minimum collection of vertice disjoint path that covers a given vertice set - algorithm

Question

Problem
Given:
A directed graph G
A source vertex s in G and a target vertex t in G
A set S of vertices of G
I want to find a collection of paths from s to t that covers S.
Then I want to partition the collection of paths into subcollections of vertex-disjoint paths.
Under these constraints, the objective is to minimise the number of subcollections.
Example
For instance, [C1 = {p1,p2,p3}, C2= {p4,p5}, C3= {p6,p7}] is a solution if:
each p_i is a path from s to t
p1,p2,p3 have no vertices in common except s and t;
p4, p5 have no vertices in common except s and t;
p6,p7 have no vertices in common except s and t;
collectively, the 7 paths cover all vertices of S.
In that case, the number of subcollections is 3.
Question
What are some good algorithms or heuristics for this optimisation problem?
I already know min cost flow, and disjoint path algos, but they don't apply in my settings.
I tried min cost flow / node disjoint paths but one run only gives one collection at a time. I don't know how to adjust cost to cover the unexplored vertices.

Answer 1

Given:
A directed graph G
A source vertex s in G and a target vertex t in G
A set S of vertices of G
I want to find a collection of paths from s to t that covers S.
Use Dijkstra's algorithm to find a path from s to every vertex in S and from every point in S to t.
Connect the paths to and from each S vertex into one path from s to t via a point in S.
You now have a collection of paths that, together, cover S. Let's call it CS.
Then I want to partition the collection of paths into subcollections
of vertex-disjoint paths.
Note that if s, the source vertex, has an out degree of sOD, there can be no more than sOD paths in each vertex disjoint collection.
Construct vVDP, an empty vector of vertex disjoint path collections
LOOP P over paths in CS
SET found FALSE
LOOP cs over collections in vVDP
IF P is vertex disjoint with every path in cs
add P to cs
SET found TRUE
BREAK out of LOOP cs
IF found == false
ADD collection containing P to vVDP
Here is a C++ implementation of this algorithm
void cProblem::collect()
{
// loop over paths
for (auto &P : vpath)
{
// loop over collections
bool found = false;
for (auto &cs : vVDP)
{
//loop over paths in collection
bool disjoint;
for (auto &csPath : cs)
{
// check that P is vertex disjoint with path in collection
disjoint = true;
for (auto vc : csPath)
{
for (auto &vp : P)
{
if (vp == vc) {
disjoint = false;
break;
}
}
}
if( ! disjoint )
break;
}
if (disjoint)
{
// P is vertex disjoint from every path in collection
// add P to the collection
cs.push_back(P);
found = true;
break;
}
}
if (!found)
{
// P was NOT vertex disjoint with the paths in any collection
// start a new collection with P
std::vector<std::vector<int>> collection;
collection.push_back(P);
vVDP.push_back(collection);
}
}
}
The complete application is at https://github.com/JamesBremner/so75419067
Detailed documentation id the required input file format at
https://github.com/JamesBremner/so75419067/wiki
If you post a real example in the correct format, I will run the algorithm on it for you.