Network Flows
Chun-Ta, Yu
Graduate Institute
Information Management Dept.
National Taiwan University
Chapter 12
ASSIGHMENTS AND
MATCHINGS
Outline
 Introduction
 Bipartite Cardinality Matching Problem
 Bipartite Weighted Matching Problem
 Stable Marriage Problem
 Nonbipartite Cardinality Matching Problem
Introduction
 Bipartite matching problem
1. The cardinality problem
2. The weighted problem
 Stable marriage problem
 Nonbipartite matching problem
Bipartite Cardinality Matching
Problem
 Transform this problem into a maximum flow
problem in a simple network
– Each arc has a unit capacity and each node has an
indegree of at most 1 or an outdegree of at most 1
– Introduce a source node s and a sink node t
 Solving this problem at worst-case O ( n m )
Bipartite Cardinality Matching
Problem
Bipartite Weighted Matching
Problem
 Assignment problem
 Given a weighted bipartite network
G  ( N1
N 2 , A) with N1  N 2 and arc weights cij
find a perfect matching of minimum weight

Minimize
( i , j ) A
c ij x ij
subject to

x ij  1
for all i  N 1

x ji  1
for all i  N 2
{ j:( i , j ) A}
{ j:( j , i ) A}
x ij  0
for all ( i , j )  A
Bipartite Weighted Matching
Problem
 The assignment problem can be viewed as
adaptation of algorithm for the minimum
cost flow problem
 Four algorithm to solve：
1.
2.
3.
4.
Successive Shortest Path Algorithm
Hungarian Algorithm
Relaxation Algorithm
Cost Scaling Algorithm
Successive Shortest Path Algorithm
 Discuss in section 9.7
 Augment 1 unit flow in every iteration
 Let S( n ,m ,C) denote the time needed to
solve a shortest path problem with
nonnegative arc lengths and n1=|N1|
 The algorithm would terminate within n1
iterations and would require O(n1S( n ,m ,C))
Hungarian Algorithm
 Discuss in section 9.7, primal-dual algorithm
 With a single supply node s* and a single
demand node t*.At every iteration ,the
primal-dual algorithm computes shortest path
distance from s* to all other nodes
 The algorithm would terminate within n1
iterations and would require O(n1S( n ,m ,C))
Relaxation Algorithm
 Closely related to the successive shortest path
algorithm
x ji  1
for all i  N 2
 Relax the constraint { j:(
j , i )A}
thus allowing any node in N2 to be assigned
to more than one node in N1
 Overall running time O(n1S( n ,m ,C))
Stable Marriage Problem
 A certain community consists of n men and n
women. Each person ranks those of the
opposite sex in accordance with his or her
preferences for a spouse.
 Unstable – if man-woman are not married to
each other but prefer each other to their
current spouses
 For any set of rankings, we can always find a
stable matching in O(n2) time
Stable Marriage Problem
 The input to the stable marriage problem
consists of two n x n matrices.
 Each rank is an integer between 1 and n
 Priority list is a vector of n elements for each
person, can be sorted using bucket sort in
O(n2) time
Stable marriage algorithm
 Propose-and-reject algorithm
 Time complexity：
– Each iteration each woman receiving a proposal
either (1) receives her first proposal (2) rejects
some proposal
– Each woman rejects any man’s proposal at most
once, so total rejection times is (n-1) for each
woman
– Total time is O(n2)
 Man-optimal matching
Nonbipartite Cardinality Matching
Problem
 Alternating Paths
– We refer to a path P=i1-i2-…ik in the graph as an
alternating path with respect to a matching M if
every consecutive pair of arcs in the path
contains one matched and one unmatched arc
ex：1-2-4-3-5
Nonbipartite Cardinality Matching
Problem
 Augmenting Paths
– We refer to an odd alternating path P with respect to
matching M as an augmenting path if the first and last
nodes in the path are unmatched
ex：1-2-4-3-5-6
– Interchanging the matched and unmatched arcs on
augmenting paths can add one more cardinality
Symmetric Difference
 Let S1 and S2 be two sets; the symmetric
difference of these sets, denoted S1⊕S2, is
the set S1⊕S2 = (S1∪S2)-(S1∩S2)
for example, S1 = {4,5,7,8} and S2 =
{2,4,8,9} , then S1⊕S2 ={2,5,7,9}
Property 12.6
 If M is a matching and P is an augmenting
path with respect to M, then M⊕P is a
matching of cardinality |M| + 1. Moreover, in
the matching M⊕P, all the matched nodes in
M remain matched and two additional nodes,
namely the first and last nodes of P, are
matched
Property 12.7
 If M and M* are two matchings, their
symmetric difference defines the subgraph
G* = (N, M⊕M*) with the property that
every component is one of the six types
shown in Figure 12.7
Augmenting Path Theorem
 If a node p is unmatched in a matching M,
and this matching contains no augmenting
path that starts at node p, then node p is
