A Maximum B-Matching Problem Arising from Median Location Models with Applications to the Roommates Problem

Home , Matching (statistics)

A MAXIMUM bMATCHING PROBLEM ARISING FROM MEDIAN

LOCATION MODELS WITH APPLICATIONS TO

THE ROOMMATES PROBLEM

Arie Tamir

Department of Statistics and Op erations Research

Scho ol of Mathematical Sciences

Sackler Faculty of Exact Sciences

Tel Aviv University

Tel Aviv Israel

Joseph S B Mitchell

Department of Applied Mathematics and Statistics

State University of New York at Stony Bro ok

Stony Bro ok NY USA

Revised Novemb er

Abstract

We consider maximum bmatching problems where the no des of the graph repre

sent p oints in a metric space and the weight of an edge is the distance b etween

the resp ective pair of p oints We show that if the space is either the rectilinear

plane or the metric space induced by a tree network then the bmatching prob

lem is the dual of the single median lo cation problem with resp ect to the given

set of p oints This result do es not hold for the Euclidean plane However we

show that in this case the bmatching problem is the dual of a median lo cation

problem with resp ect to the given set of p oints in some extended metric space

We then extend this latter result to any geo desic metric in the plane

The ab ove results imply that the resp ective fractional bmatching problems

have integer optimal solutions

We use these duality results to prove the nonemptiness of the core of a

co op erative game dened on the ro ommate problem corresp onding to the ab ove

matching mo del

Intro duction

Let G V E b e an undirected graph with a no de set V and an edge set E A

matching in G is a subset of E such that each no de of G is met by at most one edge in

the subset In Edmonds discovered a strongly p olynomial algorithm for the weighted

matching problem as well as a characterization of a linear system that denes the convex

hull of the incidence vectors of all the matchings of a graph The concept of a matching

was generalized by Edmonds to bmatchings dened as follows For each no de v V let

v denote the set of edges in E which meet v For each vector x x e E and a

0 0 0

subset E of E let xE fx e E g

Let b b v V b e a p ositive integer vector A bmatching of G is a nonnegative

integer vector x x e E satisfying the following degree constraints

x v b v V

A bmatching of G with b for all v V is a matching A bmatching is perfect if all

the degree constraints are satised as equalities Given a vector of weights c c e E

the weighted bmatching problem is

Maximize c x

e e

e2E

st x v b v V

x and integer

The linear relaxation of called the fractional bmatching problem is

Maximize c x

e e

e2E

st x v b v V

A characterization of a linear system of the convex hull of the bmatchings of a graph

was given by Edmonds and Pulleyblank and app eared in Pulleyblank Pulleyblank

has also provided a pseudop olynomial algorithm to nd a maximum weighted bmatching

Cunningham and Marsh found the rst p olynomial time algorithm for nding an

optimal bmatching and several years later Anstee presented the rst strongly

p olynomial algorithm for this problem The reader is referred to Lovasz and Plummer

and Co ok and Pulleyblank for additional results on general matchings b

matchings and capacitated bmatchings

In this pap er we study a sp ecial case of where the no des of the graph represent

p oints in a metric space and the weight of an edge is the distance b etween the resp ective

pair of p oints We rst consider in Section the case in which the metric space is

induced by a tree network Sp ecically let T V E b e an undirected tree with p ositive

edge lengths For each pair of no des v u V let dv u denote the length of the unique

simple path on T connecting v and u We consider the sp ecial instance of the weighted

bmatching problem on the complete graph G V E where for each edge e v u E

c dv u Note that G and T have the same no de set We prove that for this case

the fractional bmatching problem has an integer optimal solution where the numb er

of p ositive comp onents is at most jV j The pro of is based on a duality argument showing

that the dual of is the problem of lo cating a weighted median on the given tree T In

Section we extend the duality result to a restricted weighted median problem

We demonstrate that the ab ove integrality prop erty might not hold for the capacitated

bmatching version of where p ositive upp er b ounds are imp osed on the x variables

even if the tree T is a simple path

In Section we consider the case where the no des of the graph represent p oints

in the rectilinear plane Assuming that fb v V g is even we prove that the ab ove

integrality result holds for this case as well Again the pro of is based on showing that

the dual of the linear relaxation of is the problem of nding a weighted median of the

given set of p oints in the rectilinear plane

In Section we consider the Euclidean plane The linear relaxation of is not the

dual of the problem of nding the weighted median of the given p oints in the Euclidean

plane However using a dierent argument we prove that if fb v V g is even the

linear relaxation of has the integrality prop erty for this case as well This integrality

prop erty implies that the bmatching problem for the Euclidean plane is the dual of a

weighted median problem in some metric space extension of the Euclidean space We then

extend these results to general geo desic metrics in the plane Section In Section

we give counterexamples to the duality and integrality prop erties in three dimensions

In Section we consider an extended bmatching problem for a metric induced by

a tree Finally in Section we apply our results to the ro ommates mo del intro duced by

Gale and Shapley In the ro ommates problem there is a group of an even numb er

of b oys who wish to divide up into pairs of ro ommates Each b oy ranks all other b oys in

accordance with his preference for a ro ommate A matching of the group of b oys to pairs

is said to b e stable if under it there do es not exist a pair of b oys who would prefer to ro om

with each other rather than with their assigned ro ommates Using a simple example Gale

and Shapley demonstrated that in general the ro ommates problem may have no stable

matching The reader is referred to Guseld and Irving for recent structural and

algorithmic results on this problem Bartholdi and Trick studied the sp ecial case

where preferences are single p eaked and narcissistic and proved the existence of a unique

stable matching Roughly sp eaking this corresp onds to the case where there is a one

dimensional frame of reference eg temp erature scale or highway distance map that is

common to all b oys and alternatives are preferred less as they are farther from each b oys

ideal In contrast we consider the case where alternatives are preferred more as they are

farther from each b oys lo cation We discuss a co op erative game with side payments based

on this ro ommates problem and show that the core is nonempty for the metric spaces

discussed in Section This is to b e compared with the result of Granot who

showed that in general the core of the ro ommates problem co op erative game with side

payments may b e empty

Maximum bMatchings and Median Lo cation

The Tree Metric Space Problem

Given is an undirected tree T V E with p ositive edge lengths For each pair of

no des v u V we let dv u b e the length of the unique simple path connecting v and u

on T dv u is called the distance b etween v and u Let G V E denote the complete

undirected graph having V as its no de set Consider the maximum bmatching problem

and assume that for each edge e v u E c dv u The linear programming

dual of is

Minimize b z

v v

v 2V

st z z dv u for all pairs v u

v u

We now interpret the dual problem as a median lo cation problem on the given tree

