Math 5490 Network Flows

Transcription

1 Math 5490 Network Flows Lecture 20: Min-Cost Flow Problems Stephen Billups University of Colorado at Denver Math 5490Network Flows p.1/19

2 Preliminaries Announcements: Sign-up for presentation times before Monday. (May 2nd and April 27th are taken). Tonight: Optimal flows and Optimal Node Potentials Math 5490Network Flows p.2/19

3 Optimal flows and optimal node potentials Questions: 1. Given an optimal flow, how can we get optimal node potentials? 2. Given optimal node potentials, how can we get an optimal flow? Math 5490Network Flows p.3/19

4 Optimal flows and optimal node potentials Questions: 1. Given an optimal flow, how can we get optimal node potentials? Ans: solve shortest path problem. 2. Given optimal node potentials, how can we get an optimal flow? Ans: solve max flow problem. Math 5490Network Flows p.3/19

5 Computing Optimal Node Potentials Given an optimal flow x, 1. Construct residual network G(x ). 2. Choose a node to be the source node, say node Use shortest path algorithm to calculate shortest path distances d( ), from node 1 to s to the rest of the nodes. 4. Set π = d. Questions: 1. How do we know that the distance labels are well defined (remember, some of the arc costs c ij could be negative). 2. Why are the labels π optimal node potentials? Math 5490Network Flows p.4/19

6 Computing Optimal Node Potentials Given an optimal flow x, 1. Construct residual network G(x ). 2. Choose a node to be the source node, say node Use shortest path algorithm to calculate shortest path distances d( ), from node 1 to s to the rest of the nodes. 4. Set π = d. Questions: 1. How do we know that the distance labels are well defined (remember, some of the arc costs c ij could be negative). Ans: G(x ) doesn t have any negative cost cycles (since x is optimal (see Thm. 9.1)). 2. Why are the labels π optimal node potentials? Math 5490Network Flows p.4/19

7 Computing Optimal Node Potentials Given an optimal flow x, 1. Construct residual network G(x ). 2. Choose a node to be the source node, say node Use shortest path algorithm to calculate shortest path distances d( ), from node 1 to s to the rest of the nodes. 4. Set π = d. Questions: 1. How do we know that the distance labels are well defined (remember, some of the arc costs c ij could be negative). Ans: G(x ) doesn t have any negative cost cycles (since x is optimal (see Thm. 9.1)). 2. Why are the labels π optimal node potentials? Need to show that they satisfy one of the optimality conditions. Math 5490Network Flows p.4/19

8 Optimality of Node Potentials Recall the reduced cost optimality conditions: x and π are optimal if and only if Math 5490Network Flows p.5/19

9 Optimality of Node Potentials Recall the reduced cost optimality conditions: x and π are optimal if and only if c π ij 0 for all (i, j) G(x ). Math 5490Network Flows p.5/19

10 Optimality of Node Potentials Recall the reduced cost optimality conditions: x and π are optimal if and only if c π ij 0 for all (i, j) G(x ). Is this condition satisfied for the node potentials calculated above? Yes!! The shortest path optimality conditions imply that d(j) d(i) + c ij for all (i, j) G(x ). Letting π = d and rearranging, we get 0 c ij π(i) + π(j) = c π ij for all (i, j) G(x ). Math 5490Network Flows p.5/19

11 Example Math 5490Network Flows p.6/19

12 Obtaining Optimal Flows Given optimal node potentials π, we can obtain an optimal flow x as follows: 1. Use the complementary slackness conditions to determine flows on all the arcs with nonzero reduced costs. If c π ij > 0, x ij =. If c π ij < 0, x ij =. 2. Delete these arcs from the network, and adjust the supplies b( ) appropriately. Specifically, for each arc where c π ij < 0, increase b(j) by u ij, and decrease b(i) by u ij. Find a feasible flow on the resulting network. (How?) Math 5490Network Flows p.7/19

13 Example Math 5490Network Flows p.8/19

14 Cycle-Canceling Algorithm Main Idea: Push flow around negative-cost cycles in the residual network until there aren t any more. establish a feasible flow x in the network while G(x) contains a negative cycle do begin end identify a negative cycle W augment as much flow as possible around the cycle Math 5490Network Flows p.9/19

