Network Flows

Chun-Ta, YuGraduate InstituteInformation 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 problem1. 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

1 2 1 2( , ) and arc weights

find a perfect matching of minimum weight

ijG N N A with N N c

( , )

1{ :( , ) }

2{ :( , ) }

1

1

0 ( , )

ij iji j A

ijj i j A

jij j i A

ij

Minimize c x

subject to

x for all i N

x for all i N

x 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. Successive Shortest Path Algorithm

2. Hungarian Algorithm

3. Relaxation Algorithm

4. Cost Scaling Algorithm

Successive Shortest Path Algorithm

Discuss in section 9.7 Augment 1 unit flow in every iteration Let S( S( n ,m ,Cn ,m ,C)) denote the time needed to

solve a shortest path problem with nonnegative arc lengths and nn11=|N=|N11||

The algorithm would terminate within n1 iterations and would require O(O(nn11S( S( n ,m ,Cn ,m ,C))))

Hungarian Algorithm

Discuss in section 9.7, primal-dual algorithm With a single supply node s*s* and a single

demand node t*.t*.At every iteration ,the primal-dual algorithm computes shortest path distance from s*s* to all other nodes

The algorithm would terminate within nn11 iterations and would require O(O(nn11S( S( n ,m ,Cn ,m ,C)) ))

Relaxation Algorithm

Closely related to the successive shortest path algorithm

Relax the constraint

thus allowing any node in NN22 to be assigned to more than one node in NN11

Overall running time O(O(nn11S( S( n ,m ,Cn ,m ,C)) ))

2{ :( , ) }

1 jij j i A

x for all i N

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(O(nn22)) time

Stable Marriage Problem

The input to the stable marriage problem consists of two n x nn 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(O(nn22)) 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)(n-1) for each woman

– Total time is O(O(nn22))

Man-optimal matching

Nonbipartite Cardinality Matching Problem Alternating Paths

– We refer to a path P=iP=i11-i-i22--……iikk in the graph as an alternating path with respect to a matching MM 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 PP 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 SS11 and SS22 be two sets; the symmetric difference of these sets, denoted SS11⊕⊕SS22, is the set SS11⊕⊕SS22 = (= (SS11∪∪SS22)-()-(SS11∩∩SS22))

for example, SS11 = {4,5,7,8} and SS22 = {2,4,8,9} , then SS11⊕⊕SS22 ={2,5,7,9}

Property 12.6

If MM is a matching and PP is an augmenting path with respect to MM, then MM⊕⊕PP is a matching of cardinality ||MM| + 1| + 1. Moreover, in the matching MM⊕⊕PP, all the matched nodes in M remain matched and two additional nodes, namely the first and last nodes of PP, are matched

Property 12.7

If MM and M*M* are two matchings, their symmetric difference defines the subgraph G*G* = ( = (NN, , MM⊕⊕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 pp is unmatched in a matching MM, and this matching contains no augmenting path that starts at node pp, then node pp is unmatched in some maximum matching.

Bipartite Matching Algorithm

Start with a feasible matching MM (which might be a null matching) and then repeat the following step for every unmatched node

Try to identify an augmenting path starting at node pp. If we find such a path PP, replace MM with MM⊕⊕PP; otherwise, delete node pp and all the arcs incident to it from the graph

p N

Bipartite Matching Algorithm

Search algorithm

Bipartite Matching Algorithm

Bipartite Matching Algorithm

Time complexity：– The search algorithm execute at most nn times– For each node ii, the search procedure performs

one of the following two operations at most once (1) examine-even (2) examine-odd, the former operation require OO(|A(i)|(|A(i)|)) time, about O(O(mm))

– Total time O(O(nmnm))

Unique label property

A graph is said to possess a unique label property with respect to a given matching MM and a root node pp 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 MM and a root node pp, is a subgraph with two components：– Stem. A stem is an eveneven length alternating path that starts

at the root node pp and terminates at some node ww. We permit the possibility that p = wp = w, in which case we say that the stem is empty

– Blossom. A blossom is an oddodd length alternating cycle that starts and terminates at the terminal node ww of a stem and has no other node in common with the stem. We refer to node ww as the base of the blossom

Flower and Blossoms

Property 12.9

a) A stem spans 2l+12l+1 nodes and contains ll matched arcs for some integer l 0≧l 0≧

b) A blossom spans 2k+12k+1 nodes and contains kk matched arcs for some integer k 1. ≧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 ii 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. Introduce a new node b (pseudonode) and define its adjacency list A(A(bb)) = = A(A(ii11) A(∪) A(∪ ii22)∪)∪…… A(∪A(∪ jjkk))

2. Update the adjacency list of every node

by executing A(A(jj))==A(A(jj) {∪) {∪ bb}}

3. To be able to recover, set contracted node to “inactive” mode

( )j A b

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/2n/2 contractions

Since each contraction adds at most one element to any adjacency list (the pseudonode), and since the algorithm performs at most n/2n/2 contractions, no adjacency list will ever contain more than 3n/23n/2

Each search procedure performs one of the following operation at most once: (1)it discovers that node ii is inactive (2)examine-odd(3)examine-even, (3) require ||AAcc(i)| 3n/2, ≦(i)| 3n/2, ≦ so running time is O(O(nn22))

Total time complexity O(O(nn33))

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*T* is a minimum spanning tree

if and only if it satisfies the following cut optimality condition: For every tree arc ,

ccijij c≦c≦ klkl for every arc (k, l)(k, l) contained in the cut formed by deleting arc (i, j)(i, j) from T*T*

( , ) *i j 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, (k, l)l) of GG, ccijij c≦c≦ klkl for every arc (i, j)(i, j) contained in the path in T*T* connecting nodes kk and ll

Kruskal’s Algorithm

Time Complexity OO(mn )(mn )

Prim’s Algorithm

Time Complexity OO(m log n) (m log n)

Sollin’s Algorithm

Time Complexity OO(m log n)(m log n)

Summary of minimum spanning tree algorithm

Matroids and the Greedy Algorithm

Independent ： a subset II of objects do not form a cycle in the network

Subset System (E, (E, ζ)ζ) ： a finite set of objects E and nonempty collection ζζ of subsets of these objects

Matroid ： a subset system satisfies the growth property that if IIpp and IIp+1p+1 are independent sets containing pp and p+1 p+1 elements, we always can find an element , satisfying the property that IIpp {e}∪{e}∪ is an independent set

1p pe I I

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!