A new class of facet-inducing inequalities for a polytope associated with a constrained assignment problem.

University of Windsor

University of Windsor

Scholarship at UWindsor

Scholarship at UWindsor

Electronic Theses and Dissertations

Theses, Dissertations, and Major Papers

1-1-2007

A new class of facet-inducing inequalities for a polytope

A new class of facet-inducing inequalities for a polytope

associated with a constrained assignment problem.

associated with a constrained assignment problem.

Liping Wang

University of Windsor

Follow this and additional works at: https://scholar.uwindsor.ca/etd

Recommended Citation

Recommended Citation

Wang, Liping, "A new class of facet-inducing inequalities for a polytope associated with a constrained assignment problem." (2007). Electronic Theses and Dissertations. 7012.

https://scholar.uwindsor.ca/etd/7012

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by

Liping Wang

Thesis

Subm itted to the Faculty of G raduate Studies

through the D epartm ent of M athem atics and Statistics

in partial fulfillment of the Requirements of

the Degree of M aster of Science at the

University of Windsor

Windsor, Ontario, Canada

2007

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A b s tr a c t

Consider a constrained assignment problem where the side co n stra in t consists

o f a single e q u a lity w ith 0-1 coefficients. T h is problem has the fo llo w in g integer

p rogram m ing form u la tion :

m in i E " = i dijXij (1)

subject to £ L i x ij = 1 for all j —1, . . . , n (2)

£™ =i x ij = 1 for a lH = 1 , . . . , n (3)

Xy £ { 0 , 1 } for a lH , j = 1 , . . . , n (4)

E I L i £ J = i CijXij = r (5)

where a ll Cy are 0 o r 1 and r is an integer such th a t 0 < r < n.

L e t h = { 1, 2, . . . , m } , / 2 = { n i + 1, . . ., n } , J i = { 1, 2, . . . , n 2}, J2 = { n2 +

1 , . . . , n ) . I t was shown in [3] th a t w ith o u t loss o f generality we can assume th a t

Cij = 1 i f and o n ly i f ( i , j ) G ( i i ) x (,/[) IJ I2 x J 2. Furtherm ore, in th is case (5) is

equivalent to

x H = r u (6)

( i , j ) e / i x J i

where rq = {n\ + n2 + r — n ) /2.

D efin e

Pni,n2 ~ feasible solutions o f (2), (3), (6) and > 0, *, j = 1 , . . . , n.

T he p o lyh e d ra l stru c tu re o f was investigated in

[3],

where tw o large classes

o f facet-inducing inequalities o f Qn’[h2 were presented. In th is thesis we present a new

A c k n o w le d g e m e n ts

I w ould like to express m y m ost sincere a ppreciation to m y supervisor, D r. A b d o

A fa k ih , fo r his guidance d u rin g m y tw o years study. I learned m uch fro m h im not

o n ly m athem atics, also the way to th in k about th ing s and to do research. He also

paid much a tte n tio n to the im provem ent o f m y E nglish and checked the mistakes I

made one b y one.

Special thanks go to m y thesis readers, D r. T im T ra yn or and D r. G uoqing Zhang, for

th e ir tim e and patience to read th ro u g h m y thesis and give me constructive comments.

I w ould also like to th a n k D r. M e h di M onfraed for being the chair o f m y defense.

I w ant to acknowledge L ih u a A n , Xuem ao Zhang and W en jia L iu fo r th e ir help in m y

thesis.

Last, b u t n o t least, I w ant to express m y g ra titu d e to m y fam ily, m y parents, m y

husband, and m y son, w ith o u t th e ir su pp o rt and understanding, everyth in g w ould be

impossible.

I w ish to express m y g ra titu d e to the people w ho developed P O R T A software and

C onten ts

A b s tra c t ii

Acknowledgements iv

C hapter 1. In tro d u c tio n 1

1.1. O the r C onstrained Assignment Problems 3

C hapter 2. P relim inaries 4

2.1. Polyhedral theory 4

2.2. G raph T heory 7

C hapter 3. The Assignm ent P olytope 9

C hapter 4. K now n Facets o f Qn[,h2

4.1. F irs t Class o f Facet-Inducing Inequalities for Q "j^ 2 18

4.2. Second Class o f Facet-Inducing Inequalities for Qn’ih i 19

C hapter 5. New Class o f Facet-Inducing Inequalities for Q% [ 'n 2 21

C hapter 6. Conclusion 31

B ib lio grap h y 32

C H A P T E R 1

In tro d u ctio n

T he assignment problem (A P ) (also know n as the m in im u m w eight b ip a rtite

m atching problem ) is a classical problem w ith m any applications in operations re­

search, economics and com puter science. T he A P has been studied extensively, and

m any efficient a lgorithm s have been devised fo r solving it [15, 20, 17, 2], In p a rtic ­

ular, the assignment problem can be form u la te d as a lin e ar pro gram m in g problem ,

where the set o f feasible solutions is the w ell-know n B irk h o ff p olyto p e or the assign­

m ent p o lyto p e [24]. T he p o lyh e d ra l stru cture , such as the dim ension, extrem e points,

and facets, o f th is p olyto p e are a ll w ell know n [4, 7].

In m any p ra c tic a l applications, one is often faced w ith problem s w hich can be

posed as constrained assignment problem s[8]. T h a t is, such problem s can be form u ­ lated as assignment problem s together w ith one o r more side constraints. Whereas the

assignment problem is a p o ly n o m ia lly solvable problem , these constrained assignment

problem s are in general hard. T h is thesis is concerned w ith a constrained assignment

problem w ith one side co nstra in t. T h is side co nstra in t consists o f an e q u a lity w ith 0-1

coefficients. T h is problem was studied in [3] where tw o large classes o f facet-inducing

inequalities fo r the associated p o lyto p e were presented. N ote th a t th is p o ly ­

tope can be th o u g h t o f as a slice o f the B irk h o ff polytope. In th is thesis we present

a new class o f facet-inducing inequalities for

Qn’[,h2-T he success or fa ilu re to o b ta in a com plete description o f polytopes associated

w ith co m b in a to ria l o p tim iz a tio n problem s sheds some fig h t on the co m p u ta tio n a l

c o m p le x ity o f th e s e p ro b le m s [21, 12]. O n o n e h a n d , th e n o n - b ip a r t it e m a tc h in g problem (N B M P ) is one o f th e few genuine co m b ina to ria l o p tim iz a tio n problem s

where a com plete description o f its associated p olyto p e is know n [22, 23]. A t the

same tim e , the N B M P is also one o f the few genuine co m b in a to ria l o p tim iz a tio n

1. IN TRO DUCTIO N 2

problem s w hich are solvable by a p o lyn o m ia l tim e a lg o rith m

[11].

