1. 3.5 Minimum Cuts in Undirected Graphs
3.5.1 Global Minimum Cuts
Given connected undirected graph 𝐺=(𝑉,𝐸), capacity 𝑢∈𝑅𝐸 (𝑢𝑒>0),
find 𝑆⊆𝑉 with ∅≠𝑆≠𝑉 such that 𝑢((𝑆)) is minimum.
(When each 𝑢 𝑒 =1, the minimum is “edge-connectivity” of 𝐺.)
Def: 𝑣, 𝑤 of 𝐺 are separated by a cut (𝑆), or (𝑆) is a (𝑣, 𝑤)-cut if exactly one of 𝑣, 𝑤 is in 𝑆.
To find min (𝑣, 𝑤)-cut in an undirected graph, we can replace each edge by two oppositely directed arcs and give same capacities as the edge, then solve max flow problem. (When 𝑢 𝑒 =1  edge connectivity of 𝐺)
If we fix some node 𝑟, then any cut is (𝑟, 𝑠)-cut for some node 𝑠∈𝑉∖{𝑟}
⇒ Solve 𝑛−1 min (𝑟, 𝑠)-cut problem for 𝑠∈𝑉∖{𝑟} and take min of them.
⇒ 𝑂(𝑛4)
But other direct method exists with 𝑂(𝑛3) running time.
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2. (𝐺) : capacity of a min cut of 𝐺.
Definition:
(𝐺) : capacity of a min cut of 𝐺.
(𝐺; 𝑣, 𝑤) : capacity of a minimum (𝑣, 𝑤)-cut of 𝐺
𝐺𝑣𝑤 is a graph obtained by identifying node 𝑣 with 𝑤. ( 𝑣, 𝑤 distinct nodes)
( 𝑉(𝐺𝑣𝑤)=(𝑉∖{𝑣, 𝑤})∪{𝑥}, where 𝑥 is a new node,
𝐸(𝐺𝑣𝑤)=𝐸∖({𝑣, 𝑤});
for each 𝑒∈𝐸 and end 𝑝 of 𝑒 in 𝐺, 𝑝 is an end of 𝐺𝑣𝑤 if 𝑝≠𝑣, 𝑤,
otherwise 𝑥 is an end of 𝑒 in 𝐺𝑣𝑤.
capacities are the same)
(If multiple edges created, may replace them with an edge with total capacities. For ease of exposition, we do not do this.)
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3. b b 3 3 5 5 c a c a 2 2 2 2 4 3 4 3 h h 1 d 3 1 d 3 e e 2 2 2 2 5 5 6 g f x
Identifying nodes f, g
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4. Continue 𝑛−2 times on smaller graphs.
Prop 3.35: Every cut of 𝐺𝑣𝑤 is a cut of 𝐺. Every cut of 𝐺 that does not separate 𝑣 from 𝑤 is a cut of 𝐺𝑣𝑤.
In 𝐺𝑣𝑤, each cut corresponds to a cut in 𝐺 having 𝑣 and 𝑤 in one side of the cut.
Hence, if we can find the min cut among the cuts that separate 𝑣 and 𝑤, we can compare the value with the min cut value in 𝐺𝑣𝑤.
Straightforward implementation: solve min (𝑣, 𝑤)-cut for any 𝑣, 𝑤. Then identity 𝑣 with 𝑤 and solve min cut on 𝐺𝑣𝑤 again.
Continue 𝑛−2 times on smaller graphs.
Small improvement, but not significant.
make smart choice of 𝑣, 𝑤.
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5. Node Identification Algorithm
rationale: 𝜆(𝐺)= min ((𝐺𝑣𝑤), (𝐺; 𝑣, 𝑤))
Need to find a min cut separating 𝑣, 𝑤 (using max flow algorithm after doubling arcs, need 𝑛−1 times ).
However, choosing 𝑣, 𝑤 carefully reduces the work.
