Frugality

Ratios

And

Improved

Truthful

Mechanisms

for

Vertex

Cover

Edith

Elkind

Hebrew

University

of

Jerusalem,

Israel,

and

University

of

Southampton,

Southampton,

U.K.

Leslie

Ann

Goldberg

University

of

Liverpool

Liverpool

U.K.

Paul

Goldberg

University

of

Liverpool

Liverpool

U.K.

ABSTRACT

In

set-system

auctions,

there

are

several

overlapping

teams

of

agents,

and

a

task

that

can

be

completed

by

any

of

these

teams.

The

auctioneer"s

goal

is

to

hire

a

team

and

pay

as

little

as

possible.

Examples

of

this

setting

include

shortest-path

auctions

and

vertex-cover

auctions.

Recently,

Karlin,

Kempe

and

Tamir

introduced

a

new

definition

of

frugality

ratio

for

this

problem.

Informally,

the

frugality

ratio

is

the

ratio

of

the

total

payment

of

a

mechanism

to

a

desired

payment

bound.

The

ratio

captures

the

extent

to

which

the

mechanism

overpays,

relative

to

perceived

fair

cost

in

a

truthful

auction.

In

this

paper,

we

propose

a

new

truthful

polynomial-time

auction

for

the

vertex

cover

problem

and

bound

its

frugality

ratio.

We

show

that

the

solution

quality

is

with

a

constant

factor

of

optimal

and

the

frugality

ratio

is

within

a

constant

factor

of

the

best

possible

worst-case

bound;

this

is

the

first

auction

for

this

problem

to

have

these

properties.

Moreover,

we

show

how

to

transform

any

truthful

auction

into

a

frugal

one

while

preserving

the

approximation

ratio.

Also,

we

consider

two

natural

modifications

of

the

definition

of

Karlin

et

al.,

and

we

analyse

the

properties

of

the

resulting

payment

bounds,

such

as

monotonicity,

computational

hardness,

and

robustness

with

respect

to

the

draw-resolution

rule.

We

study

the

relationships

between

the

different

payment

bounds,

both

for

general

set

systems

and

for

specific

set-system

auctions,

such

as

path

auctions

and

vertex-cover

auctions.

We

use

these

new

definitions

in

the

proof

of

our

main

result

for

vertex-cover

auctions

via

a

bootstrapping

technique,

which

may

be

of

independent

interest.

Categories

and

Subject

Descriptors

[Theory

of

Computation]:

Analysis

of

Algorithms

and

Problem

Complexity;

[Computer

Applications]:

Social

and

Behavioral

Sciences-economics

General

Terms

Algorithms,

Economics,

Theory

INTRODUCTION

In

a

set

system

auction

there

is

a

single

buyer

and

many

vendors

that

can

provide

various

services.

It

is

assumed

that

the

buyer"s

requirements

can

be

satisfied

by

various

subsets

of

the

vendors;

these

subsets

are

called

the

feasible

sets.

A

widely-studied

class

of

setsystem

auctions

is

path

auctions,

where

each

vendor

is

able

to

sell

access

to

a

link

in

a

network,

and

the

feasible

sets

are

those

sets

whose

links

contain

a

path

from

a

given

source

to

a

given

destination;

the

study

of

these

auctions

has

been

initiated

in

the

seminal

paper

by

Nisan

and

Ronen

(see

also

We

assume

that

each

vendor

has

a

cost

of

providing

his

services,

but

submits

a

possibly

larger

bid

to

the

auctioneer.

Based

on

these

bids,

the

auctioneer

selects

a

feasible

subset

of

vendors,

and

makes

payments

to

the

vendors

in

this

subset.

Each

selected

vendor

enjoys

a

profit

of

payment

minus

cost.

Vendors

want

to

maximise

profit,

while

the

buyer

wants

to

minimise

the

amount

he

pays.

A

natural

goal

in

this

setting

is

to

design

a

truthful

auction,

in

which

vendors

have

an

incentive

to

bid

their

true

cost.

This

can

be

achieved

by

paying

each

selected

vendor

a

premium

above

her

bid

in

such

a

way

that

the

vendor

has

no

incentive

to

overbid.

An

interesting

question

in

mechanism

design

is

how

much

the

auctioneer

will

have

to

overpay

in

order

to

ensure

truthful

bids.

In

the

context

of

path

auctions

this

topic

was

first

addressed

by

Archer

and

Tardos

They

define

the

frugality

ratio

of

a

mechanism

as

the

ratio

between

its

total

payment

and

the

cost

of

the

cheapest

path

disjoint

from

the

path

selected

by

the

mechanism.

They

show

that,

for

a

large

class

of

truthful

mechanisms

for

this

problem,

the

frugality

ratio

is

as

large

as

the

number

of

edges

in

the

shortest

path.

Talwar

extends

this

definition

of

frugality

ratio

to

general

set

systems,

and

studies

the

frugality

ratio

of

the

classical

VCG

mechanism

for

many

specific

set

systems,

such

as

minimum

spanning

trees

and

set

covers.

While

the

definition

of

frugality

ratio

proposed

by

is

wellmotivated

and

has

been

instrumental

in

studying

truthful

mechanisms

for

set

systems,

it

is

not

completely

satisfactory.