T V E Supp ose that the edges of the tree are rectiable and the tree is emb edded

in the plane Each edge is a line segment of the appropriate length We refer to interior

p oints on an edge by their distance on the line segment from the two no des of the edge

The continuum set of all p oints of these segments is a metric space that we denote by

AT Supp ose that each no de v of the tree is asso ciated with a nonnegative weight b

and consider the problem of nding a p oint in AT that will minimize the sum of weighted

distances from the p oint to all no des It is well known that if b is a p ositive real for all

v V then the set of optimal p oints known as weighted medians of T is either a single

no de or the set of p oints on a single edge of T In particular there is always one weighted

median which is a no de The next theorem follows from the general results in Erkut

Francis Lowe and Tamir on monotonic tree lo cation problems It states that is

a reformulation of the ab ove weighted median problem

v V b e Theorem Supp ose that b is nonnegative for each v V Let z z

an optimal solution to problem Then there exists a weighted median of T v V

with resp ect to the weights fb g such that z dv v for all v V

We will need the following characterization of weighted medians due to Sabidussi

Zelinka Kang and Ault and Kariv and Hakimi

Theorem Let v b e a no de of T Then v is a weighted median with resp ect to the

nonnegative weights fb g if and only if the sum of the weights of the no des in each connected

comp onent of T obtained by the removal of v do es not exceed fb v V g

We are now ready to prove the main result of this section

Theorem Supp ose that b is a p ositive integer for each v V The linear program

has an integer optimal solution x where the numb er of its p ositive comp onents is at

most jV j

Pro of It is sucient to show that there is an integer feasible solution to with at most

jV j p ositive comp onents whose ob jective value is equal to the optimal value of the linear

program Let v b e a weighted median of T Consider the connected comp onents of

the tree obtained by the deletion of v From Theorem the total no de weights in each

comp onent do es not exceed B where B fb v V g By aggregating comp onents

1 2 3

we can obtain a partition of the no de subset V fv g into three subsets fV V V g

satisfying the following prop erties

j j

i B fb v V g B for j

ii Every simple path connecting a pair of no des v u where v is in some V and u is in

V V must contain the median v

We will now construct a bmatching x of the complete graph G V E with the

following two prop erties

iii No pair of no des b elonging to the same subset V are matched together

e v g b iv For each no de v v fx

Therefore if the edge e connects the pair of no des v and u G then the contribution

of the variable x to the ob jective is x dv u x dv v dv u Moreover from iv

e e e

it follows that the ob jective value of any bmatching satisfying iiiiv is fb dv v

v V g which is exactly the ob jective value of the weighted median problem To

1 2 3

construct the ab ove bmatching supp ose without loss of generality that B B B

4 4

Next we replicate each no de v and For convenience denote V fv g and B b

j j

replace it by b copies quasinodes Thus we assume that V contains B quasino des

i j

j By saying that we match p pairs from V V we will mean that a subset

i j

of p quasino des from V are paired with a subset of p quasino des from V

Consider the following cases

1 2 1 2 3

I B B and B B B

2 1 2

Match B pairs from V V

3 1 3

B pairs from V V and

1 2 3 1 4

B B B pairs from V V

1 2 1 2 3 2 3 1

I I B B B B B and B B B is even

1 2 3 1 2

Match B B B pairs from V V

1 2 3 1 3

B B B pairs from V V and

1 2 3 2 3

B B B pairs from V V

1 2 1 2 3 2 3 1

I I I B B B B B and B B B is o dd

1 2 3 1 2

Match B B B pairs from V V

1 2 3 1 3

B B B pairs from V V

1 2 3 2 3

B B B pairs from V V and

1 4

pair from V V

1 2 3

IV B B and B is even

1 3 1 2

Match B B pairs from V V

3 1 3

B pairs from V V and

3 2 3

B pairs from V V

1 2 3

V B B and B is o dd

1 3 1 2

Match B B pairs from V V

3 1 3

B pairs from V V

3 2 3

B pairs from V V and

1 4

pair from V V

It is easy to check that each one of the ve cases constructs a bmatching satisfying

prop erties iiiiv ab ove For example consider Case I Each matched pair involves a

1 1 4

no de from V The only questionable matching is from V V However this matching

1 2 3 4

is feasible if B B B is b etween and B Since we fo cus on Case I the nonnegativity

1 2 3 1 1 2 3 4

of B B B is ensured The second constraint is equivalent to B B B B B

1 2 3 4

which follows from prop erty i ab ove Note that B B B B B

It remains to prove that in each one of the ve cases the bmatching can b e imple

mented using at most jV j edges Consider for example Case I Similar arguments can b e

applied to the other cases Let V b e a minimal subset of V with resp ect to contain

2 3 2 3 2 3

ment such that fb v V g B B Match B B pairs from V V V

v 1 1

Using a basic solution argument eg the classical NorthWest corner rule applied to

the transp ortation problem we conclude that this matching can b e implemented using

2 3

at most jV j jV j jV j edges From the minimality of V all no des in V but

1 1 1

p ossibly one say v are p erfectly matched ie they satisfy the equality in prop erty iv

1 2 3 1 4

Again this match ab ove Next we match B B B from V V fv g V

1 4

ing can b e implemented using at most jV j jV j jV j edges To conclude

1 2 3

jV j jV j jV j jV j edges will suce to implement the bmatching in Case I As

noted ab ove similar arguments can b e used for the other four cases This completes the

pro of

We note that recently Gerstel and Zaks considered the case where all edges are

of unit length and b for each v V They did not discuss the fractional matching

problem However they did prove that the optimal value of the resp ective integer

program is equal to the optimal value of the unweighted median problem It is clear

that if b for each v V there is an optimal integer solution where the numb er of

p ositive variables is at most jV j

Theorem and the duality of problems imply that in our case an optimal

integer bmatching can b e found in O jV j time The complexity b ound is dominated by

the eort to nd a weighted median of a tree The latter task can b e p erformed in linear

time see Goldman Kariv and Hakimi For comparison purp oses recall

that the b est complexity b ound known for the general maximum weighted bmatching

problem is sup er cubic in jV j Anstee The b ound is cubic when b for all

v V Lawler If in addition the edge weights fc g equal to the Euclidean

distances b etween p oints in the plane which represent the no des of the graph then some

sub cubic algorithms are known Vaidya presented such an algorithm for nding

a minimumweight geometric p erfect matching Marcotte and Suri considered the

case where the representing p oints are on the b oundary of a simple p olygon in the plane