Def: A legal ordering of 𝐺 is an ordering 𝑣1, 𝑣2, …, 𝑣𝑛 of the nodes of 𝐺 such that, where 𝑉 𝑖 denotes {𝑣1, …, 𝑣 𝑖 }
𝑢 𝛿 𝑉 𝑖−1 ∩𝛿 𝑣 𝑖 ≥𝑢 𝛿 𝑉 𝑖−1 ∩𝛿 𝑣 𝑗 for 2≤𝑖<𝑗≤𝑛
Choose any node as 𝑣1 and at step 𝑖 choose 𝑣𝑖 that has the largest total capacity of edges joining it to the previously chosen nodes.
Can be found in 𝑂(𝑛2) (refer Prim’s or Dijkstra’s algorithm)
Thm 3.36: If 𝑣1, 𝑣2, …, 𝑣𝑛 is a legal ordering of 𝐺, then (𝑣𝑛) is a minimum ( 𝑣 𝑛 , 𝑣 𝑛−1 )-cut of 𝐺.
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6. Legal ordering beginning with a is : a, b, c, d, e, h, g, f3 5 c a 2 2 4 3 h 1 d 3 e 2 2 5 6 g f
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7. Node Identification Minimum Cut Algorithm
Initialize 𝑀 to be ∞, 𝐴 to be undefined
While 𝐺 has more than one node
Find a legal ordering 𝑣1, 𝑣2, …, 𝑣𝑛 of 𝐺;
If 𝑢((𝑣𝑛))<𝑀
Replace 𝑀 by 𝑢((𝑣𝑛)), 𝐴 by (𝑣𝑛);
Replace 𝐺 by 𝐺 𝑣 𝑛−1 𝑣 𝑛 ;
Return 𝐴.
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8. (𝐺;𝑝, 𝑞)≥ min ((𝐺;𝑟, 𝑞), (𝐺;𝑝, 𝑟))
In proving the theorem, we do not assume 𝐺 is connected. (“Legal ordering” does not assume this)
Lemma 3.37: If 𝑝, 𝑞, 𝑟∈𝑉, then
(𝐺;𝑝, 𝑞)≥ min ((𝐺;𝑟, 𝑞), (𝐺;𝑝, 𝑟))
(pf) Choose a min (𝑝, 𝑞)-cut (𝑆) with 𝑝∈𝑆.
If 𝑟∈𝑆, then (𝑆) is an (𝑟, 𝑞)-cut so 𝑢((𝑆))≥(𝐺;𝑞, 𝑟).
Otherwise, (𝑆) is a (𝑝, 𝑟)-cut, so 𝑢((𝑆))≥(𝐺;𝑝, 𝑟). 
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9. (1) Let 𝐺′ denote 𝐺 with edge 𝑒 deleted. Then 𝑣1, …, 𝑣𝑛 still legal
(Proof of Thm 3.36)
Need to show 𝑢((𝑣𝑛))≤(𝐺; 𝑣 𝑛−1 , 𝑣 𝑛 ). Use induction on the number of edges and nodes. (trivial if |𝑉|=2 or |𝐸|=0)
Consider two cases: (1) 𝑣 𝑛 𝑣 𝑛−1 is an edge 𝑒 of 𝐺 (2) 𝑣 𝑛 , 𝑣 𝑛−1 not adjacent in 𝐺.
(1) Let 𝐺′ denote 𝐺 with edge 𝑒 deleted. Then 𝑣1, …, 𝑣𝑛 still legal
ordering of 𝐺′
𝑢((𝑣𝑛))=𝑢(′(𝑣𝑛))+𝑢𝑒=( 𝐺 ′ ; 𝑣 𝑛−1 , 𝑣 𝑛 )+𝑢𝑒=(𝐺; 𝑣 𝑛−1 , 𝑣 𝑛 )
(2) In this case, (𝑣𝑛) is a ( 𝑣 𝑛 , 𝑣 𝑛−1 )-cut, need to show that it is min
( 𝑣 𝑛 , 𝑣 𝑛−1 )-cut. From lemma 3.37, enough to show that
𝑢((𝑣𝑛))≤(𝐺; 𝑣 𝑛−2 , 𝑣 𝑛 ) and 𝑢((𝑣𝑛))≤(𝐺; 𝑣 𝑛−2 , 𝑣 𝑛−1 )
For the first case, apply induction on 𝐺′=𝐺∖ 𝑣 𝑛−1 .