unmatched in some maximum matching.
Bipartite Matching Algorithm
 Start with a feasible matching M (which
might be a null matching) and then repeat the
following step for every unmatched node p  N
Try to identify an augmenting path starting at
node p. If we find such a path P, replace M
with M⊕P; otherwise, delete node p and all
the arcs incident to it from the graph
Bipartite Matching Algorithm
 Search algorithm
Bipartite Matching Algorithm
Bipartite Matching Algorithm
 Time complexity：
– The search algorithm execute at most n times
– For each node i, the search procedure performs
one of the following two operations at most once
(1) examine-even (2) examine-odd, the former
operation require O(|A(i)|) time, about O(m)
– Total time O(nm)
Unique label property
 A graph is said to possess a unique label
property with respect to a given matching M
and a root node p if the search procedure
assigns a unique label to every labeled node
irrespective of the order in which it examines
labeled nodes
 Bipartite network satisfy it; nonbipartite
network doesn’t satisfy it
Flower and Blossoms
 A flower, defined with respect to a matching M and
a root node p, is a subgraph with two components：
– Stem. A stem is an even length alternating path that starts
at the root node p and terminates at some node w. We
permit the possibility that p = w, in which case we say
that the stem is empty
– Blossom. A blossom is an odd length alternating cycle
that starts and terminates at the terminal node w of a stem
and has no other node in common with the stem. We refer
to node w as the base of the blossom
Flower and Blossoms
Property 12.9
a) A stem spans 2l+1 nodes and contains l
matched arcs for some integer l≧0
b) A blossom spans 2k+1 nodes and contains k
matched arcs for some integer k≧1. The
matched arcs match all nodes of the
blossom except its base
c) The base of blossom is an even node
Property 12.10
 Every node i in the blossom (except its base) is
reachable from the root (or from the base of the
blossom) through two distinct alternating paths. One
has even length and the other has odd length
Contracting a Blossom
1.
2.
3.
Introduce a new node b (pseudonode) and define its
adjacency list A(b) = A(i1)∪A(i2)∪… ∪A(jk)
Update the adjacency list of every node j  A(b)
by executing A(j)=A(j)∪{b}
To be able to recover, set contracted node to “inactive”
mode
Nonbipartite matching algorithm
Nonbipartite matching algorithm-Find a augmenting path
Complexity of nonbipartite matching
algorithm
 Lemma 12.13
– During an execution of the search procedure, the algorithm performs
at most n/2 contractions
 Since each contraction adds at most one element to any
adjacency list (the pseudonode), and since the algorithm
performs at most n/2 contractions, no adjacency list will ever
contain more than 3n/2
 Each search procedure performs one of the following
operation at most once: (1)it discovers that node i is inactive
(2)examine-odd(3)examine-even, (3) require |Ac(i)|≦3n/2,
so running time is O(n2)
 Total time complexity O(n3)
Chapter 13
MINIMUM SPANNING TREES
Outline
 Introduction
 Optimality Condition
 Kruskal’s Algorithm
 Prim’s Algorithm
 Sollin’s Algorithm
Introduction
Optimality Condition
 Cut Optimality Conditions
– A Spanning tree T* is a minimum spanning tree
if and only if it satisfies the following cut
optimality condition: For every tree arc (i , j )  T * ,
cij≦ckl for every arc (k, l) contained in the cut
formed by deleting arc (i, j) from T*
Optimality Condition
 Path Optimality Conditions
– A spanning tree T* is a minimum spanning tree if
and only if it satisfies the following path
optimality conditions: For every nontree arc (k, l)
of G, cij≦ckl for every arc (i, j) contained in the
path in T* connecting nodes k and l
Kruskal’s Algorithm
 Time Complexity O(mn )
Prim’s Algorithm
 Time Complexity O(m log n)
Sollin’s Algorithm
 Time Complexity O(m log n)
Summary of minimum spanning tree
algorithm
Matroids and the Greedy Algorithm
 Independent：a subset I of objects do not form a
cycle in the network
 Subset System (E, ζ)：a finite set of objects E and
nonempty collection ζ of subsets of these objects
 Matroid：a subset system satisfies the growth
property that if Ip and Ip+1 are independent sets
containing p and p+1 elements, we always can find
an element e  I p1  I p , satisfying the property that
Ip∪{e} is an independent set
Matroids and the Greedy Algorithm
 Maximal independent set：an independent
set I satisfying the property that we cannot
add any other element e to I and produce
another independent set
 Greedy algorithm：
Thank You!