O n the o ther

hand, no com plete description o f the p olyto p e associated w ith any N P -h a rd problem

is known.

T he constrained assignment problem w ith 0-1 side constraint th a t we investigate

in th is thesis has the fo llo w in g integer pro gram m in g form u la tion :

min E L i E " = i d ijX ij (7)

subject to E I L i x ij = ^ for all j = 1 , . . . , n (8)

Y ! j= 1x ij - 1 for a lH = 1 , . . . , n (9)

Xy € {0,1} for a li i , j = 1 ,. . . , n (10)

- E I L i E ”= i CijXij = r (11)

where a ll are 0 o r 1 and r is an integer such th a t 0 < r < n. A n example o f such problem s arises in the core management o f pressurized w ater nuclear reactors

[6, 14].

N ote th a t the above problem where Cjj are general integers is N P -h a rd [10].

L e t G — (V i U V2, E ), |Vi| = |V2I = n, be a com plete b ip a rtite graph where each edge o f G is colored e ithe r red o r blue. T hen any feasible so lu tion o f (8) - ( l l ) can be in te rp re te d as a perfect m atching on G w hich uses exactly r red edges where an edge

( i , j ) o f G is colored red i f and o n ly i f ct] = 1. Define:

P n,r = Set o f feasible solutions o f (8), (9), (11) and > 0, i , j = 1 ,n . Q n’r — integer h u ll o f P n’r .

Therefore, the above problem is the problem :

n n

m in E E dijXij : € Q n,r. (12)

i= i j= 1

N ote th a t P n,r is th e intersection o f th e B irk h o ff p o lyto p e w ith a hyper-plane. In

general P n,r has fra c tio n a l extrem e points thus P " r # Q n,r. In C h a pte r 4, i t is shown

1.1. O TH ER CONSTRAINED ASSIGNMENT PROBLEMS 3

m in E E dijXij : E Q n\^l2, (13)

»=l j =i

where w hich is defined in C h apter 4, is a special case o f Q n r . In th is thesis

we present a new class o f facet-inducing inequalities for

1.1. O ther C on strained A ssign m en t P rob lem s

Leclerc

[16]

considered the fo llo w in g problem know n as the 2-edge restriction

matching problem. G iven a b ip a rtite graph G — (V i U V2, E ), le t W C V\ U V2. F in d a perfect m atching M such th a t \M C \8(W )\ = 2, where <5 ( IT ) is the set o f edges in cid en t

w ith e xa ctly one node in W . He presented an 0 ( n4 5) a lg o rith m fo r th is problem where

n is the num ber o f nodes o f G. N e xt we show th a t th is 2-edge re s tric tio n m a tching

problem is a special case o f problem ( 7 ) - ( ll) .

L e t W n Hi = I u W n V2 = Ji and le t I2 = V1\ I U J2 = V2 \ Ji. Then, the 2-edge

re s tric tio n m atching problem reduces to the problem o f fin d in g a feasible so lu tion o f

the follo w in g system:

X T = i x ij = 1 for a ll j = 1, • • •, n

YTj= 1 x ij = 1 for a l i i = 1,

Cij —

€ { 0 , 1 } for a lH , j = 1 , . . . , n

s =iE;=iw = 2

where

' 1 i f ( i , j ) € ( h x J 2) U ( I2 x

0 i f { h j ) € { I \ x J\ ) U ( I2 x J2) ■

A b o u d i and Nemhauser [1] studied a constrained assignment problem w ith m side constraints o f the form :

T2fc-I,2fc—1 - x2k,2k = 0 for k = 1, . . . , m.

T h ey presented a class o f facet-inducing inequalities for the associated p olytope,

and the y showed th a t th is class provides a com plete description o f the associated

C H A P T E R 2

P relim in aries

In th is chapter, we present basic d e fin itio n s and relevant results fro m polyhe d ral

the o ry and graph th e o ry th a t w ill be needed in th is thesis.

2 .1 . P o ly h e d r a l t h e o r y

Vectors (or points) x 1, . . . , x k € R " are said to be lin e a rly ind e p e n d e n t i f Ai =

• • ■ = A*, = 0 is the unique so lu tion o f the system

k

Y k x 1 =

0.

i = 1

T hen i t easily follow s th a t n is the m a xim u m num ber o f lin e a rly independent p oints

in R n . O n the o th e r hand, vectors (or points) x 1, . . . , x k € R " are said to be

a ffin e ly ind e p e n d e n t i f Ai = • ■ • = A* = 0 is the unique solution o f the system

k

£ > ‘ = 0,

i= 1

k

£ > = o.

*=1

Note th a t n + 1 is the m a xim u m num ber o f affinely independent points in R n . C learly,

the notions o f affine and linear independence arc related. L inear independence im plies

affine independence, b u t th e converse m ay n o t be true. T he next lem m a establishes

the exact re la tio n between these tw o notions.

L e m m a 2 .1 .1 . x1, . . . , x k €= R ” are a ffin e ly in d e p e n d e n t i f f x 1 — x 1, . . . , x k — x l are lin e a rly independent.

A set S C R " is said to be convex i f the line segment jo in in g any tw o points

2.1. POLYHEDRAL THEORY 5

A t1 + (1 - A )x2 € S fo r a ll 0 < A < 1. G iven a set S = { x 1, x2, . . . , xm} C R ", a p o in t

x G R n is said to be a convex combination o f x 1, x 2, . . . , x k, i f there exist nonnegative

scalars A i, A2, . . . , A*,, Ya= 1\ — such th a t x = Y li= i ■*-n p a rtic u la r, x is a convex com b ina tio n o f x 1. x2 i f x lies in the closed line segment jo in in g x1 and x 2.

The convex hull o f S, denoted b y conv(S'), is the set o f a ll p o ints th a t are convex

com binations o f points in S. G iven a set S C M X). the integer hull o f S is the convex

h u ll o f the in te g ra l points in S. T he follo w in g result, due to Caratheodory, is w ell

know n [20].

