewpoint Interrupt the flow from s to t destroy some arcs s 6 t cost proportional to the capacities

8 s-t cut Given a directed graph G =(N,A) and two nodes s, t N (integer) capacities u ij 0, (i, j) A Cut (N s,n t ) N s N t = N,N s N t =,s N s,t N t U(N s,n t )= Capacity of the cut (i,j) A:i N s,j N t u ij x(n s,n t )= Flow traversing the cut (i,j) A:i N s,j N t x ij (j,i) A:j N t,i N s x ji A + (N s,n t ) A (N s,n t )

9 Properties Given a feasible flow x ij, (i, j) A for any cut (N s,n t ) the flow going from s to t equals the flow traversing the cut v = x(n s,n t ) the flow traversing the cut is less than or equal to the capacity of the cut x(n s,n t ) U(N s,n t ) Can we prove that when x = is optimum the equality holds?

10 Computation tool Given a feasible flow s 6 0 t Can we improve the solution? What are the possible changes?

11 Residual graph Consider G =(N,A) (integer) capacities s, t N u ij 0, (i, j) A feasible flow x ij, (i, j) A G R (x) =(N,A(x) =A + (x) A (x)) A + (x) ={(i, j) A : x ij <u ij } the flow can be increased by at least one unit A (x) ={(j, i) :(i, j) A : x ij > 0} the flow can be decreased by at least one unit

12 Combine the changes Path from s to t in G R (x) flow augmentation Can we prove that if there are no paths the flow is maximum?

13 Very important property Given a directed graph G =(N,A) and two nodes s, t N either there is a path p st or there is a cut (N s,n t ),s N s,t N t such that each arc of the cutset is (i, j) :i N t,j N s

14 Augmenting_paths (G, s, t, x) foreach (i,j) A do x[i,j] 0 repeat G R Residual graph(x); Graph_search(G R,s,P); if P[t] 0 then Augment_flow(P,x); until P[t]=0 Augment_flow(P,x) ϑ Residual_capacity(P,x) foreach (i,j) P do if (i,j) A + then x[i,j] x[i,j]+ϑ else x[j,i] x[j,i]-ϑ

15 Complexity issues How to select the augmenting path Edmonds and Karp Select the augmenting path with the minimum number of arcs O(m n) More efficient algorithms Design and Analysis of Algorithms

16 Assignment problem workers jobs Compatibility Assign jobs to workers (at most one worker per job) Maximize the number of pairs

17 Maximum cardinality assignment Given a bipartite graph A S T G =(S, T, A) Matching M A no two arcs are incident in the same node The problem can be easily reduced to a Maximum Flow unit capacities how does the complexity change?

18 10 A waste disposal problem cost, capacity landfill pile of waste 55 burner pile of waste processing center recycling center

19 Road construction excess of material 4 level of the road lack of material Cost of moving material from one site to an adjacent one = 1 per truck Distribute the material at minimum cost

20 Minimum cost flow problem Given a graph G =(N,A) b i, i N node balances b i < 0 offering node b i > 0 b i =0 demanding node transfer node w.l.o.g. let us assume that i N b i =0 costs capacities c ij, (i, j) A u ij 0, (i, j) A send the required flow (flow balance constraints) capacity constraints minimize the cost of the flow

21 Generalization Max Flow and Shortest Path Tree problem can be seen as special cases of the Minimum Cost Flow Problem min c ij x ij (i,j) A x ji x ij = b i i N (j,i) BS(i) (i,j) FS(i) 0 x ij u ij (i, j) A

22 A simple yield management problem Rome Florence Bologna Milan p number of seats b ij i<j persons willing to go from i to j f ij i<j ticket price for going from i to j Accept the reservations so as to maximize the earning

23 Changing a feasible solution Given a feasible flow -4 1, 1 0, 4 0 1, 1 1, 5 i cost, capacity j 0, 4 4 Can we change the solution? What changes preserve the feasibility? What is the effect on the objective function of sending one unit of flow along a cycle?

24 Residual graph Consider G =(N,A) (integer) capacities costs u ij 0, (i, j) A c ij, (i, j) A feasible flow x ij, (i, j) A G R (x) =(N,A(x) =A + (x) A (x)) A + (x) ={(i, j) A : x ij <u ij } the flow can be increased by at least one unit cost: c ij, (i, j) A + (x) A (x) ={(j, i) :(i, j) A : x ij > 0} the flow can be decreased by at least one unit cost: c ij, (j, i) A (x)

25 Characterizing the optimal solution If in the residual graph there is a negative cost cycle the flow can be improved We can prove that If there are no negative cost cycles the flow is optimal

26 Negative cycle elimination x Feasible_flow; GR Residual graph(x); while Negative_cost_cycle(C, G R ) ϑ Residual_capacity(C, x); Update_flow(ϑ, C, x); Update(G R ) Update_flow(C, ϑ, x) foreach (i,j) C do if (i,j) A + then x[i,j] x[i,j]+ϑ else x[j,i] x[j,i]-ϑ

27 Detecting negative cost cycles From the complexity analysis we know that a node cannot be inserted into Q more than n-1 times If a node is inserted n times negative cost cycle

28 Starting feasible flow Disregard the arc costs Consider the flow in the original arcs, , 1 5, 1, 10, s - 5 4, 6 7, 6 1, -, 4 Add a source and a sink Connect the source to all offer nodes Connect the demand nodes to the sink 4 1 1, 6 t Move the balances on the capacities Compute the maximum flow from s to t If the flow = i:b i >0 b i then there is a feasible solution

29 Complexity issues initialization max flow at each iteration: SPT_L_queue update the flow update the graph O(mn) O(n) How many iterations at worst? Selecting the cycle according to a particular rule O(m n log n) iterations

30 A distribution problem oranges , , 10 1, 10 1, , , 10 1, 10 1, 10 pears , Multicommodity flow A set of flow variables for each commodity Flows of different commodities concur in the use of common resources (arc capacities)

31 A multi terminal routing problem We want to route traffic the intranet network of Politecnico unit routing costs 15 LC capacities (in both directions) Origin/Destination matrix o/d MiL MiB CO LC CR PC MN CO MiL MiB CR MN MiB CO LC MiL 1 5 CR PC PC MN Problem:send the required flows from origins to destinations, satisfy the capacities, minimize the overall cost