𝑣1, .. , 𝑣 𝑛−2 , 𝑣𝑛 is a legal ordering of 𝐺′. Hence
𝑢((𝑣𝑛))=𝑢(′(𝑣𝑛))=( 𝐺 ′ ; 𝑣 𝑛−2 , 𝑣 𝑛 )≤(𝐺; 𝑣 𝑛−2 , 𝑣 𝑛 )
For the second case, apply induction on 𝐺′=𝐺∖𝑣𝑛
𝑣1, .. , 𝑣 𝑛−1 is a legal ordering of 𝐺′. Hence
𝑢((𝑣𝑛))≤𝑢(( 𝑣 𝑛−1 ))=𝑢(′( 𝑣 𝑛−1 ))=( 𝐺 ′ ; 𝑣 𝑛−2 , 𝑣 𝑛−1 )≤(𝐺; 𝑣 𝑛−2 , 𝑣 𝑛−1 ) 
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10. Random Contraction Algorithm
randomized algorithm for minimum cut problem
Random Contraction Algorithm
While 𝐺 has more than two nodes
Choose an edge 𝑒 of 𝐺 with probability 𝑢𝑒/𝑢(𝐸);
Where 𝑒=𝑣𝑤, replace 𝐺 by 𝐺𝑣𝑤 (edge contraction);
Return 𝐴, the unique cut of 𝐺.
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11. Then the probability that an edge from 𝐴 is chosen at step 𝑖+1 is
Thm 3.38: Let 𝐴 be a minimum cut of 𝐺. Then the random contraction algorithm returns 𝐴 with probability at least 2/𝑛(𝑛−1).
(pf) Algorithm will return 𝐴 provided that none of its edges is chosen to be contracted. (some edges might be lost even though not contracted, but it causes no problem)
Suppose 𝑖 edges have been chosen, none from 𝐴. Let the current graph be 𝐺′=(𝑉′, 𝐸′), with 𝑉 ′ =𝑛−𝑖.
Since 𝐴 is min cut of 𝐺′, its capacity is at most the average of capacities 𝑢(′(𝑣)), 𝑣∈𝑉′ ( 𝑢(𝐴)≤𝑢(′(𝑣)) ). Thus, 𝑢(𝐴)≤2𝑢(𝐸′)/(𝑛−𝑖).
Then the probability that an edge from 𝐴 is chosen at step 𝑖+1 is
𝑢(𝐴)/𝑢(𝐸′)≤2𝑢(𝐸′)/(𝑛−𝑖)𝑢(𝐸′)=2/(𝑛−𝑖)
⇒ The probability that no edge of 𝐴 is chosen at step 𝑖+1 is at least 1−2/(𝑛−𝑖)=(𝑛−𝑖−2) / (𝑛−𝑖),
so the probability that no edge of 𝐴 is ever chosen is at least
𝑛−2 𝑛 ⋅ 𝑛−3 𝑛−1 ⋅ 𝑛−4 𝑛−2 ⋅⋅⋅ 3 5 ⋅ 2 4 ⋅ 1 3 = 2 𝑛(𝑛−1) . 
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12. Cor 3.39: Let 𝐴 be a minimum cut of 𝐺 and let 𝑘 be a positive integer.
The probability that the random contraction algorithm does not return 𝐴 in one of 𝑘𝑛2 runs is at most 𝑒 −2𝑘 .
(pf) By thm 3.38, the probability is at most
1− 2 𝑛(𝑛−1) 𝑘 𝑛 2 ≤ 1− 2 𝑛 2 𝑘 𝑛 2 ≤ 𝑒 − 2 𝑛 𝑘 𝑛 2 = 𝑒 −2𝑘 (since 1−𝑥≤ 𝑒 −𝑥 )
ex: run the algorithm for 10𝑛2 times : ( 1− 𝑒 −20 )≥
Reference: Randomized Algorithms, Rajeev Motwani and Prabhakar Raghavan, Cambridge, 1995
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13. 3.5.2 Cut-Trees Multiterminal Cut Problem:
Given an undirected graph 𝐺=(𝑉, 𝐸), 𝑢𝑒≥0 for all 𝑒∈𝐸, and a set of terminals 𝐾⊆𝑉, find a minimum (𝑟, 𝑠)-cut for each pair of nodes 𝑟, 𝑠∈𝐾.