They gave an O jV j log jV j time algorithm for the minimumweight p erfect matching

problem which has b een improved to O jV j log jV j by Hershb erger and Suri

and an O jV j log jV j time algorithm for the maximumweight p erfect matching problem

They also presented improved b ounds for the case where the p olygon is convex

We p oint out another interesting prop erty of our mo del From Theorem it follows

that an optimal weighted median dep ends only on the top ology of the tree and the no de

weights fb g but not on the edge lengths Due to the ab ove duality we conclude that an

optimal bmatching is also indep endent of the tree distances fdv ug

Remark Theorem shows that there is an optimal solution x to the bmatching

problem considered ab ove where the numb er of edges e with a p ositive variable x is at

most jV j We note that the resp ective b ound for general bmatching problems is higher

than jV j It is shown in Pulleyblank that there is an optimal solution to the general

bmatching problem where the subset of edges with a p ositive variable contains no even

cycles simple or nonsimple Therefore all cycles are o dd and no de disjoint In particular

there are at most jV j such cycles in the subgraph induced by the ab ove subset of edges

By removing one edge from each cycle this subgraph reduces to a forest Thus the total

numb er of edges in the subgraph is at most jV j Simple examples demonstrate

that this b ound is tight

Remark From the pro of of Theorem we observe that the strict p ositivity

assumption on the fb g co ecients can b e weakened Generally it is not p ossible to replace

the inequality with equality in the degree constraints since the problem with equality

constraints might b e infeasible Indeed in the optimal solution constructed in the ab ove

pro of the degree of the median v is equal to zero in Cases II and IV and it is equal to

one in Cases I I I and V In Case I the median degree is guaranteed to b e b ounded ab ove

by b Thus the only cases of the pro of where the p ositivity is required are III and

Therefore V Furthermore even in these two cases we only need the p ositivity of b

the p ositivity assumption can b e replaced by the assumption that all the co ecients are

nonnegative and the co ecient of a weighted median is p ositive Alternatively it suces

to require that the sum of all the co ecients is even since this will exclude cases III and

V The integrality result of Theorem may not hold if neither of the two assumptions

is satised Consider the example of a no de tree where one of the no des the center is

connected by an edge to each one of the other no des leaves Supp ose that each edge is of

length one Finally set the b co ecient of each leaf to b e equal to and let the resp ective

co ecient of the center b e equal to Generally to nd the b est integral solution when

the sum of the fb g co ecients is o dd and the co ecient of the weighted median is zero

we rst subtract from the p ositive co ecient of a no de which is closest to the weighted

median amongst all no des with p ositive co ecients and then solve the mo died problem

as ab ove We also note that if the given tree is a general path then Theorem holds

even if the b co ecients are allowed to take on a zero value

If we imp ose p ositive integer upp er b ounds on the x variables in problem the

integrality result may not hold even if the tree is a path

Example Consider the path dened by the following p oints no des on the real

line v v v v and v Let b b b b and

1 2 3 4 5 v v v v

1 5 2 4

Supp ose that the ten x variables corresp onding to the edges of the complete graph b

with the no de set V fv v v v v g are all b ounded ab ove by The b est integer

1 2 3 4 5

solution satisfying the constraints of and the ab ove upp er b ound yields the ob jective

value However the following fractional solution sp ecied by its nonzero comp onents

yields the value

x x x

(v v ) (v v ) (v v )

1 5 1 4 2 5

and

x x x x x

(v v ) (v v ) (v v ) (v v ) (v v )

1 2 2 4 4 5 5 3 3 1

We note that Theorem cannot b e extended to general distance functions where we

replace the assumption that the numb ers fdv ug represent tree distances by the weaker

assumption that they only satisfy the triangle inequality This is obvious when the numb er

of no des is o dd and b for each v since no p erfect matching is feasible Consider

a no de triangle with unit length edges More interestingly assuming b for each

v V the result do es not hold for the even case for which a p erfect matching is feasible

even when jV j as demonstrated by the next example

Example Consider the complete graph with V fv v v v v v g Sup

1 2 3 4 5 6

p ose that b for i Let dv v i j b e equal to if i j is in

v i j

f g and b e equal to otherwise The optimal ob jective value of prob

lem is while the b est integer solution satisfying the constraints of yields the

value

The Rectilinear Planar bMatching Problem

The duality result b etween the weighted median problem in the tree metric space

AT and the resp ective bmatching problem do es not hold for general metric spaces see

next section We now prove that it do es hold for the rectilinear plane

1 2 2

Let V fv v g v v v i n b e a collection of p oints in R Let

1 n i

i i

b i n b e a p ositive integer weight The rectilinear weighted median problem in

1 2

the plane is to nd a p oint x x x which minimizes the function

2 2 1 1

j j jx v jx v b

i i i

i=1n

The rectilinear integer bmatching problem on the complete graph G V E is the

version of problem where for each edge e v v

i j

2 2 1 1

j a v j jv v c dv v jv

e i j

j i j i

Theorem Consider the rectilinear weighted median problem in the plane and supp ose

v V g is even Then the optimal solution value of this problem is equal that fb

i v

to the optimal solution value of the resp ective rectilinear integer bmatching problem

Moreover the former is equal to the optimal ob jective value of problem while the

latter is equal to the ob jective value of problem where fc g and fdv v g are dened

e i j

ab ove in a

Pro of Using continuity and p erturbation arguments it is sucient to prove the theorem

for the case when the n p oints fv v g are distinct b for i n and no

1 n v

pair fv v g i j lie on a vertical or horizontal line in R The supp osition in the

i j

theorem implies that n is even An optimal solution to the rectilinear problem is given by

1 2 1 1 2

the p oint x x x where x is a median of the set fv g and x is a median of the set