T h e o r e m 2.1.2. Let S C R n, then every point x G conv(S) can be represented as

a convex combination of n + 1 points from S. i.e. fo r every point x G conv(S), there

exist A i, A2, . . . , An+i > 0, X ^ i.1 A, = 1 such that x = ■

A set H C R " o f the fo rm {x € R n, pTx = a 0, p 0, a0 G R } is called a

hyperplane. E very hyperplane H divides the space in to tw o halfspaces:

H + — {x G R ” : pTx > cco}

H ~ = {x G R " : pTx < a 0}

I t is easy to see th a t b o th halfspaces H + , H ~ are convex sets. A set P C R n is

a polyhedron i f i t is the intersection o f a fin ite num ber o f halfspaces. E quivalently,

a polyhedron is the set o f points th a t satisfy a fin ite num ber o f lin e ar inequalities;

O bviously, a polyhedron is a convex set. A polyhedron P C R " is bounded i f there

exists a positive scalar 00 such th a t P C {x G R " : —a; < X j < o j fo r j = 1 , . . . , n } .

Bounded p olyhedra are called poly topes. We say a polyhedron P is o f dim ension k,

denoted by d im (P ) — k, i f the m a xim u m num ber o f affinely independent p o in ts in P

is k + 1. In a d d itio n , a polyhedron P G R n is said to be full-dimensional i f d im (P )= ra .

I f P is n o t full-dim ensional, the n a t least one o f the inequalities plx < ai describing

2.1. POLYHEDRAL THEORY 6

valid inequality for P i f it is satisfied by a ll points in P . E quivalently, pTx < Qo is a

va lid in e q u a lity o f P i f and o n ly i f P lies in the half-space {x € Mn : pTx < a o }[2 6 , 13].

I f pTx < Oq is a va lid in e q u a lity o f P , then F = {x G P : pTx = « o } is called a

face o f P , and we say th a t (pT , a 0) represents F . N ote th a t F is a polyhedron and P

and $ are faces o f P . A face F is said to be proper i f F ^ <& and F ^ P . T he face F

represented by (pT , ao) is nonem pty i f and o n ly i f m ax {pTx : x G P } = cxq. W hen F

is nonem pty, we say th a t th e hyperplane pTx = a0 supports P . I f F' is a p roper face o f P , the n d im (P ) < d im (P ). In p a rtic u la r, the dim ension o f F is k i f the m a xim u m

num ber o f affinely independent p oints th a t lie in F is k + 1.

A face F o f P is called a facet o f P i f d im (F ) = d im (P ) — 1, and a face F o f P

is called an edge o f P i f d im (F ) = 1. G iven a polyhedron P = { i 6 R " : A x < b},

one is interested in fin d in g out w hich o f the inequalities alx < bi are necessary in the

d escription o f P and w h ich are redundant.

Facets, w hich have the highest dim ension among a ll p roper faces, are cru cia l for

the descriptio n o f a polyhedron in th e sense th a t, for each facet F o f P , at least one

o f the inequalities representing F is necessary in the description o f P . I f P is fu ll­

dim ensional, the n fo r each facet o f P , there exists a unique (up to a m u ltip lic a tio n by

a scalar) in e q u a lity representing it. However, i f P is not full-dim ensional, then there

are more th a n one in e q u a lity representing each facet.

P olyhedra can also be represented in term s o f th e ir extrem e points. G iven a convex

set S, x € S is said to be an extreme point o f S, i f i t is im possible to represent x as a

proper convex com b ina tio n o f tw o o ther points in S. i.e. x is an extrem e p o in t o f S

if f whenever x = \ x l + (1 — X) x2, x 1, x2 £ 5, 0 < A < 1, we m ust have x1 = x2 = x.

A p olyhedron P has a fin ite num ber o f extrem e points. Let x l , x2 be tw o d is tin c t

extrem e points o f P , the n re1, x2 are said to be adjacent i f the line segment [x 1, x 2] is

an edge o f P . N ote th a t a face F o f P is an extreme point i f and o n ly i f d i m( F ) = 0.

T he fo llo w in g w ell-know n result shows th a t a p olyto p e can be expressed as the convex

2.2. G RAPH THEORY 7

Th e o r e m 2.1.3 (W e y l-M in k o w s k i). Let P be a nonempty polytope, and l et x 1, . . . , x k be its extreme points, then

k k

P = { x € ffin : x = Ai X1, where \ = 1, Aj > 0 fo r i = 1 ,. . ., k}

i = l i = 1

Th e o r e m 2 .1 .4 . Let P be a polyhedron defined by P = { x € M n : A x = b, x > 0 } .

Then dimension P < n — ra n k (A ).

2.2. G raph T h eory

In th is section, we review basic d e fin itio n s and results fro m graph theory.

A graph G = (V ,E ) consists o f a fin ite set V o f vertices and a collection E o f un­

ordered pairs o f vertices called edges. T w o or more edges th a t jo in the same p a ir o f

d is tin c t vertices are called parallel edges. A n edge represented by an unordered p a ir

in w hich the tw o vertices are th e same is know n as a loop. A simple graph is a graph

w ith no parallel edges and loops. T he complete graph K n is a graph w ith n vertices

in w hich there is an edge jo in in g every p a ir o f vertices. A bipartite graph, denoted by

G = (Vi U V2, E ), is a graph in w h ich the set o f vertices can be p a rtitio n e d in to tw o

subsets V\ and V2 such th a t every edge has one end node in V\ and the o ther in V2. The

complete bipartite graph is the graph (VJ. U V2, E ) in w hich there is an edge between

every ve rte x in Vi and every ve rtex in V2. A walk in G is a fin ite nonem pty sequence

W — v q, C], v-\, 6 2, v2, ■ ■ ■ Cfc, v^, whose term s are a lte rn a te ly vertices and edges, such

th a t, for 1 < * < k, the ends o f e,; are V i-\ and vt. I f the edges e i, e-2, ■ ■ ■ ■, ek o f a w a lk

are d is tin c t, W is called a trail, in a d d itio n , i f the vertices v0, v i , . . . ,Vk are d is tin c t,

W is called a path. A graph G = (V , E) is connected i f for each tw o vertices vt , Vj o f

G, there exists a p a th fro m vl to Vj. A p a th is closed i f its o rig in and term inus are

the same. A closed p a th containing a t least one edge is a cycle. The le n gth o f a p a th

o r a cycle is the num ber o f edges in it. T he fo llo w in g theorem characterizes b ip a rtite

graphs in term s o f cycles.

