Closed torressa closed 1 year ago
Happy if we want to further split this into three issues, but thinking about the common formulation probably leads to a cleaner implementation.
Yes such network flow problems certainly deserve a nup/nups. How exactly this should be realized in the backend? Many options, we should brainstorm a little before starting this. My current thinking is "simplicity first", but what exactly that means remains to be determined.
In the short term I agree, simplicity first. It would be good to just implement various specific network flow problems without any dependency on one another.
Long term - as an educational tool it would be great to show a heirarchy of how various problems reduce to network flows, then each specific case can use the network flow nup as it's backend.
Cool! Will split this into three at some point this week.
How exactly this should be realized in the backend?
In the backend complicated version, I was thinking of just having a single private network flow formulation for these problems nupstup._network_flow
and then using the table above to formulate and solve the problem depending on the call.
# nupstup.min_cost_flow would return
nupstup._network_flow(G, costs, capacities, demand, source, sink)
# nupstup.shortest_path
nupstup._network_flow(G, costs, [1 for _ in arcs], [1, 1], source, sink)
# max_flow returns (need to define V' for this one)
nupstup._network_flow(
G,
[-1 for e in edges if e[0] == source],
capacities,
None,
source,
sink,
Vd=[n for n in nodes if n not in [source, sink]],
)
Except for min-cut where max-flow has to be run then we have to process the cutsets using the solution but this we will have to do anyway.
I think this sort of grouping would help maintenance, but yeah I agree it is a bit too much for the beginning.
network_flow
might as well be public: it could make a good nup itself if there is more information we want to convey about the formulation
Closing as base functionality is already in; follow-up in #51.
Why this Nup?
Flow problems are present in many applications. Particularly, the following share the same formulation (with some small changes):
All single source/sink.
Given a digraph $G=(V, E)$, with source $s$ and sink $t$, we can formulate these as follows:
$$ \begin{alignat}{2} \min \quad & \sum{(i, j) \in E} c{ij} x{ij} \ \mbox{s.t.} \quad & \sum{j \in \delta^+(i)} x{ij} - \sum{j \in \delta^-(i)} x_{ji} = bi & \forall i \in V' \ & 0 \leq x{ij} \le B_{ij} & \forall (i, j) \in E \ \end{alignat} $$
Where $\delta^+(\cdot)$ ( $\delta^-(\cdot)$ ) denotes the outgoing (incoming) neighours, and
$$ b_i = \begin{cases} D & \text{if } i = s\ -D & \text{if } i = t \ 0 & \text{ow} \end{cases} $$
Does it fall under an existing category?
Graphs
What will the API be?
Additional context
Well-known graph problems, so graph theory terminology is fine.
There are many real-world applications and other graph problem transformations for these, so would be good to have some of these in there as well.
Problem 1 -> Minimum weight bipartite matching Problem 3 -> Maximum cardinality bipartite matching, closure problem
If we go up another dimension and add $x_{ij}^k$ (also $D^k$) for a commodity $k$ we can model multicommodity flows with this same formulation as well.