15 Example (Figure 9.8) Math 5490Network Flows p.10/19

16 Integrality Property Theorem If all arc capacities and supplies/demands of nodes are integer, the minimum cost flow problem always has an integer minimum cost flow. Proof (Main Ideas) Recall from Max-Flow theory that we can find an integer solution to the feasible flow problem (Application 6.1 and Theorem 6.5). Using the cycle cancelling algorithm, we can show that If the flow is integral after k iterations, then the residual capacity of any arc is integral, the capacity of any cycle in the residual network is integral, an integral amount of flow will be augmented at the k + 1st iteration. Thus, the flow will be integral after k + 1 iterations. By induction on k, the flow will be integral after every iteration of the algorithm (and in particular, will be integral when the algorithm terminates, yielding an integer minimum cost flow). But how do we know the algorithm terminates? Math 5490Network Flows p.11/19

17 Termination and Run-time Every iteration reduces the objective value by at least 1. Upper bound on initial feasible flow = mcu. Lower bound on optimal flow = mcu. O(mCU) iterations. (Finite termination guaranteed.) O(nm) time per iteration. Total run-time = O(nm 2 CU) (Not polynomial). Math 5490Network Flows p.12/19

18 Variations of Cycle Canceling Network Simplex Method One of the fastest algorithms available in practice. But not polynomial (because of degeneracy). Polynomial Algorithms Augment flow on the negative cycle that will give the maximum improvement. But finding the max-improvement cycle is NP-complete! (so some modifications are needed). Augment flow along a cycle with minimum mean cost. Might not yield the max-improvement, because the capacity of the cycle might be small. Math 5490Network Flows p.13/19

19 Successive shortest path algorithm Overview: Maintains a pseudoflow x and potentials π that satisfy the reduced cost optimality conditions. Works toward feasibility of x by pushing flow along shortest paths from excess nodes to deficit nodes. Review: A pseudoflow is a solution x that satisfies the capacity and nonnegativity constraints, but not necessarily the flow balance constraints. Math 5490Network Flows p.14/19

20 Key properties Then, Suppose that x is a pseudoflow satisfying the reduced cost optimality conditions for some choice of node potentials π. Let s be any node, and let d represent the shortest path distances from s to all other nodes in G(x), with c π ij as the length of arc (i, j). Lemma x satisfies the reduced cost optimality conditions with respect to the node potentials π = π d. 2. c π ij Lemma 9.12 = 0 for any arc (i, j) that is in the shortest path from s to any other node. Let x be formed from x by sending flow along a shortest path from node s to some other node k; then x also satisfies the reduced cost optimality conditions. Note: the potentials that satisfy the optimality conditions are π. Math 5490Network Flows p.15/19

21 Proof 1. x satisfies c π ij 0 for all (i, j) G(x). 2. d satisfies shortest path optimality conditions: d(j) d(i) + c π ij for all (i, j) G(x). 3. Substituting c π ij = c ij π(i) + π(j) and rearranging yields: c ij (π(i) d(i)) + (π(j) d(j)) 0. or c π ij 0. (establishing statement 1 of Lemma 9.11). 4. For each arc (i, j) in a shortest path P from s to l, d(j) = d(i) + c π ij = d(i) + c ij π(i) + π(j) = c π ij = 0. (statement 2 of Lemma 9.11) 5. Augmenting flow along any arc in (i, j) in P might add the reverse arc (j, i) to the residual network. But, since c π ij = 0, then cπ ji = 0, so reduced cost optimality conditions are still satisfied. (establishing Lemma 9.12). Math 5490Network Flows p.16/19

22 Algorithm Outline The lemmas above justify the following strategy: 1. Start with x = 0 and π = 0. x is a pseudoflow. Why? x and π satisfy the reduced cost optimality conditions. Why? 2. Identify an excess node k and a deficit node l. Push as much flow as possible along shortest path from k to l. The resulting flow still satisfies the reduced cost optimality conditions (but we need to update the node potentials to satisfy the conditions). 3. Update potentials π = π d (where d are the shortest path distances from k). Now the updated x and π satisfy the reduced cost optimality conditions. And we have moved closer to satisfying the flow balance constraints. 4. Repeat Steps 2 and 3 until there are no more excess nodes. Math 5490Network Flows p.17/19

23 Example (Figure 9.10) Math 5490Network Flows p.18/19

24 Generic Successive Shortest Path Algorithm x := 0, π = 0 e(i) := b(i) for all i N set E := {i : e(i) > 0} and D := {i : e(i) < 0} while E do select k E and l D determine shortest path distances d from node k to all other nodes in G(x) with respect to the reduced costs c π ij. let P denote shortest path from node k to node l. update π := π d δ := min [e(k), e(l), min{r ij : (i, j) P }] augment δ units of flow along the path P update x, G(x), E, D, and the reduced costs end (while) Math 5490Network Flows p.19/19