2.2. GRAPH THEORY 8

G iven a graph G = ( V , E ) , a matching M is a subset o f edges no tw o o f w hich are

in cid en t w ith a com m on vertex. V ( M ) denotes the set o f vertices in cid en t to an edge

in a m atching M . A m atching is said to be perfect i f V ( M ) = V , th a t is, every node

is matched.

T w o o f the m ost studied problem s concerning m atchings are th e maximum cardi­

nality matching problem and the minimum weight matching problem [9]. T he m a x i­

m um c a rd in a lity m atching problem is concerned w ith fin d in g a m a xim u m c a rd in a lity

m atching in a given graph. One o f its m any applications is the problem o f assigning

students to tw o-person d o rm ito ry rooms. In p a rtic u la r, given a lis t o f pairs o f s tu ­

dents w ho w ould be w illin g to share a room , th is problem asks fo r an assignment o f

students to room s so as to m axim ize the num ber o f room m ates w ho are acceptable

to each other.

T he m in im u m w eight m a tching problem is the problem o f fin d in g a perfect m a tch ­

ing w ith m in im u m w eight in an edge-weighted complete b ip a rtite graph. I t is also

know n as the assignment problem since i t models the fo llo w in g problem . G iven n

men, n jobs and a cost dkj o f m an i p e rfo rm ing jo b j , how should these men be

assigned to jobs in order to m inim ize the to ta l cost. The feasible region o f the lin ­

ear p ro gram m in g fo rm u la tio n o f the assignment problem , know n as the assignment

C H A P T E R 3

T h e A ssig n m en t P o ly to p e

In th is chapter, we review the properties o f th e assignment p olytope, w hich is also

know n as the B irk h o ff polytope. In p a rtic u la r, we present know n results concerning

its dim ension, facets, and extrem e points.

T he assignment problem is concerned w ith fin d in g a m in im u m w eight perfect

m atching in a b ip a rtite graph. G iven a b ip a rtite graph G = (V i U V2, E ) and |V i| =

|V2I — n, le t us associate w ith each edge ( i , j ) E E a weight dtl and a b in a ry variable such th a t x t:) = 1 i f ( i , j ) belongs to a m atching and x tJ = 0 otherwise. T hen the

assignment problem can be form u la te d as the fo llo w in g integer p ro gram m in g problem :

m in Y S ^ i d i j X ij (14)

s.t. J 2 j= i x ij = 1 f ° r a ll * = 1, • • •, n (15)

Y T i= i = 1 fo r all j = 1, . . . ,n (16)

E { 0 , 1 } for a lH , j = 1 , . . . , n (17)

As i t w ill be shown la ter, c o n d itio n (17) can be replaced by

x^ > 0 fo r a ll i , j = l , . . . , n , (18)

since the co n stra in t m a trix o f th e assignment problem is Totally Unim odular(TU). A

m a trix A is said to be T U i f the d ete rm ina n t o f every square s u b -m a trix o f A is 0,1

o r —1.

T he assignment polyto p e o f order n, denoted by Pn , is the set o f a ll feasible

solutions o f the assignment problem , i.e. the set o f a ll x = ( x^) sa tisfying (15), (16),

and (18). A non-negative n x n m a trix is called a doubly stochastic i f the sum o f

the entries in each row and in each colum n is equal to 1. T he sim plest exam ple o f

3. TH E ASSIGNMENT POLY TO PE 10

stochastic m atrices are p e rm u ta tio n m atrices. A permutation m atrix P is a square

m a trix w ith e xactly one ’1’ in each row and in each colum n (the rest o f the entries being zero). T hus by considering the variables as the entries o f an n x n m a trix ,

Pn can be equivalently defined as the set o f a ll d o u b ly stochastic m atrices o f order

n. F urtherm ore, there is one-to-one correspondence between feasible assignments and

p e rm u ta tio n m atrices o f the same order.

L e t I = { l , . . . , n } and J = { l , . . . , n } . In some cases we w ill fin d it conve­

nient to represent variables x t] o f a feasible assignment b y (i , j ) t h cells in the tw o

dim ensional a rray I x J w ith the values o f the variables entered in th e ir associated

cells. In o ther cases a feasible assignment w ill be represented by a p e rm u ta tio n

( a ( i i ) , a ( i 2) , . . . , a ( i n)), such th a t x Ul = l , x2i2 = l , . . . , x nin = 1, and = 0 o th ­

erwise. For example, the diagonal assignment is represented by the p e rm u ta tio n

(1,2, . . . , n).

T he fo llo w in g result is well know n[7]. We present a p ro o f fo r completeness.

T h e o r e m 3.0.2. Let Pn be the assignment polytope of order n. Then the dimen­

sion of Pn is (n — l ) 2.

P r o o f: Since the ra n k o f the co nstra in t m a trix in (15-16) is 2n — 1, the n by Theorem

2.1.4, we have th a t dim Pn < n2 — (2n — 1) — (n — l ) 2. N e xt, we w ill show th a t

dim Pn > (n — l) 2 by e x h ib itin g (n — l) 2 + 1 affinely independent assignments in Pn

thus p ro ving the theorem.

Represent each assignment e ithe r as a p e rm u ta tio n (c r(l), <r(2),. . . , ar(n)) or as a per­

m u ta tio n m a trix .

Step 1: F irs t, le t x1 = (1, 2 , . . . ,n ) . See Table 3.1

T hen b y sw itch in g colum n 1 and colum n k in assignment x 1, for k — 2 , . . . , n. We

o b ta in assignments: x 1,2, x 1,3, . . . , x 1,n, where x l 'k — (k,2, 3 , . . . , k — 1,1, k + 1, . . . , n) for k = 2 , . . . , n. In th is step we have a to ta l o f n — 1 new assignment.

Step 2, N ow le t x2 = x 1'2. See table 3.2

B y sw itch in g colum n 1 and colum n k in x2, fo r a ll k 3, we o b ta in the assignments:

3. TH E ASSIGNMENT POLYTOPE 11

T hus generating (n — 2) new assignment.