2 1 2

fv g Due to the ab ove assumptions the median p oint x x x can b e selected such

that no p oint v lies on the vertical and horizontal lines passing through x Consider the

four quadrants dened by this pair of lines and let n k denote the numb er

of p oints in the set fv g that lie in the k th quadrant We have n n n n n

i 1 2 3 4

and n n n n n Thus n n and n n Consider a pair of p oints v

2 3 1 4 1 3 2 4 i

and v where v is in the st quadrant nd quadrant and v is in the rd quadrant

j i j

th quadrant Then dv v dv x dv x Therefore if we now match the n

i j i j 1

n p oints in the st nd quadrant with the n n n n p oints in the rd

2 3 1 4 2

th quadrant we obtain a p erfect matching with an ob jective value of dv x To

complete the pro of we now show that the matching constructed is optimal Consider the

pair of linear programs where dv v is the rectilinear distance dened ab ove

i j

Problem is certainly a relaxation of the integer matching problem Problem can b e

viewed as the following relaxation of the rectilinear median problem Consider the complete

undirected graph with no de set fv v g fv g Each edge v v is asso ciated with a

1 n i j

length which is equal to the rectilinear distance dv v The problem is to assign lengths

i j

g to the edges fv v g such that the triangle inequality is satised for all edges of the fz

i v

v V g is minimized The rectilinear median problem in the plane graph and fz

i v

g are the rectilinear distances from corresp onds to the sp ecial case where the lengths fz

some p oint in the plane to the p oints fv g

The p erfect matching constructed ab ove and the rectilinear median x corresp ond

resp ectively to feasible solutions to problems Moreover as noted ab ove they

yield the same ob jective value to b oth problems Therefore from duality theory b oth are

optimal solutions to the resp ective problems This concludes the pro of of the theorem

The Euclidean Planar bMatching Problem

In the previous sections we proved that the problem of nding a weighted median of

a set of p oints fv v g in a tree metric space or in the rectilinear plane is the dual

1 n

of the resp ective bmatching problem when fb v V g is even We used this result

v i

to conclude that the resp ective fractional bmatching problem has an integer solution

Example demonstrates that this duality result do es not hold even in the Euclidean

plane

Example Consider the vertices fv v v g of an equilateral triangle with unit

1 2 3

edge lengths in the Euclidean plane Let b for i The optimal solution

value to the bmatching problem is The weighted median problem in this case is to

nd a p oint in the plane that minimizes the sum of weighted Euclidean distances to the

which is greater than vertices The optimal value is

Note however that for any metric space the optimal value of the weighted median problem

is greater than or equal to the optimal value of the resp ective bmatching problem

In spite of the lack of duality in the Euclidean planar case we prove in this section

that the resp ective fractional bmatching problem has an integer solution We will then

demonstrate in Section that the integrality prop erty of the rectilinear and Euclidean

planar cases do es not hold for higher dimensions

We will need the following lemma dealing with the general bmatching problem

Lemma Let G V E denote the complete undirected graph having V as its

no de set Each edge e v u E is asso ciated with a nonnegative real c and each no de

v V is asso ciated with a p ositive integer b Consider the bmatching problem and

its linear relaxation the fractional bmatching problem

i There exists an optimal solution x to such that all no des but p ossibly

one are saturated ie there exists a no de v and x v b for each v V fv g

0 v 0

ii Supp ose that the edge weights fc g satisfy the triangle inequality If has no

integer optimal solution then has an optimal solution x such that all no des are

saturated and each comp onent of x is an integer multiple of

Pro of The rst prop erty follows directly from the nonnegativity of the edge weights

fc g To prove the second prop erty consider a basic optimal solution of x It is well

known see Pulleyblank that each comp onent of x is an integer multiple of If

all no des are not saturated we assume that no de v is the only unsaturated no de in V We

also assume that x has the smallest saturation gap amongst all optimal solutions that

are integer multiples of If x for all edges e v u v v u v then x is

0 0

integer since x b for each edge e v v Thus supp ose that there exists a pair of

v 0

no des v u v v u v and x e v u is p ositive The saturation gap at v is a

0 0 0

P P

p ositive integer since it is equal to fb v V g fx e E g

Finally consider the solution of obtained from x by subtracting from x

e v u and adding to b oth x e v v and x e u v Since the saturation

0 0

e e

gap at v is at least one this solution is feasible Since the edge weights satisfy the triangle

inequality the new solution is certainly optimal Moreover all no des but p ossibly v are

saturated and the saturation gap at v has decreased This contradicts the fact that x

has the smallest p ossible gap

While the duality result do es not in general hold for the Euclidean plane we now

show that an integrality result do es hold

We will need a few denitions Let ab denote the directed closed line segment joining

two distinct p oints a and b in the plane A closed polygonal walk P v v is the

1 n

union of the line segments edges v v v v v v v v joining n distinct p oints v

1 2 2 3 n1 n n 1 i

If is the line containing ab then the two rays that comprise the set n ab are called

ab ab

the extension rays or simply extensions of ab We say that two line segments ab and

cd are in convex position if the lines containing these two segments intersect at a p oint

that is not contained in either of the two segments ie an extension of ab intersects an

extension of cd

We will need the following lemma which may have indep endent interest

Lemma Let P and Q b e closed p olygonal walks in the plane each having an o dd

numb er of vertices Then there exists an edge of P and an edge of Q such that the two

edges are in convex p osition

Pro of For simplicity we assume that no three vertices of P Q are colinear in partic

ular then the vertices of P Q are distinct and that no edge of P is parallel to an edge

of Q the lemma remains true in degenerate cases but the details are tedious

Let us b egin with some notation The lines of Q are simply the lines through edges of

Q For a p olygonal walk P v v we say that vertex v induces the closed convex

1 n i

cone C whose b oundary consists of the two rays with ap ex v that are the extensions of

i i

directed edges v v and v v If a line intersects a cone in a ray then we say that the

i1 i i+1 i

line and the cone are incident

The pro of of the lemma is based on two simple claims

Claim Let P b e a closed p olygonal walk having an o dd numb er of vertices Then any

line that is not parallel to an edge of P must b e incident on an o dd numb er of cones

induced by P In particular this implies that any line must b e incident on some induced

cone of P

Pro of Without loss of generality assume that is the y axis Since is not parallel

to any edge of P a vertex of P can b e classied as one of three typ es according to the

lo cal geometry of the two incident edges left b oth incident edges are to the right of

the vertex right vice versa or middle otherwise There must b e an even numb er of

leftright vertices since b etween any left resp right vertices there is a right resp

left vertex Thus since P is o dd there are an o dd numb er of middle vertices But for any

middle vertex the induced cone necessarily includes the vertical direction so the y axis

will intersect such a cone in a ray

Claim Let C b e a cone induced by P If C is incident on an o dd numb er of lines of

i i

Q then one of the two rays b ounding C must intersect an extension ray of Q

Pro of Assume that there are an o dd numb er of lines of Q incident on C Consider a

line of Q that is incident on C Consider the two vertices of Q that are on If neither

vertex lies in C or if b oth vertices lie in C then an extension ray of Q namely one of

i i

the two on must cross the b oundary of C and we are done Thus we can assume that

for any line of Q that is incident on C there is exactly one vertex v of Q on that

lies within C Each vertex of Q has exactly two lines through it thus there is a second

