Linear Programming Optimization

1
3.3 Applications of Maximum Flow and Minimum Cut
Bipartite Matchings and Covers

• A cover of an undirected graph G is a set C of
  nodes such that every edges of G has at least one
  end in C.
• For any matching M and any cover C, each edge in
  M has an end in C and the corresponding nodes in
  C are all distinct.
• gt M ? C (primal-dual relation)
• Thm 3.14 (Königs Theorem, 1931)
• For a bipartite graph G,
• max M M a matching min C C a
  cover

2
(Transform to max flow problem)
P
Q
a
a
A
A
b
b
B
B
c
c
C
C
r
s
d
d
D
D
f
F
f
F
h
h
H
H
urp1
uqs1
I
I
i
i
upq?
3
(Maximum matching and corresponding flow)
a
a
A
A
b
b
B
B
c
c
C
C
r
s
d
d
D
D
f
F
f
F
h
h
H
H
urp1
uqs1
I
I
i
i
upq?
node cover
4
(Identifying x-augmenting path)
a
a
A
A
b
b
B
B
c
c
C
C
r
r
s
s
d
d
D
D
f
F
f
F
h
h
H
H
urp1
uqs1
urp1
uqs1
I
I
i
i
upq?
upq?
node reachable from r
5
(blue nodes defines cut flow value)
a
A
a
A
b
b
B
B
c
c
C
C
r
r
s
s
d
D
d
D
f
F
f
F
h
h
H
H
urp1
uqs1
urp1
uqs1
I
I
i
i
upq?
upq?
cut arcs
node reachable from r
Note that no cut arc from blue in P ? green in Q
6
a
A
a
A
b
b
B
B
c
c
C
C
r
r
s
s
d
D
d
D
f
F
f
F
h
h
H
H
urp1
uqs1
urp1
uqs1
I
I
i
i
upq?
upq?
Cover C (pink) (P\A) ? (Q?A) Capacity of cut
P\AQ?AC
Let A blue nodes\r
7
Q\A
P\A
s
r
P?A
Q?A

• no edge from P?A to Q\A
• Cover C (P\A) ? (Q?A)
• Capacity of cut P\AQ?AC
• Running time O(mn)

8
(Halls Theorem)

• System of distinct representatives
• Given a set Q and a family (S1, S2, , Sk) of
  subsets of Q, a system of distinct representative
  (SDR) is a set q1, , qk of distinct elements
  of Q such that qi ?Si for 1 ? i ? k.
• Halls Theorem
• The family (S1, S2, , Sk) has an SDR if and
  only if for every subset I of 1, , k we have
  ?i ? I Si ? I .
• (The family (S1, S2, , Sk) does not have an SDR
  if and only if there is a set I such that ?i ?
  I Si lt I . )

9
q1
q1
S1
S1
q2
q2
S2
S2
q3
q3
S3
S3
q4
q4
S4
S4
q5
q5
S5
S5
q6
q6
The family has a SDR iff the corresponding
bipartite graph has a matching of size k.
10
q1
q1
S1
S1
q2
q2
S2
S2
q3
q3
S3
S3
q4
q4
S4
S4
q5
q5
S5
S5
q6
q6
Circled nodes have ?i ? I Si lt I , hence
no SDR.
11
Dilworths Theorem

• Chains and Antichains in Partially Ordered Sets
• A list of pairs x ? y satisfying the following
  conditions is called a partial order
• (i) no x ? x
• (ii) x ? y and y ? z together imply x ? z
• A set whose elements appear in a partial order
  is called a partially ordered set
• A subset C of a partially ordered set is called
  a chain if
• x ? y or y ? x whenever x, y ? C and x ?
  y
• (The elements can be enumerated as z1, z2, , zr
  in such a way that z1 ? z2, z2 ? z3, , zr-1 ?
  zr )
• A subset A of a partially ordered set is called
  an antichain if
• x ? y for no x, y ?A

12

• Dilworths Theorem
• In every partially ordered set P, the smallest
  number of chains whose union is P equals the
  largest size of an antichain
• application guided tour, airplane assignment
• pf) text exercise 3.37. We consider a different
  approach here.
• ex) Set X a, b, c, d, e, f, g, tours
• partial order a ? c, a ? d, a ? f, a
  ? g, b ? c, b ? g, d ? g, e ? f, e ? g
• ( need 3 guides, a ? d ? g, b ? c, e ? f
  )
• Construct a bipartite graph by doubling
  each node v as v and v.
• There exists an edge xy iff x ? y.