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1

Table 3.1 x1

0 1 0 0 o 0 0 0

1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

o 0 0 0 0 0 0 1

Table 3.2 (x2 = x 1,2)

Step 3: N ow le t x3 = x 1,3. B y sw itch in g colum n 1 and colum n k in x3 for a ll k 3,

we o b ta in the assignment x 3,2. x 3,4, x 3,5, . . . , x 3,n. Thus generating (n — 2) new assign­

ments.

3. TH E ASSIGNMENT POLYTOPE 12

X4-2 , X4’3 X4’5 , X4’6 , X4 ’"

x 5>2 , x 5’3 , x 5’4 , 2-5.6,

I " - 1’2 , X rfVrt—1,3 . , x n “ M , ■jXt — l , n —2 —l,ro

X™’2 rrTl,X 3, x " ’4 , . . . . ,£.71,71 — 2 ,£,71,71 — 1

Therefore, the to ta l num ber o f assignment generated is 1 + (n — 1) + (n — 2 )(n — 1) =

1 + (n — l ) 2.

N e xt we show th a t these assignments are affinely independent. Let x1,1 denote the diagonal assignment x 1. A rra n ge a ll these (n — l) 2 + 1 assignments in the order the y were generated.. Thus fo r a ll these assignments we have: the i j th com ponent o f as­

signment x1'3 is equal to 1, w h ile the i j th com ponent o f a ll assignments generated

before x1'3 is equal to 0. Therefore, a ll these assignments are affinely independent,

and the results follow s.■

T h e follo w in g is an im m ediate co ro lla ry to the p ro o f o f the previous theorem .

Co r o l l a r y 3.0.3. Let Pn be the assignment polytope of order n, then Xij > 0 is

a facet-inducing inequality of Pn fo r all i, j = 1 , . . . , n.

From the d e fin itio n o f d ou b ly stochastic m atrices it im m e d ia te ly follow s th a t a

convex co m b ina tio n o f p e rm u ta tio n m atrices is a dou b ly stochastic m a trix . The

converse, nam ely th a t every d o u b ly stochastic m a trix can be expressed as a convex

com bination o f p e rm u ta tio n m atrices was independently proven by B irk h o ff and Von

Neum ann[25],

T h e o r e m 3.0.4. (Birkhoff-Von Neumann theorem) Let A be a doubly stochastic

matrix of order n, then A can be written as a convex combination of permutation

m ,atrice s o f o rd e r n .

Proof:

Since A = is a d o u b ly stochastic m a trix , a ll entries are non-negative.

Let P l be a p e rm u ta tio n m a trix such th a t Ai = m in {a tJ : P\ 7 = 1 } is positive. T hen

3. THE ASSIGNM ENT POLYTOPE 13

the num ber o f zero entries o f R1 is a t least one m ore tha n those o f A. Repeating th is argum ent on R1 and n o tin g th a t A has a t m ost n2 non-zero entries, a fte r a fin ite , say

k, steps we have

A = A1P1+ , • • •, +AfcPfc

where each Pi is a p e rm u ta tio n m a trix , A, > 0 and Y li= 1 = 1- ®

F ollow ing is an example o f the decom position process used in the above p ro o f o f

B irk h o ff-V o n Neum ann theorem.

G iven the stochastic m a trix

( 1

2

A =

1

3

1 1

2 2

1 1 \

6

0

0 ₆

6

/

The m in im u m positive e n try in A is so le t Ai be and

Pi

T hen P x = A — A] Pi is nonnegative and has equal row and colum n sums.

R i

( I I q \ 2 3 u

I

I

0

0 0

I /

Now the m in im u m positive e n try in P i is so le t A2 = | , and

P

2

is so

' 0 1

1 0

0 0

3. THE ASSIGNMENT POLYTOPE 14

I t'2 = R \ — A2P2 is again nonnegative and has equal row and colum n sums.

The m in im u m positive e n try in R2 is so le t A3 = and

0 1 0

0 0 1

/

A fte r th is decom position, A = Ai P i + X2P2 + A3P3, where each Pi is a p e rm u ta tio n

m a trix , and each Ai; for i= l, 2 , 3 and Ai + A2 + A3 = | | | = 1.

Because o f the B irk h o ff-V o n Neum ann Theorem , the extrem e points o f the assign­

ment p olyto p e Pn are e xa ctly the n x n p e rm u ta tio n m atrices [5]. A n o th e r way to

arrive at th is result is by using the n o tio n o f to ta l u n im o d u la rity [9]. I t is easy to prove

th a t the co nstra in t m a trix o f the assignment problem is T U . Therefore, a ll extrem e

p oints o f th e assignment p o lyto p e are integral. Because o f this, co nd itio n (17 ) can be

replaced by co n d itio n (18) in th e integer program m ing fo rm u la tio n o f the assignment

problem .

A djacency on the assignment polyto p e is characterized in th e fo llo w in g theorem .

[18]

T h e o r e m 3 .0 .5 . Let M i and M 2 be two distinct assignments. Then M i and M 2 a rc a d ja c e n t on th e a s s ig n m e n t polyt.opa i f f ( M i \ M 2) U ( M 2 \ M i ) fo r m s one. cycle.

R elated to th e n o tio n o f adjacency o f extrem e points is the n o tio n o f diam eter

o f a polytope. T he distance between a p a ir o f extrem e p o ints in a p o lyto p e is the

3. TH E ASSIGNMENT POLYTOPE 15

a p olyto p e is the greatest distance between any p a ir o f extrem e points in the polytope.

The fo llo w in g theorem establishes the diam eter o f the assignment p olytope. [4]

Th e o r e m 3 .0 .6 . The assignment polytope has diameter 2.

T h is theorem im plies th a t any tw o d is tin c t feasible assignments are either adja­

cent on the B irk h o ff P olytope or are b o th adjacent to some feasible assignment.

T h e fo llo w in g is an exam ple o f the ch aracterization of adjacency on the assignment

polytope.

L e t M i, M2 be the sets o f edges corresponding to the assignment (2 ,1 ,3 ,4 ) and (1 ,2 ,3 ,4 ). ( M i \ M2) U ( M2 \ M i) form s one cycle, thus M i and M2 are adjacent. Now let M3 be the set o f edges o f assignment (1 ,2 ,4,3 ), th e n ( M i \ M 3) U ( M3 \ M i) form s tw o cycles. Hence, M i and M3 are n o t adjacent.