0 0

line of Q through v If is not incident on C then the extension ray of Q that has

ap ex v and lies on line must intersect the b oundary of C and we are done Thus we

can assume that each vertex of Q that lies within C has b oth of its lines incident on C

i i

But since each line of Q that is incident on C must have exactly one of its two vertices

within C this implies that there are an even numb er of lines of Q incident on C a

i i

contradiction

Now by Claim each line of Q is incident on an o dd numb er of cones induced by P

Since there are an o dd numb er of lines of Q this implies that the total numb er of linecone

incidences is o dd

This implies that some cone C of P is incident on an odd numb er of lines of Q But

then by Claim a ray b ounding C must intersect an extension ray of Q implying that

there is a crossing of extensions This concludes the pro of of the lemma

Remark Janos Pach has suggested to us an alternate but logically equivalent

pro of of this lemma this same pro of was previously discovered but not published by

M Perles The problem arises in the study of geometric graphs where the following

Turantyp e question arises What is the smallest numb er k n such that in any geometric

graph on n p oints in the plane that has at least k n edges there must exist a pair of edges

in convex p osition A simple construction shows that k n n Kupitz

conjectured that k n n Recently Katchalski and Last have shown that

k n n

Theorem Let V b e a nite set of p oints in R Each p oint v in V is asso ciated

with a p ositive integer b Supp ose that fb v V g is even Let G V E denote

v v

the complete undirected graph having V as its no de set For each edge pair of p oints

e v u let c denote the Euclidean distance b etween v and u Then the fractional

bmatching problem has an integer optimal solution where the numb er of its p ositive

comp onents is at most jV j

Pro of We assume without loss of generality that the p oints in V are distinct Other

wise identical p oints can b e aggregated without aecting the evenness assumption on the

sum of the no de weights

Moreover for the sake of brevity in order to avoid complicated and messy p ertur

bation arguments we assume that the p oints in V satisfy the following general p osition

assumption no three p oints of V are colinear

If problem has no integer solution then by Lemma there exists an optimal

solution to which is an integer multiple of and all no des are saturated

We thank Noga Alon for p ointing out the existence of Perles unpublished result and

for acquainting us with the relevant literature

We will now prove the claim that if problem has a saturated optimal solution

then every basic optimal solution that is saturated is integer

Let x b e a saturated basic optimal solution Supp ose that x is not integer x is an

integer multiple of

Let E b e the subset of edges asso ciated with integer comp onents of x Since x is not

integer E is a prop er subset of E For each no de v V let v b e the subset of edges

of E that are incident to v Consider the instance of problem obtained by subtracting

x v from b for each v V Refer to this instance as the reduced problem Let y

b e the vector obtained from x by equating to zero each comp onent x such that e E

Then y is an optimal solution to the reduced problem and all its nonzero comp onents

are o dd integer multiples of To prove the claim it is now sucient to contradict the

optimality of y

Consider the subset E of edges asso ciated with p ositive comp onents of y Since x

0 0

is a basic solution it follows from Pulleyblank that the subgraph G V E has

no even cycles In particular all cycles are o dd and they are no de disjoint If G has no

cycles it is a forest Since all no des are saturated y is integer If G has exactly one cycle

say C then all edges which are not on C must b e asso ciated with integer comp onents of

y But y has no p ositive integer comp onents Therefore all edges of G are on C Since

P P

e E C g e C g is o dd This is not p ossible since fx C is an o dd cycle fy

e e

is even and

X X X

fb v V g fy e C g fx e E C g

e e

Thus G contains two o dd cycles C and C which are no de disjoint The two cycles

1 2

corresp ond to two closed p olygonal walks P and P in the plane each having an o dd

1 2

numb er of vertices From Lemma there exist an edge of P and an edge of P such

1 2

that the two edges are in convex p osition

To simplify the notation supp ose that V fv v v g and that P v v v

1 2 n 1 1 2 k

and P v v Also assume that the edges v v and v v are in convex

2 k +1 p 1 2 k +1 k +2

p osition This implies that the four p oints fv v v v g are on the b oundary of their

1 2 k +1 k +2

convex hull p olygon where v is adjacent to v and v is adjacent to v Without loss

2 1 k +2 k +1

of generality supp ose that their order along the b oundary is given by v v v v

1 2 k +1 k +2

Consider the solution z obtained from y by subtracting from y and from

(v v )

1 2

y and adding to the comp onents y and y

(v v ) (v v ) (v v )

k+1 k+2 1 k+1 2 k+2

Then z is a saturated feasible solution that is an integer multiple of From the

assumption that the p oints in V are in general p osition it follows that the convex hull of

the ab ove four p oints is not a line segment Therefore it is easy to see that the ob jective

value at z is larger than that of y contradicting the optimality of y This completes

the pro of of the claim From the ab ove claim it follows that problem has an integer

optimal solution We can also use the ab ove claim to show that there is an integer optimal

solution where the numb er of its p ositive comp onents is at most jV j

If there is a saturated optimal solution it follows directly from the ab ove claim that

every saturated basic optimal solution is integer If there is no saturated integer optimal

solution then there is an integer optimal solution x and a no de v such that all no des

but v are saturated Moreover the saturation gap at v is even Consider the problem

0 0

This mo died problem obtained from by reducing the even saturation gap from b

is a saturated optimal solution Therefore by the ab ove claim each one of its saturated

basic optimal solutions is integer This completes the pro of of the theorem

Remark The pro of ab ove applies not only to the Euclidean metric but to any

metric that ob eys the simple prop erty that a straight path b etween p and q is an instance

of a shortest path in the metric In particular this geometric pro of gives an alternative

metho d of pro of for the integrality result of Theorem

Remark We have already noted ab ove that the planar Euclidean median problem

is not the dual of the resp ective matching problem However we can use the integrality

result of Theorem to dene an appropriate dual Problem can b e viewed as the

following relaxation of the weighted planar Euclidean median problem Consider a metric

space extension of the Euclidean plane which preserves the Euclidean distances b etween

the p oints in V The extended weighted median problem is to nd a p oint w in such a

space so that the sum of the weighted distances with resp ect to the new metric of w

from all p oints in V is minimized For each v V z will denote the distance from w to v

