CSL 356: Analysis and Design of
Algorithms
Ragesh Jaiswal
CSE, IIT Delhi
Network Flow: Application
Min-flow with lower bounds
Network Flow: Applications
 Problem (Min-flow with lower bounds): Given a network
graph where each edge e has an integer capacity lower bound
l(e). Find an s-t flow with minimum value such that
following constraints are met:
 Lower bound: For all edge e, f(e) ≥ l(e).
 Flow conservation: For all v≠s,t, fin(v) = fout(v).
1
1
s
100
50
1
t
1
Network Flow: Applications
 Let f be an s-t flow (not necessarily minimum such flow)
such that all the lower bounds are met.
 Consider the graph G’ where all the edges are reversed and a
reversed edge e has a capacity of (f(e) – l(e)).
 Let f ’ be maximum value t-s flow in G’.
 Note we have usual capacity constraints now.
l(e)
f(e)-l(e)
Network Flow: Applications
 Let f be an s-t flow (not necessarily minimum such flow)
such that all the lower bounds are met.
 Consider the graph G’ where all the edges are reversed and a
reversed edge e has a capacity of (f(e) – l(e)).
 Let f ’ be maximum value t-s flow in G’.
 Claim 1: g=(f-f ’) is a “lower bound” s-t flow.
Network Flow: Applications
 Let f be an s-t flow (not necessarily minimum such flow)
such that all the lower bounds are met.
 Consider the graph G’ where all the edges are reversed and a
reversed edge e has a capacity of (f(e) – l(e)).
 Let f ’ be maximum value t-s flow in G’.
 Claim 1: g=(f-f ’) is a “lower bound” s-t flow.
 Proof:
 Lower bound: For all edges e, g(e)=(f(e) – f ’(e)) ≥ l(e).
 Flow conservation: For all v≠s,t, gin(v) = gout(v).
Network Flow: Applications
 Let f be an s-t flow (not necessarily minimum such flow)




such that all the lower bounds are met.
Consider the graph G’ where all the edges are reversed and a
reversed edge e has a capacity of (f(e) – l(e)).
Let f ’ be maximum value t-s flow in G’.
Claim 1: g=(f-f ’) is a “lower bound” s-t flow.
Claim 2: g=(f-f ’) is a “lower bound” s-t flow of minimum
value.
Network Flow: Applications
 Let f be an s-t flow (not necessarily minimum such flow)




such that all the lower bounds are met.
Consider the graph G’ where all the edges are reversed and a
reversed edge e has a capacity of (f(e) – l(e)).
Let f ’ be maximum value t-s flow in G’.
Claim 1: g=(f-f ’) is a “lower bound” s-t flow.
Claim 2: g=(f-f ’) is a “lower bound” s-t flow of minimum
value.
 Proof:
 For the sake of contradiction assume there is a flow g’ that is a s-t lowerbound flow and v(g’) < v(g).
 Consider the flow f ’’ = (f – g’). (f is large enough to ensure that
for all e, f(e)-g’(e)>0.)
 Claim 2.1: f ’’ is a t-s flow in G’.
 Claim 2.2: v(f ’’) > v(f ’).
Network Flow: Applications
Maximum Cohesiveness
Network Flow: Applications
 Problem (Maximum cohesiveness): Given a graph G=(V,E),
find a non-empty subset S of vertices such that e(S)/|S| is
maximized.
 e(S): Number of edges that have both end vertices in S.
e(S)/|S| = 1
Network Flow: Applications
 We will give an algorithm that checks if there is subset with
cohesiveness at least ∆.
 We will then use this algorithm to find the subset with
maximum cohesiveness. Note the algorithm will run in
polynomial time.
 We construct the following network graph G’.
 There is a vertex corresponding to every vertex in G.
 There is a vertex vij corresponding to each edge in G.
 There is a source s and a sink vertex t.
 Vertex vij has edges to vertex i and j. This has capacity ∞.
 There is an edge from s to all vertices vij. These edges have
capacity 1.
 There is an edge from every vertex v to t with capacity ∆.
Network Flow: Applications
 We construct the following network graph G’.






There is a vertex corresponding to every vertex in G.
There is a vertex vij corresponding to each edge in G.
There is a source s and a sink vertex t.
Vertex vij has edges to vertex i and j. These have capacity ∞.
There is an edge from s to all vertices vij. These have capacity 1.
There is an edge from every vertex v to t with capacity ∆.
12
1
3
2
2
3
34
5
35
4
G
1
24
s
1
23
∞
G’
45
4
5
∆
t
Network Flow: Applications
 Let (A,B) be a min-cut in G’. Let S be the vertices on the right that
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
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are in B.
Claim 1: If vij is in A, then both i and j are in A.
Claim 2: If i and j is in A, then vij is in A.
Claim 3: c(A,B) = |E| - e(S) + |S|*∆
Claim 4: There is a subset S with cohesiveness > ∆ if and only if the
min-cut in G’ has capacity < |E|.
12
1
3
2
2
3
34
5
35
4
G
1
24
s
1
23
∞
G’
45
4
5
∆
t
End