T he existence o f m any efficient alg orith m s fo r solving the assignment problem

is due, in p a rt, to the s im p lic ity o f its p olytope. In the fo llo w in g chapters, th is

m otivates our p o lyhe d ral in ve stiga tio n o f the p olyto p e Q%[n2 obtained b y intersecting

i-C H A P T E R 4

K n ow n F acets o f

Qn,Tl

_ni,n2

R ecall th a t our problem is

m in L I U £ " = i dijx ij ( 19)

subject to £ I L i = 1 fo r all j —1, . . . , n (2 0)

E j = i x ij = 1 fo r a lH = 1 , . . . , n (21)

e {0,1} for all i, j = 1, . . . , n (2 2)

= r

(23)

where a ll cy are 0 o r 1, and r is an integer such th a t 0 < r < n.

L e t G = (Vi U V2, E), | Vj | = | V2I = n be a colored complete b ip a rtite graph, where edges are colored e ithe r red o r blue. T hen any feasible so lu tion to (20)-(23) can be

in terprete d as a perfect m atching on G w hich uses e xactly r red edges, where an edge

( i , j ) is colored red i f and o nly i f c,? = 1. L e t us represent each edge ( i , j ) by a cell

(i, j ) in a tw o dim ensional a rray I x J where I = J = { 1 , 2 . , n}.

We say th a t problem (19)-(23) belongs to a special case called th e partitioned case

i f there exist p a rtitio n s / = I \ U I2 and J = J\ U such th a t cell ( i , j ) is red i f and

only i f ( i , j ) S (R x J j) U (I2 x J2). In th is p a rtitio n e d case, the cells o f the I x J

array are p a rtitio n e d in to 4 blocks: B \ = R x J{, B2 = h x J2, B :i = I2 x J 2, and

B4 = I2 x Ji. L e t |J i| = n \ and |J i| = n 2. T hen it was shown in [19] th a t in the

p a rtitio n e d case, c o n stra in t (23) is equivalent to

X I x ij = ri> (24)

(i , j ) e B i

4. KNOWN FACETS OF Q lpn2 17

I t is n o t d iffic u lt to show th a t (24) is also equivalent to either one o f the follo w in g

constraints:

= r 2, (25)

where r2 = n\ — rq ,

Xii = r3 > (26)

where r3 = n — n2 — r 2,

Xii = r4 > (27)

( i , j ) e s 4

where r± = n2 —'r\.

T h e follo w in g theorem was proved in [3].

T h e o r e m 4.0.7 (A lfa k ih et al [3]). The problem of solving (19)-(23) polynomially

reduces to a problem of the same type belonging to the partitioned case.

Therefore, w ith o u t loss o f generality we assume th a t o ur problem belongs to the

p a rtitio n e d case.

Define:

P n f f2 = Set o f feasible solutions o f (20), (21), (24) and Xtj > 0, i , j = 1 ,n.

Q r n k = integer h u ll o f P f [ f 2.

T h e o r e m 4.0.8 (A lfa k ih a t al [3]). Suppose r4 > 1 f o r i = 1 , . . . , 4 and Q ^ f k 7^ 0.

Then dimension = dimension P f f k ~ n2 ~ 2n.

T w o large classes o f facet-inducing inequalities for Q^’f k were Presented in [3].

Before we present these tw o classes we re m a rk th a t th e facet-inducing inequalities for

-4.1. FIR ST CLASS OF FACET-INDUCING INEQUALITIES FOR 18

4 .1 . F irst C lass o f F acet-Ind ucing Ineq ualities for Q”fn2

Facet-inducing inequalities fo r Qn’[h2 o f the firs t class are characterized by a p ri­

m a ry defining cell a non-em pty subset o f row indices K r , and a non-em pty

subset o f colum n indices K c

-T he defining cell (p, q) for the firs t class can be any cell in the array. Suppose it is

in block B \. T hen the defining subset of row indices K r m ust be a n on-em pty proper

subset o f I2, and the defining subset of column indices K c m ust be a non-em pty

proper subset o f J2, and together th e y have to satisfy \ K r \ + \Kc\ = 1 + r.3.

P

Kr

Figure 4.1 Facets of the First class

+ + + *•• +

B i b2

+

+

+

b4

4.2. SECOND CLASS OF FACET-INDUCING INEQUALITIES FOR 19

T H E O R E M 4.1.1 (A lfa k ih et al [3]). Let (p ,q ) be the defining cell and K R and K c be the defining subsets of row and column indices selected as discussed above. Then

x pg

t

y

]

x pj

+

y

]

X iq —

y

'

x VJ

< 1

j e K c i ^ h \ K R , j e J 2\ K c

is a facet-inducing inequality fo r Qnfh2 ■

N ote th a t all coefficients in th is facet-inducing in e q u a lity are —1,1 or 0. T h is

in e q u a lity is shown in Figure4.1 where a + ( —) sign in cell ( i , j ) means th a t the

coefficient o f in the in e q u a lity is +1( —1).

4.2. Second C lass o f F acet-Ind u cing In eq u alities for

Qn,r_{Til ,712} 1

Facet-inducing inequalities o f th e second class are characterized by tw o defining

cells called the primary and the secondary defining cells, and by tw o defining subsets

o f row indices, and tw o defining subsets o f colum n indices.

T he p rim a ry defining cell, (p, q) can be any cell in the array. Suppose i t is con­

tained in block B \, the n th e second class o f facet-inducing inequalities fo r Qnffn 2

exists o n ly i f the numbers r-2 and r4 are b o th > 2 . I f th is co n d itio n is satisfied, the

secondary defining cell (m, I) can be any cell in block B2 o r B4 such th a t I q.

Suppose th a t ( m , l ) G B 4. T he defining subsets o f colum n indices K c , K q can

be any nonem pty d is jo in t p roper subsets o f J2. T he defining subset K R can be any n onem pty subset o f J2\ { m } , and the defining subset K f{ can be any nonem pty subset o f I \ . These defining subsets also m ust satisfy \Kc\ + \ K r \ = 1 + r s , and

\ K C\ + \K r \ = r 4.