The duality of and the integrality result of Theorem imply that the extended

weighted median problem is the dual of the bmatching problem for the Euclidean planar

case

Geo desic Metrics in the Plane

We will next show that the integrality result of Theorem holds if we replace the

Euclidean metric with the geodesic metric induced by a set of p olygonal obstacles in the

plane More formally let F denote a b ounded closed multiply connected region in R

whose b oundary consists of a union of a nite numb er of b ounded line segments Such

a set F is commonly called a polygonal domain We refer to the set F as the free space

the complement R n F is a nite set S of connected p olygonal obstacles Since F is

b ounded and connected the obstacle set S consists of one unb ounded op en p olygon and

h b ounded op en simple p olygons where h is the numb er of holes in F Note that we

consider the free space F to b e closed it includes its b oundary Let W denote the nite

vertex set for F and hence for S

The geodesic distance with respect to F d p q b etween p oint p F and p oint q F

is dened to b e the Euclidean length of a shortest obstaclefree path from p to q ie the

length of a shortest path joining p to q within the free space F A shortest path within F

is called a geodesicF path It is well known see Alt and Welzl that a geo desicF

path b etween any two p oints of the free space is a simple nonselfintersecting p olygonal

path whose vertices b end p oints are among the set W of obstacle vertices In general

there may b e many shortest paths b etween two p oints in the sp ecial case that F is a

simple p olygon without holes there is a unique shortest path

Theorem Let F b e a p olygonal domain free space in the plane having vertex

set W Let U fu u g and V fv v g b e nite sets of p oints in the

0 k 1 0 l1

free space having o dd cardinalities k and l Let P resp Q b e a closed p olygonal walk

linking up the p oints U resp V using geo desicF paths u u resp v v

i i+1 j j +1

Then there exists a closed p olygonal walk R linking up the p oints U V using geo desicF

paths such that the length of R is greater than or equal to the sum of the lengths of P

and of Q

Pro of We make the following nondegeneracy assumptions ab out the input F

equals the closure of the interior of F No three p oints of V U V W are colinear

There is a unique geo desicF path b etween any pair of p oints V U V W and

No p oint w U V lies on some geo desic path b etween a pair of p oints w w U V

1 2

w w w This is analogous to the assumption in the Euclidean plane without

1 2

obstacles that no three p oints are colinear By a p erturbation argument the pro of can

b e extended to handle the cases when these assumptions do not hold but the details are

tedious and omitted here

Let VG denote the visibility graph whose no de set is V U V W and whose edges

called VGedges connect two no des if and only if the straight line segment joining the two

no des is contained in the free space F By our nondegeneracy assumption a VGedge

intersects no p oints of V other than at its endp oints Since F is closed the VGedges

include each edge line segment of the b oundary of F Any geo desic path b etween two

no des in V must lie on the visibility graph ie consist of a union of segments that are

VGedges see Alt and Welzl In particular the geo desic paths u u and

i i+1

v v that constitute the edges call them gedges of the walks P and Q must lie

j j +1

on the visibility graph

For a set of three distinct p oints z z z U V let z z z denote the closed

1 2 3 1 2 3

oriented Jordan cycle going from z to z along z z from z to z along z z and

1 2 1 2 2 3 2 3

then from z back to z along z z The fact that the three paths z z z z

3 1 3 1 1 2 2 3

and z z form a closed Jordan curve follows from our nondegeneracy assumption

3 1

which ensures that any two geo desicF paths that intersect do so in a connected set ie

a single subpath

From our assumptions we know that z z z separates R into two regions a

1 2 3

b ounded region which we call a geodesic triangle and an unb ounded region We say that

z U V lies to the left of z z if the region to the left of z z z is b ounded a

3 1 2 1 2 3

geodesic triangle The region will necessarily have nonempty interior from our nondegen

eracy assumptions and We let Lz z denote the set of all such p oints z U V

1 2 3

Similarly we say that z U V lies to the right of z z if the region to the right of

3 1 2

z z z is b ounded and we let Rz z denote the set of such p oints z U V

1 2 3 1 2 3

The following denitions apply to distinct p oints z z z z U V We say that

1 2 3 4

z z points at z z if z Lz z and z Rz z or z Rz z and

1 2 3 4 3 1 2 4 1 2 3 1 2

z Lz z

4 1 2

We say that z z and z z cross each other if they intersect in a connected

1 2 3 4

subpath and the ordering of the p oints ab out the union z z z z is z z z z

1 2 3 4 1 3 2 4

in either clo ckwise or counterclo ckwise traversal of the union Note that z z and

1 2

z z cross if and only if z z p oints at z z and z z p oints at z z

3 4 1 2 3 4 3 4 1 2

We say that gedge z z and gedge z z are in convex position if z z do es

1 2 3 4 1 2

not p oint at z z and z z do es not p oint at z z To motivate this denition

3 4 3 4 1 2

note that it is equivalent to saying that the gedges do not cross and the geo desic convex

hull of the endp oints A fz z z z g CH A has all four endp oints on it CH A

1 2 3 4

is the simplyconnected closed region having minimumlength p erimeter such that A

CH A F

Consider a gedge e u u of P and a gedge f v v of Q There are

i i+1 j j +1

four cases for any pair e f

e p oints at f and f p oints at e implying that e and f cross or

e p oints at f and f do es not p oint at e

f p oints at e and e do es not p oint at f

e do es not p oint at f and f do es not p oint at e implying that e and f are in convex

p osition

Now if P and Q are b oth o dd cycles then there are an o dd numb er of pairs e f

Our goal is to prove that the numb er of pairs e f corresp onding to case ab ove is odd

which in particular will show that there is at least one such pair in convex p osition Thus

it suces to show that each of the cases arise an even numb er of times This is done

in the following two claims

Claim There are an even numb er of pairs of crossing gedges e u u of P and

i i+1

f v v of Q

j j +1

Pro of Since any pair of gedges intersect in a connected set or not at all we can

p erturb the gedges of P and of Q so that in the arrangement of p erturb ed gedges of P

and Q any pair of crossing gedges share exactly one p oint any pair of adjacent gedges

along P Q share exactly one p oint while all other pairs are disjoint In particular two

nonadjacent gedges that coincide along a subpath but do not cross can b e p erturb ed so

that they are disjoint Since P is a closed walk the ab ove p erturbation implies that the

planar arrangement of P is a planar graph with all of its no des having an even degree