13
A matching M defines k n - M chains (guides)
ex) (i) a ? d ? g (ii) b ? c
(iii) e ? f In the matching, each tour has at
most one successor and at most one predecessor.
Otherwise, M would not be a matching. By
concatenating successive tours, we obtain a set
of chains (A chain may consists of a single
tour) Let k be the number of chains and ri be the
length (number of nodes) of i-th chain. Then
a
b
a
c
d
b
c
e
f
d
g
e
f
Converse can be shown too, i.e. k chains define n
k sized matching M.
g
14

• Different reasoning using Königs Theorem
• Let M be a largest matching and C be min cover.
  Define a set A of trips as follows
• trip z ?A iff z?C and z? C.
• No x ? y with x, y ?A is possible. (Otherwise, ?
  xy with x?C, y?C, hence xy not covered
  by C, contradiction to C being cover.)
• ? no guide can take care of two different trips
  in A. (e.g., c ? f)
• need at least A guides.
• But A has at least n - C elements (each node
  in C makes at most one tour ineligible for
  membership in A)
• Hence every schedule must use at least
• A ? n - C n - M different guides.
• ? The schedule defined by M, using only n -
  M guides, is optimal. (from earlier)

Cover C
a
A
b
a
c
d
b
c
e
f
d
g
e
f
g
15

• Proof of Dilworths Theorem
• The partially ordered set P may be thought of as
  a set of tours. Chains are those sets of tours
  that can be taken care of by one guide. The
  described procedure gives the smallest k of
  chains. Along with the chains, the procedure
  also finds an antichain A of size at least k.
• Since every antichain A shares at most one
  element with each of the chains C1, C2, , Ck ,
  it must satisfy A ? k ? A. So A is a
  largest antichain and its size is k (plug in A
  for A). ?
• Reference Linear Programming, Vasek Chvatal,
  1983, Freeman, 338-341.

16
Optimal Closure in a Digraph

• Situation If we want to choose project v, then
  we must choose project w also. We want to choose
  the set of projects which gives maximum profit
• Model as a problem on a digraph. Use directed
  arc vw if project w precedes project v. Find
  the maximum benefit set C ? V such that ?(C) ?.
  ( closure of G)
• Example open pit mining problem.
• Partition the volume to be considered for
  excavation into small 3-dimensional blocks.
• Block v produces profit of bv (profit of ore in
  v cost of excavating v )
• If we need to excavate block w to excavate block
  v , we include arc vw in G.
• Want to find a closed set in G which maximizes
  profit.

17

• Convert to min-cut problem (max-flow problem)
• Divide the nodes into two sets A and B.
• v ?A if bv ? 0 and v ? B if bv lt 0.
• Add a source r and a sink s to the graph and
  add arcs rv for each v ?A and arcs vs for each
  v ?B.
• Capacity on the arcs are urv bv for each v
  ?A and uvs - bv for each v ?B.
• The upper bounds on the original arcs are
  infinite.
• A set C of nodes is closed if and only if the cut
  C ? r has finite capacity.
• If the capacity of C ? r is finite, then it
  equals

Since ?v ?A bv is a constant, minimizing the
capacity of C ? r amounts to maximizing
?v ?C bv .
18
(Example)
2
-7
?
2
?
C
7
4
r
4
1
?
?
-1
3
?
3
2
3
s
?
-3
?
1
2
?
-1
19
Mengers Theorems

• Ref Graph Theory with Applications, J. A. Bondy,
  U. S. R. Murty
• Lemma Let G be a digraph in which each arc has
  unit capacity. Then
• (a) The maximum (r, s) flow is equal to the
  maximum number m of arc disjoint directed (r,
  s)-paths in G and
• (b) The capacity of a minimum (r, s)-cut is
  equal to the minimum number n of arcs whose
  deletion destroys all directed (r, s)-paths in G.
• Thm Let r, s be two nodes of a digraph G. Then
  the maximum number of arc-disjoint directed (r,
  s)-paths in G is equal to the minimum number of
  arcs whose deletion destroys all directed (r,
  s)-paths in G.
• (pf) From above Lemma and max-flow and min-cut
  theorem.

20

• Thm Let r, s be two nodes of a graph G. Then
  the maximum number of edge-disjoint (r, s)-paths
  in G is equal to the minimum number of edges
  whose deletion destroys all (r, s)-paths in G.
• (pf) Replace each edge of G by two oppositely
  oriented arcs and apply the previous results. ?
• Lemma A graph G is k-edge-connected iff any two
  distinct nodes of G are connected by at least
  k-edge-disjoint paths.

21

• (node connectivity)
• Thm Let r, s be two nodes of a digraph G, such
  that r is not joined to s. Then the maximum
  number of node-disjoint directed (r, s)-paths in
  G is equal to the minimum number of nodes whose
  deletion destroys all directed (r, s)-paths in G.
• (pf) Construct G as follows ( except r and s)

22

• Thm Let r and s be two nonadjacent nodes of a
  graph G. Then the maximum number of
  node-disjoint (r, s)-paths in G is equal to the
  minimum number of nodes whose deletion destroys
  all (r, s)-paths.
• Cor A graph G with n ? k1 is k-node connected
  iff any two distinct nodes of G are connected by
  at least k node-disjoint paths.