W ith those assum ptions m entioned above we have

Th e o r e m 4.2.1 (A lfa k ih et al [3 ]). Let (p ,q ), (m , l ), K R, K R, K c and K c be as discussed above. Then,

%pq

+

y ^ Xpj

+

X iq

y

^

■

j e K c i6K R i e I2\ ( K R U { m } ) j e J 2\ K c j £J 2\ ( Kc UKc )

y ]

x v

_

y

^

x u

— 1

4.2. SECOND CLASS OF FACET-INDUCING INEQUALITIES FOR Q " ^ 20

q l Kg Kc

p

Hr

m

k r

is a facet-inducing inequality fo r Qnfh2 ■

T h is in e q u a lity is shown in Figure 4.2. As was pointed o u t in

[3],

these tw o

classes o f facets do n o t present a com plete description o f Q rr\ f \ l 2. In the next chapter

we present a new class o f facet-inducing inequalities fo r Qnfh2 ■

+ + + ••• +

B i E 2

_

---+

+

+

-b4

:

E*3

C H A P T E R 5

N e w C lass o f F a cet-In d u cin g In eq u a lities for

Q

^ n2

In th is chapter, we present a new, i.e. a th ird , class o f facet-inducing inequalities

for Qn'[h2 ■ Facet-inducing inequalities in th is new class are characterized by a primary

defining cell (p ,q), three secondary defining cells (l, q) , (m ,q) and (m ’ ,q); and by

fou r n onem pty d is jo in t defining subsets o f colum ns K c , K c, K c , K c , and b y one

nonem pty defining subset o f rows K r .

T h e p rim a ry defining cell, (p,q), can be any cell in the array. Suppose i t is in

block B i, th is new class o f facet-inducing inequalities for o n ly exists i f

r2 > 2 and r4 > 3, (28)

or

r2 > 3 and r& > 2. (29)

I f (28) holds, then the three secondary defining cells can be in B lo ck B4. O n the other hand, i f (29) holds, then the three secondary defining cells can be in B lo ck B 2.

Suppose th a t (28) holds and th a t the secondary defining cells are in B 4. T hen

K r C B \ { p } , K c , K q C J A M and K c , K c C J2 (see Figure 5 .1 ). We require th a t

these defining subsets o f rows and colum ns satisfy

\K c \ + \K c \ = r 2, (30)

\ K c \ + \ K c \ = r l + l , (31)

5. NEW CLASS OF FACET-INDUCING INEQUALITIES FOR Ql'^„ 22

(I Kc K c

< > < —> k c h'< p

Kn

I

m

'm!

+

+ + ••• +

--- E

*1

- b

2

+

---

---R,

D4

D3

--- --- --- -

. .

,

----Figure 5.1 Facets of the third class

L e m m a 5 .0 .2 . Let K c , K c , K c , K c , Kr be as discussed above and assumer 4 > 3

and?~2> 2. Then

x._PQ+ y~^ x Pj + x tq y~^ •x *.?

j£J2\(KcUKc )

E

=

iefQi, jeA c je A c

y ~ i x v ~ X,rry

»e/i\KnU{p}, j e K c ?:e/i\K'/iu{p}. j gKc j& Kc j e l < c

y ^

x m ' j

_ y ^

x m ' j

~

y .

j e ^ c j e A c i £ l2\ { l , m , m ' } , j e K c U K c U K c U K c

(3 3 )

5. NEW CLASS OF FACET-INDUCING INEQUALITIES FOR Q l ' ^ 23

P ro o f: For any assignment x G Q™'[R2, the sum

Xpq"I- %lqT 'y ^ X p j, (34)

j e J i M K c u K c )

is equal to 0,1 , or 2. I f (34) is equal to e ithe r 0 or 1 the lem m a tr iv ia lly holds.

Therefore, assume th a t i t is equal to 2. T h is holds when xiq = 1 and xn{] = 1 for

some j0 € ^ \ ( - ^ c U K c ).

For ease o f n o ta tio n le t B c = J2\ { K C U K c ), A c = J i \ ( { q } U R c U K c ) , A R =

I i \ ( { p } L ! K R), and B R = I2\ { l , m , m'}(see Figure 5.2). Thus i t follows fro m (30)-(32)

th a t

p

Kr

Ar

I

m

m1

Br

q

K c K c Ac Be K „

Kc

< >< > < > < >

+

+

B i

Ba

Bs

B,

5. NEW CLASS OF FACET-INDUCING INEQUALITIES FOR 24

\Bc \ = r 3, (35)

\ A c \ = u - 2 , (36)

\Ar \ - \ R C\ = \Rc\ - I- (37)

We say a sub-block (X x Y ) has k allocations i f there exists an assignment x €

Q l ’unz such th a t E ( i j)e (x x y ) = k

-R ecall th a t in any assignment x <E Q%[h2, blocks B x, B 2, B3 and R4 m ust have

allocations r i , r 2, r3 and r4 respectively. Four cases w ill be considered (see Figure

5.2):

C ase 1:

x mj =

0

and x m/j =

0.

jeKc j£ Rc

R ecall th a t xiq = 1. Since xP ] 0 = 1 fo r some jo G R c , sub-block ( / 2 x Be)

can have a t m ost r3 — 1 allocations. B u t fo r any assignment x in Qn’[,h2:

B lo ck B3 should have r3 allocations. Therefore, a t least some cell in B lo ck

B3 w ith a negative sign m ust have an allo catio n and the result follows.

C ase 2:

x m>j1 =

1

fo r some j x € K c and x mj =

0.

I f some cell ( i , j ) w ith a negative sign in B lo ck B3 has an a llo ca tio n then

we are done. So assume th a t sub-block (J2 x Be) has r3 — 1 allocations, i.e.,

Xij = 1 fo r a ll j e Be- Thus sub-block ( ( K R iJ A R) x B e ) has no allocations.

N ow we have tw o subcases.

Su b case 2a:

x mn = 1 fo r some j2 G

Be-T hen the sub-block ( {m , m'} x J i \ { q } ) can n o t have any allocations.

T herefore sub-block ( Br x J i \ { q } ) m ust have r4 — 1 allocations in any feasible assignment x. However, sub-block ( Br x Ac) can have a t most

r4 — 2. T hus one cell w ith a negative sign in sub-block ( B R x ( K cU K c j)

m ust have an a llo ca tio n and the result follows.

5. NEW CLASS OF FACET-INDUCING INEQUALITIES FO R 25

T he sub-block ( ( Kr U Ar) x ( ( K c \ { j i } ) U K c)) m ust have r 2 — 1 al­

locations. Now i f any cell w ith a negative sign in sub-blocks ( Kr x

( K c \ { j i } ) ) or ( Ar x K c ) has an allocation, we are done. So assume

th a t sub-blocks ( Kr x K c) and ( Ar x Kc) have no allocations. Now

fro m (30) it follows th a t sub-blocks ( Kr x Kc) and ( Ar x K c) have

r2 — 1 allocations. Hence, a ll colum ns in K c and K c have allocations. Recall th a t j x€ K c has an a llo ca tio n in cell ( m ' , ji).