Thus the planar dual of this arrangement with no des corresp onding to the faces is

bipartite and hence can b e colored Now if we start at the vertex v of Q and walk

along the closed curve Q until we return to v we will cross from face to face in the

arrangement of P If we cross from a face of one color to a face of the same color then

the total numb er of gedges of P that are crossed is even We call such a crossing an even

crossing This can happ en only if the two faces share a common vertex of the arrangement

of P but do not share a common face If we cross from a face of one color to a face of

the other color then the numb er of gedges of P that are crossed is o dd Such a crossing

is called an odd crossing The total numb er of o dd crossings must b e even since we will

end up back in the cell where we started namely the face containing v Thus the

total numb er of crossings of gedges of P by the curve Q is even Now the claim follows

directly from the fact that a gedge of Q can interesect a gedge of P in at most one p oint

Claim A gedge f v v of Q p oints at an even numb er of gedges u u

j j +1 i i+1

of P Similarly a gedge e u u of P p oints at an even numb er of gedges

i i+1

v v of Q

j j +1

Pro of A vertex u of P is either left of f v v or right of f v v

i j j +1 j j +1

Here we are invoking the nondegeneracy assumptions Starting with vertex u

of P consider walking along P until we return to u Each time that we go from a vertex

that is left resp right of v v to a vertex that is right resp left of v v

j j +1 j j +1

the corresp onding gedge of P is p ointed at by v v Since P is a closed walk it is

j j +1

easy to see that the numb er of such transitions is even implying the claim

This allows us to conclude that

Lemma There exist gedges e u u of P and f v v of Q that are

i i+1 j j +1

in convex p osition

Finally we claim that

Claim If e u u and f v v are in convex p osition then either

i i+1 j j +1

a u v intersects v u or

i j +1 j i+1

b u v intersects u v

i j i+1 j +1

Pro of Since e and f are in convex p osition we know that e do es not p oint at f and

f do es not p oint at e Assume without loss of generality that v v Lu u If

j j +1 i i+1

a and b b oth fail to hold then we must have a planar emb edding of the two geo desic

triangles u u v and u u v Since they share the gedge e u u

i i+1 j +1 i i+1 j i i+1

we must have either v contained in the region b ounded by u u v or v con

j i i+1 j +1 j +1

tained in the region b ounded by u u v But in either case this implies that u

i i+1 j i

and u do not lie on the same side of f v v b oth in Lv v or b oth in

i+1 j j +1 j j +1

Rv v contradicting the fact that f do es not p oint at e

j j +1

From the ab ove claim and the lemma we know that there exist gedges e P and

f Q such that either a or b holds in particular we may assume that e u u

i i+1

and f v v exist such that u v intersects v u at some p oint c

j j +1 i j +1 j i+1

P Q By p erforming an exchange at p oint c we can construct a new closed walk R

as follows Starting at u follow P to u then follow u v to v then follow Q

0 i i j +1 j +1

to v then follow v u to u then follow P back to u The fact that R is longer

j j i+1 i+1 0

than or equal to the sum of the lengths of P and of Q follows from the triangle inequality

In particular the exchange has saved length

du u dv v

i i+1 j j +1

while adding length

du v dv u du cdc v dv cdc u du u dv v

i j +1 j i+1 i j +1 j i+1 i i+1 j j +1

Theorem can b e used to generalize Theorem to the case where for each edge

pair of p oints e v u c is the Euclidean length of a geo desic path connecting v and

u Following the pro of of Theorem we observe that in the case of the geo desic metric

Theorem enables us to successively replace a pair of o dd cycles by an even cycle

whose length is equal to or greater than the sum of the lengths of the two o dd cycles A

solution with no o dd cycles can b e represented as a convex combination of integer solutions

For the sake of brevity we skip the formal technical details For completeness we state the

generalized theorem

Theorem Let F b e a p olygonal domain free space in R and let V b e a nite set

of p oints in F Each p oint v in V is asso ciated with a p ositive integer b Supp ose that

fb v V g is even Let G V E denote the complete undirected graph having V as

its no de set For each edge pair of p oints e v u let c denote the Euclidean length

of a geo desicF path b etween v and u Then the fractional bmatching problem has

an integer optimal solution where the numb er of its p ositive comp onents is at most jV j

Higher Dimensions

We now consider the higher dimensional versions of the problems discussed ab ove

First we note that the duality result of Theorem do es not extend to the case where

the p oints fv g are in R and the metric is rectilinear This is demonstrated by the next

example

Example Consider the four p oints v v v

1 2 3

and v Let b i The solution value to the bmatching prob

4 v

lem is while the optimal solution value to the resp ective median problem is

Furthermore we also show that the integrality prop erty do es not hold in R even if

there are an even numb er of no des and the weights fb g are all ie even for the case of

maximum weight p erfect matching on an even set of no des Example considers the

Euclidean case while Example considers the rectilinear case

Example Consider the following six p oints in R v v

1 2

v v v v Then each of the two o dd

3 4 5 6

cycles P v v v and Q v v v has Euclidean length implying

1 2 3 4 5 6

that the fractional solution having x for edges e in P or Q and x otherwise

e e

On the other hand it is easy to check that every p erfect has ob jective value

matching on these six p oints has length less than

Example Consider the following six p oints in R v v

1 2

v v v v Then each of the two o dd cycles

3 4 5 6

P v v v and Q v v v has rectilinear length implying that the fractional

1 2 3 4 5 6

solution having x for edges e in P or Q and x otherwise has ob jective value

e e

On the other hand it is easy to check that every p erfect matching on these six p oints

has length at most

An Extended bMatching Problem

In this section we generalize the results of Section and consider the following

extended mo del

Referring to problem ab ove where c is dened as the resp ective tree distance

dv u supp ose that the degree co ecient b of each no de v can b e increased linearly

provided that a certain nonnegative p enalty p is paid The ob jective is now to maximize

the net reward obtained by subtracting the total p enalty from the weighted matching

Sp ecically consider the following problem

X X

Maximize c x p y

e e v v

e2E v 2V

st x v b y v V

v v

x y

We will extend Theorem and show that problem has an integer optimal solution

The dual linear program of is given by

Minimize b z

v v

v 2V

st z z dv u for all pairs v u

v u

z p for all no des v

v v

Again using the general results in Erkut Francis Lowe and Tamir we note

that problem is a reformulation of the following restricted weighted median problem