Procedure:
Choose some pair of terminal nodes 𝑟, 𝑠∈𝐾. Find min (𝑟, 𝑠)-cut and denotes the two node sets in each side of the cut as 𝑅 and 𝑆. 𝑅 𝑃 𝑆 𝑆 p
𝑓(𝑟,𝑠)
𝑓(𝑝,𝑞)
𝑓(𝑟,𝑠) q 𝑄
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14. General Procedure B3 B3 B2 B1 C1 B2 B1 C1 f(y,z) A Y Z E2 E1 D1 E2 E1
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15. Tree satisfying the first property: K-flow-equivalent to 𝐺.
Thm 3.40: For any 𝑟, 𝑠∈𝐾, the capacity of min (𝑟, 𝑠)-cut equals the minimum label 𝑓𝑒 in the (𝑟, 𝑠)-path in tree 𝑇. Moreover, min (𝑟, 𝑠)-cut is obtained by the bipartition of the tree obtained after eliminating the minimum weight edge 𝑒 ∗ from 𝑇.
Tree satisfying the first property: K-flow-equivalent to 𝐺.
Tree that also satisfies the second property : Gomory-Hu K-cut-tree ( Gomory-Hu cut-tree)
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16. Remarks
Def: Let 𝑁 be a finite set, and let 𝑓 be a real-valued function on the subsets of 𝑁. Then 𝑓 is called submodular if 𝑓 𝑆 +𝑓 𝑇 ≥𝑓 𝑆∪𝑇 +𝑓(𝑆∩𝑇) for 𝑆,𝑇⊆𝑁.
Prop: 𝑓 is submodular if and only if
𝑓 𝑆⋃ 𝑗 −𝑓 𝑆 ≥𝑓 𝑆∪ 𝑗,𝑘 −𝑓(𝑆∪ 𝑘 ), for 𝑗,𝑘∈𝑁, 𝑗≠𝑘, and 𝑆⊆𝑁∖ 𝑗,𝑘 .
(function of diminishing marginal returns)
Cut function of a graph (also for directed graph), matroid rank function (no details here), etc are examples of submodular functions.
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17. (pf) Let (𝑋) be a min (𝑣, 𝑤)-cut. May assume that 𝑠∈𝑋.
Lemma 3.41: Let (𝑆) be a minimum (𝑟, 𝑠)-cut for some node 𝑟, 𝑠∈𝑉, and let 𝑣, 𝑤∈𝑆. Then there exists a minimum (𝑣, 𝑤)-cut (𝑇) such that 𝑇⊂𝑆.
(pf) Let (𝑋) be a min (𝑣, 𝑤)-cut. May assume that 𝑠∈𝑋.
Consider 2 cases: (1) 𝑟∈𝑋 (2) 𝑟∈𝑉\X
(1) From submodular inequality (Exercise 3.66), we have
𝑢((𝑆∩(𝑉\X)))+𝑢((𝑆∪(𝑉\X)))≤𝑢((𝑆))+𝑢((𝑉\X))
Now 𝑢((𝑆∪(𝑉\X)))≥𝑢((𝑆)) since (𝑆∪(𝑉\X)) is an (𝑟, 𝑠)-cut
⇒ 𝑢((𝑆∩(𝑉\X)))≤𝑢((𝑉\X))=𝑢((𝑋))
So (𝑆∩(𝑉\X)) is a min (𝑣, 𝑤)-cut.
Similarly for (2), can show (𝑆∩X) is a min (𝑣, 𝑤)-cut.  S w
(2) r
(1) r s v X
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18. (pf) True for the initial two node tree.
Let 𝑇 be a tree produced at some stage of Gomory-Hu procedure and 𝑒 is an edge joining the sets(identified nodes) 𝑅 and 𝑆.