Now in B lo ck B xi f any cell in sub-block ( K Rx

Kc)

o r sub-block ( Ar x

K c ) has an a llo ca tio n the n we are done. Therefore assume th a t sub­

blocks ( Kr x K c ) and ( Ar x Kc) have no allocations. T h is im plies

th a t sub-blocks ( Kr x ( Kc U Ac)) and ( Ar x ( Kc U A c j ) m ust have 7T allocations since the colum n q already has an a llo catio n in the cell

(l,q) £ B 4. N o w we have to consider tw o subcases depending on w hether

or not a colum n in A c has an a llo ca tio n in B lo ck B x.

Su bcase i:

Xij3 = 1 fo r some i £ K r U A r and some j 3£

Ac-In th is case, S ub-block ( ( { m } U B r ) x ( « / l \ { < 7 } ) ) m ust have r 4 — 1

allocations. However, i t follows fro m (36) th a t ( ( { m } U B r ) x A c )

can have at m ost r 4 — 3 allocations. Hence, Sub-block ( ( { m } U

B r ) x ( K c U

Kc))

m ust have at least 2 allocations. T h is im plies th a t sub-block ( B r x ( K c U K c ) ) m ust have at least 1 a llo ca tio n and the result follows.

Su bcase ii:

S ub-block ( ( K r U A r ) x A c ) has no allocations.

In th is case, Sub-blocks ( K r , K c ) and ( A r x ( K c ) ) m ust have

7T allocations. Now i t follow s fro m (32) and (37) th a t \ K r \ —

\ K C \= \ K C \ ~ 1 and \ A R \- \ K c \ { j i } \ = \ K C\- Hence, S ub-block

( K r x K c ) has \ K c \ —1 allocations and Sub-block ( A r x K c ) has \ K c \ allocations. Therefore, the Sub-block ( ( { m } U B r ) x ( K c U

5. NEW CLASS OF FACET-INDUCING INEQUALITIES FO R 26

\Ac\ — r 4 ~ 2. Thus th e Sub-block ( ( { m } U Br) x Kc) m ust have an a llo ca tio n and the result follows.

C ase 3:

x rnH =

1

fo r some

j 4

G and x m >j =

0.

K K c

T h is case is s im ila r to Case 2.

C ase 4:

x mj, =

1

for some j?,

€

K c and x m>:j6 =

1

fo r some jo

€

T h is

case is s im ila r to Case 2a.

T h e o r e m 5.0.3. The valid inequalities of the form(33) are facet-inducing inequal­

ities fo r Q ^ n2.

T he fo llo w in g lem m a is cru cia l fo r the p ro o f o f the above theorem . I t allows us

to lif t a facet-inducing in e q u a lity fo r Qnfh2 in to another facet-inducing in e q u a lity for

(Qn+l.n nn+l,n 1,71+1 j ✓yi+l.n ^Tti,ri2 ’ ^vni+l,ri2’ ^vni4-l,ri2+l’ cmu ^m,ra2+l*

LEM M A 5.0.4 (A lfa k ih et al [3]). Let X n = i X q = i au x u C «-o be a non trivial

facet-inducing inequality fo r Qn’[h2 and let A* = (a*j) be the (n + 1) x (n + 1) matrix derived

from A = (aij) such that:

A A, 0 \

Aio. 0

)

fo r any io G {n f + 1 , . . . , n } and any jo G { n2 + 1 , . . . , n } satisfying aioj0 = 0, where

A,j0 and Ai0. denote, respectively, the joth column and the ioth row of A. Then

X q = i a *jx ij — ao is a facet-inducing inequality for provided that it is a

valid inequality fo r it.

P r o o f o f T h eorem 5.0.3:

Consider the problem where n = 7, n4 = 3, n2 = 4, rq = 1. T hen r2 = 2, r3 = 1 a n d r 4 = 3 (see F ig u r e 5 .3 ) .

L e t (p,q) = (1 ,1 ), I = 4, m = 5 and m' = 6. F urther, le t B e = 5, K c = { 2 } ,

K c = { 3 } , K C = {6} , K C = { 7 } and K R = {2}. T hen

5. NEW CLASS OF FACET-INDUCING INEQUALITIES FOR 27

q

_{K c K c K c K c}

p

+

+

Kr -

--

-I

+

m

-

-m f

-

-- - -

-Figure 5.3 Facets for the

Q7

/ A

is a facet-inducing in e q u a lity o f Q l ’^, since i t is a va lid in e q u a lity o f Q I ’I by Lem m a

5.0.2 and since the follo w in g 35 feasible assignments, represented as p erm utations,

are affinely independent and satisfy (38) as an equality. Recall th a t d im Q34 = 35. x 1 = ( 1 , 5 , 6 , 2 , 7 , 3 , 4 ) , x 2 = ( 5 , 1 , 6 , 2 , 7 , 3 , 4 ) , x 3 = (5, 1, 6, 7, 2, 3,4)

x 4 = (1, 7 , 6 , 5 , 2 , 3 , 4 ) , x5 = (1, 7, 6,2, 5 ,3,4) , x 6 = (2, 7 , 6 , 1 , 5 , 3 , 4 )

x 7 = ( 5 , 4 , 6 , 1 , 2 , 3 , 7 ) , x 8 = ( 5 ,4 ,6, 1, 7,3, 2), x 9 = (5, 4, 6, 2, 7,3,1)

x 10 = ( 5 ,2 ,6 ,1 , 7, 3,4), x 11 = ( 5, 3 , 6, 1 , 7, 2,4), x 12 = (5, 3, 6, 1, 7, 4,2)

x 13 = ( 5,3,6, 7 , 2 , 1, 4 ), x 14 = ( 5, 3 , 6, 1 , 2, 7,4), x 15 = ( 7 , 4 , 6 , 1 , 2 , 3 , 5 )

x 16 = (4, 7 ,6 ,1 ,2 ,3 , 5), x 17 = (3, 7,6,1, 2 ,4 ,5 ), x 18 = (3, 7 ,6 ,1 ,2 , 5,4)