1. Algorithms and Networks
Minimum Cost Flow
Algorithms and Networks

2. This lecture The minimum cost flow problem: statement and applications
The cycle cancelling algorithm
A polynomial time variant of cycle cancelling
The successive shortest paths algorithm
Algorithms and Networks: Minimum Cost Flow

3. Minimum Cost Flow I Edges have Cost of flow f:
Capacity: c(u,v): bound on amount of flow that can go through the edge
Cost: cost(u,v): cost that must be paid per unit of flow that goes through the edge.
Cost of flow f:
Algorithms and Networks: Minimum Cost Flow

4. Minimum cost flow problem
Given: Network G, c, cost, s, t, and a target flow value r.
Question: Find a flow from s to t with value r, with minimum cost.
Algorithms and Networks: Minimum Cost Flow

5. Unbounded capacities Some edges may have unbounded capacities
If there is a cycle of negative cost with only edges with unbounded capacity:
Arbitrary small cost (degenerate case)
Finding such a cycle: shortest paths algorithms (see lectures by Johan)
Otherwise: simple transformation to bounded capacities
E.g., set each unbounded capacity to sum of all bounded capacities
Algorithms and Networks: Minimum Cost Flow

6. Separate demands
Similar to max-flow with multiple sources and multiple sinks
Or: work with “demand”, which can be positive or negative G s1 t1 t s sk tr
Algorithms and Networks: Minimum Cost Flow

7. Nonnegative arc costs
It is possible to transform the network to case where all costs are nonnegative.
In case of negative costs: assume bounded capacity. Modify network to equivalent one with nonnegative costs:
Cost -r
Or work
with separate
demands b a
Capacity c
Cost 0
Cost 0
Cost r s b a t
Capacity c
Capacity c
Capacity c
Algorithms and Networks: Minimum Cost Flow

8. Applications Transport problems Minimum cost matchings
Reconstruction of Left Ventricle from X-ray projections
Image: 2d bit array; known are sums of columns, rows; probabilities for each bit
Look for image with correct row and column sums of maximum probability
Can be modelled as minimum cost flow problem
Algorithms and Networks: Minimum Cost Flow

9. Application: Optimal loading of hopping airplane1 2 n
bij weight units (or passengers) can be transported from i to j
Each gives a profit of fij
Plane can never carry more than p units
How much units do we transport of each type for maximum profit?
Capacity i ® j: p
Node ij has supply bij
Cost from ij to i: - fij
Node i has demand sum over all bji 14 24 13
All other arcs
infinite cap.
All other costs 0 12 23 34 1 2 3 4
Algorithms and Networks: Minimum Cost Flow

10. Cycles and circulations; paths and flows
We can view:
Cycles as a special type of circulations
Paths as a special type of flows
If we have a directed cycle c, then we can see it as a circulation:
c(e) = 1 when e is on c
c(e) = 0 otherwise
If we have a path p from s to t, we can see it as a flow with value 1:
p(e) = 1 when e is on p
p(e) = 0 otherwise
Algorithms and Networks: Minimum Cost Flow

11. First: useful theorems on flows and circulations
Theorem Let f be a circulation. Then f is a nonnegative linear combination of cycles in G.
Proof.
Ignore lower bounds.
If f(v,w) = 0 everywhere, then it clearly holds.
Otherwise, there is a directed cycle c in G with each edge e on c: f(e)>0
Let r be the minimum flow of an edge on c, i.e., r = min {f(e) | e on c}
Use induction: the theorem also holds for f – cr:
f-cr(e) = f(e) when e is not on c
f-cr(e) = f(e) – r when e is on c …
A&N: Maximum flow

12. Corollary for flows
Corollary: Let f be a flow from s to t. f is a nonnegative linear combination of cycles in G and paths from s to t.
Proof: Add edge from t to s with flow equal to value(f). Now, we have a circulation. Apply previous theorem. Cycles that use (t,s) correspond to a path from s to t. … QED
Algorithms and Networks: Minimum Cost Flow

13. More on these linear combinations
If circulation f is integer, then f can be written as an`integer scalared’ linear combination of cycles in G
A flow f with each f(e) an integer is the integer scalared linear combination of cycles and paths from s to t
Algorithms and Networks: Minimum Cost Flow