Find a p oint in AT whose distance from each no de v is at most p and such that the sum

of weighted distances of all no des to the p oint is minimized The optimal solution is called

a restricted weighted median If an unrestricted median satises the distance constraints

it is optimal Otherwise supp ose that the set of all p oints on the tree which satisfy the

distance constraints is nonempty It is easily observed that this feasible set is a connected

set closed subtree of AT The optimal restricted weighted median is the closest p oint

to the unrestricted weighted median in the feasible set

Theorem Supp ose that b is a p ositive integer for each v V If problem has a

nite optimal value then there is an optimal integer solution x y to problem where

g is at most jV j and the numb er of p ositive variables the numb er of p ositive variables fx

g is at most fy

Pro of Let t b e a p oint on the tree where the optimal restricted weighted median

is lo cated In particular using the ab ove duality the optimal value of and is

fb dv t v V g Supp ose without loss of generality that t is a p oint on some

edge v u of T and t is not the unrestricted weighted median v Let v b e such that

0 0 0

dv t p and the p oint t is on the unique path connecting v and v Let V denote

the set of no des in V having the p oint t on the unique simple path connecting them to v

Dene

0 0

B fb v V g

0 0

0 0 0

to the weight b B B Add y By denition we have B B Next dene y

v v v

It then follows from Theorem that each p oint on the edge v u which contains t is

an optimal unrestricted median with resp ect to the mo died set of weights fb g v v

0 0

fb y g Using Theorem for the mo died weights we conclude that there exists an

v v

integer matching x with at most jV j p ositive comp onents such that

x v b v v

0 0

B B x v b

and

X X

0 0

c x b dv t B B dv t

e v

e2E v 2V

Dene y by

0 0

B B if v v

otherwise

Then x y is a feasible solution to problem with ob jective value of

X X

0 0 0

B B p B B dv t c x c x

v e e

e e

e2E e2E

From this ob jective value is equal to the optimal value of the restricted weighted

median problem problem Using the duality of problems we conclude that

x y satises the statement of the theorem

The Ro ommates Problem

Consider the ro ommates problem dened in the Intro duction Let V indicates the

group of n b oys The b oys preferences are dened according to the following scheme

Each b oy is represented by some p oint v of a given metric space X For each pair of

b oys v u let dv u denote the distance b etween them For simplicity supp ose that the

nn distances b etween all pairs of b oys v u v u are distinct We consider the

case where alternatives are preferred more as they are farther from each b oys lo cation

Each b oy ranks all other b oys by their distances from him where the highest preference is

given to the farthest one The ro ommate problem with these preferences is easily observed

to have a unique stable matching First match the unique pair of b oys whose distance is

the diameter of the set of n representing p oints ie the pair v u corresp onding to the

largest distance Remove this pair of p oints from the set and rep eat the pro cess with the

remaining set Having established the existence of a stable matching we follow Granot

and analyze a co op erative game asso ciated with the ro ommates problem We

consider the multiple version of this ro ommates problem This game with side payments

V f is dened by its characteristic function f R Supp ose that G V E is

the complete undirected graph with no de set V Each no de v V represents a b oy

For each nonempty subset S of V coalition let

f S Max c x st x v b for each v S

e e v

e2E

x v for each v V S

x and integer

The core of the co op erative ro ommates problem game C V f is dened by

C V f y y v V fy v S g f S S V

v v

and fy v V g f V

Granot considered the co op erative game corresp onding to the classical match

ing mo del where each player b oy v is to b e paired with a single ro ommate ie b for

each v V and demonstrated that the core might b e empty when the fc g co ecients are

arbitrary nonnegative reals The games corresp onding to Example and Example

show that the core might b e empty even when these co ecients satisfy the triangle inequal

ity and jV j The core of any game with jV j is nonempty if the triangle inequality

holds We now use the ab ove integrality results to prove that the core is nonempty if the

fc g co ecients are dened by any of the distance functions discussed in Section

Theorem Let G V E b e the complete undirected graph with no de set V Each

edge e E is asso ciated with a nonnegative weight c and each no de v V is asso ciated

with a p ositive integer b Supp ose that the fractional bmatching problem has an

integer optimal solution Let z z v V b e an optimal solution to problem

Then the vector y y b z v V is in the core C V f of the resp ective

v v

ro ommates problem game

Pro of Given the game V f dened ab ove we consider the fractional ro ommates prob

lem game V f where for each S V

c x st x v b for each v S f S Max

e e v

e2E

x v for each v V S

Since we have dropp ed the requirement that x should b e integer we have f S

f S for all S V Moreover since has an integer optimal solution f V f V

Therefore C V f C V f Since the game V f can b e viewed as a linear

pro duction game it follows from Owen that if z is an optimal solution to problem

the vector y y b z v V is in the core of the game V f This completes

v v

the pro of of the theorem

Remark If b is not constant for all v V the core C V f may contain vectors

that cannot b e generated from the solutions to problem by the ab ove pro cess As an

b b example consider the case of a no de tree V fv v v g where b

v v 1 2 3 v

3 2 1

and dv v dv v The vector is in the core but there is no optimal

1 2 2 3

The example given in Remark shows that the solution z to problem with z

core C V f may b e empty if the sum of all fb g co ecients is o dd and the b co ecient

of the weighted median is zero

Theorem Let G V E b e the complete undirected graph with no de set V Each

edge e E is asso ciated with a nonnegative weight c and each no de v V is asso ciated

with a p ositive integer b Assume that b b for all v V for some p ositive integer

v v

b Supp ose that the fractional bmatching problem has an integer solution If y is a

vector in C V f then the vector y b is an optimal solution of

Pro of We prove that if b b for each v V then for each vector y C V f

the vector z y b is an optimal solution to problem If y is in the core then it

follows from the duality of that fy v V g is equal to the optimal ob jective

P P

value of problem But fb z v V g fy v V g Thus it is sucient

v v

to prove that z satises the constraints of problem The monotonicity of f implies

that y is nonnegative Therefore by denition z is nonnegative We have to prove that

z z dv u for each pair of no des v u Indeed consider the coalition S fv ug

v u

0 0 0

Since b z is in the core we have b z b z f S But f S is easily observed to b e equal

v u

to b dv u

It may happ en that the core of the game discussed in Theorem has no stable

matching with resp ect to the preferences dened ab ove This is demonstrated by the next

example

Example Consider a no de path with dv v dv v i

1 2 i i+1

Then add a new no de v and connect it to v with an edge of length The no de v

7 2 3

is the unique median of this tree The no des v and v should b e matched in any stable

1 7

matching However this is not the case in a core solution

Acknowledgements We thank Noga Alon Esther Arkin Micha Sharir and Emo

Welzl for useful discussions on p ortions of this research We also thank the referees for

several suggestions that improved the presentation of this pap er Jo e Mitchell thanks

Noga Alon and Micha Sharir for hosting his visit to the Sp ecial Semester in Computa

tional Geometry at Tel Aviv University which provided the opp ortunity for this research

collab oration

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