We say that terminal nodes 𝑟∈𝑅 and 𝑠∈𝑆 are representatives for 𝑒 if 𝑓𝑒=𝑓(𝑟,𝑠).
Lemma 3.42: At every stage of the Gomory-Hu procedure, there exists representatives for each edge of the tree 𝑇
(pf) True for the initial two node tree.
Suppose we split a node 𝐴 into 𝑋 and 𝑌, based on an (𝑥, 𝑦)-cut. (𝑥∈𝑋, 𝑦∈𝑌)
𝑥 and 𝑦 are representatives for the edge joining 𝑋 and 𝑌 and other edges, except the ones incident to 𝐴 earlier, are not affected.
Suppose 𝐵 is a node joined to 𝐴 (using edge ℎ) before the iteration is executed and assume 𝐵 is joined to node 𝑋 in the new tree.
Then there exist 𝑎∈𝐴, 𝑏∈𝐵 which are representatives for ℎ such that 𝑓ℎ=𝑓(𝑎, 𝑏). Now consider 2 cases: (1) 𝑎∈𝑋 (2) 𝑎∈𝑌.
(1): 𝑎 and 𝑏 continue to be representatives for ℎ.
(2): can show that 𝑥 and 𝑏 are representatives for ℎ in the new tree. 
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19. b a B y B h Y h A a x X Proof of Lemma 3.42
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20. Hence, the second assertion follows from the first.
Thm 3.40: For any 𝑟, 𝑠∈𝐾, the capacity of min (𝑟, 𝑠)-cut equals the minimum label 𝑓 𝑒 in the (𝑟, 𝑠)-path in tree 𝑇. Moreover, min (𝑟, 𝑠)-cut is obtained by the bipartition of the tree obtained after eliminating the minimum weight edge 𝑒 ∗ from 𝑇.
(Proof of Theorem 3.40)
By construction of 𝑇, each edge 𝑒 in 𝑇 corresponds to a cut in 𝐺 specified by the bipartition of 𝑉 obtained by deleting 𝑒 from 𝑇, and this cut has capacity 𝑓𝑒.
Hence, the second assertion follows from the first.
Let 𝑉0, 𝑒1, 𝑉1, …, 𝑒𝑘, 𝑉𝑘 be the path that joins 𝑟 and 𝑠.
Then 𝑓(𝑟, 𝑠)≤ min ( 𝑓 𝑒 1 ,…, 𝑓 𝑒 𝑘 ) since each of the cuts corresponding to the edges 𝑒1, …, 𝑒𝑘 separates 𝑟 and 𝑠.
Let 𝑣𝑖 be the unique terminal node in 𝑉𝑖 for 𝑖=0, 1, …, 𝑘. Have 𝑣0=𝑟, 𝑣𝑘=𝑠.
By Lemma 3.42, 𝑓 𝑒 𝑖 =𝑓( 𝑣 𝑖−1 , 𝑣 𝑖 ), for 𝑖=1, …, 𝑘
By Exercise 3.69, 𝑓(𝑟, 𝑠)≥ min ( 𝑓 𝑒 1 ,…, 𝑓 𝑒 𝑘 )
(𝑓( 𝑣 0 , 𝑣 𝑘 )≥𝑚𝑖𝑛𝑖𝑚𝑢𝑚(𝑓 𝑣 0 , 𝑣 1 ,𝑓 𝑣 1 , 𝑣 2 ,…,𝑓 𝑣 𝑘−1 , 𝑣 𝑘 ) for any choice of nodes 𝑣 0 , 𝑣 1 ,…, 𝑣 𝑘 )
⇒ 𝑓(𝑟, 𝑠)= min ( 𝑓 𝑒 1 ,…, 𝑓 𝑒 𝑘 ) 
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21. A variant of Gomory-Hu procedure can be used to identify the violated odd set constraint for the matching problem
(Ref: M. W. Padberg and M. R. Rao (1982), Odd Minimum Cut-Sets and b-Matchings, Mathematics of Operations Research 7, )
More efficient implementation: D. Gusfield, "Very simple methods for all pairs network flow analysis," SIAM Journal on Computing 19 (1990)
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