14. Algorithms for finding minimum cost flow
Here: two simple algorithms, based on the residual network
Better algorithms exist, see also the exercise sets
Algorithms and Networks: Minimum Cost Flow

15. Residual network Define the residual network Gf of a flow f
Capacities as for maximum flow algorithms
If f (u,v)>0, then costf (u,v) = cost(u,v), and costf (v,u) = – cost(u,v).
Algorithms and Networks: Minimum Cost Flow

16. Example Capacity 5, cost 3 Suppose we send 1 flow from a to b a b
In Gf:
Capacity 1, cost -3
Algorithms and Networks: Minimum Cost Flow

17. Cycle cancelling algorithm
Make a feasible flow f in the network
while Gf has a negative cycle do
Find a negative cycle C in Gf
Let D be the minimum residual capacity cf of an edge on C
Add D units of flow to each edge on C: this is a new feasible flow of smaller cost
Output f.
Algorithms and Networks: Minimum Cost Flow

18. Cycle cancelling algorithm is correct
Theorem: a flow f has minimum costs, if and only if Gf has no negative cycle.
Proof:
If Gf has negative cycle, then we can improve f to one with smaller cost.
Suppose f is a flow, and f’ is an optimal flow. f’ – f is a circulation in Gf, hence a linear combination of cycles, and if f is not optimal, then the total cost of these cycles is negative, so there is a negative cycle in this set: it is a cycle in Gf.
Algorithms and Networks: Minimum Cost Flow

19. More on the cycle cancelling algorithm
No guarantee that it uses polynomial time.
Corollary: if all capacities and target flow value are integral, then there is an optimal integer minimum cost flow.
The cycle cancelling algorithm finds an integer flow in this case.
Variant: using always the minimum mean cost cycle gives a polynomial time algorithm!
Algorithms and Networks: Minimum Cost Flow

20. Minimum mean-cost circulation algorithm
Cycle cancelling algorithm but find always the minimum mean cost cycle and use that.
O(nm) time to find the cycle.
A theorem shows that O(nm2 log2 n) iterations are sufficient.
O(n2m3 log2 n) algorithm.
Uses algorithm from exercise set.
Algorithms and Networks: Minimum Cost Flow

21. Lower bounds Cycle cancelling algorithm also works with lower bounds:
First find a feasible flow, respecting lower bounds (see lecture on max flow)
Remaining algorithm the same, with the residual network defined as for flows with lower bounds
Algorithms and Networks: Minimum Cost Flow

22. Successive shortest paths
Start with flow f with f(u,v)=0 for all u,v.
repeat until value(f ) = r
Find the shortest path P in Gf from s to t
Let q be the minimum residual capacity of an edge on P.
Send min(q,r – value(f)) additional units of flow across P.
Algorithms and Networks: Minimum Cost Flow

23. Time and correctness Time: possibly exponential time …
Correctness: only if G has no cycle with negative cost
If G has a negative cycle, the algorithm may give an incorrect answer, e.g., when we have target flow r=0
Proof of correctness: next
Algorithms and Networks: Minimum Cost Flow

24. Correctness of successive shortest paths algorithm
Theorem: If G has no negative cycle, then the successive shortest paths algorithm gives the minimum cost flow.
Proof follows from
Claim: The ssp-algorithm maintains the following invariant: f has minimum cost among all flows with value value(f).
Algorithms and Networks: Minimum Cost Flow

25. Proof of claim
Suppose we obtain f’ from f by sending flow over a path P.
From invariant: f has minimum cost for flows with value equal to value(f)
Suppose f’’ is a minimum cost flow with same value as f’.
f’’ – f is a flow in Gf. Hence, it can be written as a linear combination of cycles in Gf and paths from s to t in Gf.
All cycles in Gf have cost at least 0: otherwise f is not of minimum cost
All paths in this combination have cost at least the cost of P: this is how the algorithm works (P is shortest path)
It follows that cost(f’’ – f) >= cost (f’ – f)
So, cost(f’’) >= cost(f’) which show minimality of cost of f’.
QED
Algorithms and Networks: Minimum Cost Flow

26. Finally More efficient algorithms exist Some use scaling techniques:
Scaling on capacities
Scaling on costs
Scaling will be discussed in the chapter on shortest paths
Algorithms and Networks: Minimum Cost Flow