Consider,

for

example,

the

graph

of

Figure

with

the

costs

cAB

cBC

A

B

C

D

Figure

The

diamond

graph

cCD

cAC

cBD

This

graph

is

and

the

VCG

payment

to

the

winning

path

ABCD

is

bounded.

However,

the

graph

contains

no

A-D

path

that

is

disjoint

from

ABCD,

and

hence

the

frugality

ratio

of

VCG

on

this

graph

remains

undefined.

At

the

same

time,

there

is

no

monopoly,

that

is,

there

is

no

vendor

that

appears

in

all

feasible

sets.

In

auctions

for

other

types

of

set

systems,

the

requirement

that

there

exist

a

feasible

solution

disjoint

from

the

selected

one

is

even

more

severe:

for

example,

for

vertex-cover

auctions

(where

vendors

correspond

to

the

vertices

of

some

underlying

graph,

and

the

feasible

sets

are

vertex

covers)

the

requirement

means

that

the

graph

must

be

bipartite.

To

deal

with

this

problem,

Karlin

et

al.

suggest

a

better

benchmark,

which

is

defined

for

any

monopoly-free

set

system.

This

quantity,

which

they

denote

by

ν,

intuitively

corresponds

to

the

value

of

a

cheapest

Nash

equilibrium.

Based

on

this

new

definition,

the

authors

construct

new

mechanisms

for

the

shortest

path

problem

and

show

that

the

overpayment

of

these

mechanisms

is

within

a

constant

factor

of

optimal.

Our

results

Vertex

cover

auctions

We

propose

a

truthful

polynomial-time

auction

for

vertex

cover

that

outputs

a

solution

whose

cost

is

within

a

factor

of

of

optimal,

and

whose

frugality

ratio

is

at

most

where

Δ

is

the

maximum

degree

of

the

graph

(Theorem

We

complement

this

result

by

proving

(Theorem

that

for

any

Δ

and

n,

there

are

graphs

of

maximum

degree

Δ

and

size

Θ(n)

for

which

any

truthful

mechanism

has

frugality

ratio

at

least

This

means

that

the

solution

quality

of

our

auction

is

with

a

factor

of

of

optimal

and

the

frugality

ratio

is

within

a

factor

of

of

the

best

possible

bound

for

worst-case

inputs.

To

the

best

of

our

knowledge,

this

is

the

first

auction

for

this

problem

that

enjoys

these

properties.

Moreover,

we

show

how

to

transform

any

truthful

mechanism

for

the

vertex-cover

problem

into

a

frugal

one

while

preserving

the

approximation

ratio.

Frugality

ratios

Our

vertex

cover

results

naturally

suggest

two

modifications

of

the

definition

of

ν

in

These

modifications

can

be

made

independently

of

each

other,

resulting

in

four

different

payment

bounds

TUmax,

TUmin,

NTUmax,

and

NTUmin,

where

NTUmin

is

equal

to

the

original

payment

bound

ν

of

in

All

four

payment

bounds

arise

as

Nash

equilibria

of

certain

games

(see

the

full

version

of

this

paper

the

differences

between

them

can

be

seen

as

the

price

of

initiative

and

the

price

of

cooperation

(see

Section

While

our

main

result

about

vertex

cover

auctions

(Theorem

is

with

respect

to

NTUmin

ν,

we

make

use

of

the

new

definitions

by

first

comparing

the

payment

of

our

mechanism

to

a

weaker

bound

NTUmax,

and

then

bootstrapping

from

this

result

to

obtain

the

desired

bound.

Inspired

by

this

application,

we

embark

on

a

further

study

of

these

payment

bounds.

Our

results

here

are

as

follows:

We

observe

(Proposition

that

the

four

payment

bounds

always

obey

a

particular

order

that

is

independent

of

the

choice

of

the

set

system

and

the

cost

vector,

namely,

TUmin

NTUmin

NTUmax

TUmax.

We

provide

examples

(Proposition

and

Corollaries

and

showing

that

for

the

vertex

cover

problem

any

two

consecutive

bounds

can

differ

by

a

factor

of

n

where

n

is

the

number

of

agents.

We

then

show

(Theorem

that

this

separation

is

almost

best

possible

for

general

set

systems

by

proving

that

for

any

set

system

TUmax/TUmin

n.

In

contrast,

we

demonstrate

(Theorem

that

for

path

auctions

TUmax/TUmin

We

provide

examples

(Propositions

and

showing

that

this

bound

is

tight.

We

see

this

as

an

argument

for

the

study

of

vertexcover

auctions,

as

they

appear

to

be

more

representative

of

the

general

team

-selection

problem

than

the

widely

studied

path

auctions.

We

show

(Theorem

that

for

any

set

system,

if

there

is

a

cost

vector

for

which

TUmin

and

NTUmin

differ

by

a

factor

of

α,

there

is

another

cost

vector

that

separates

NTUmin

and

NTUmax

by

the

same

factor

and

vice

versa;

the

same

is

true

for

the

pairs

(NTUmin,

NTUmax)

and

(NTUmax,

TUmax).

This

symmetry

is

quite

surprising,

since,

e.g.,

TUmin

and

NTUmax

are

obtained

from

NTUmin

by

two

very

different

transformations.

This

observation

suggests

that

the

four

payment

bounds

should

be

studied

in

a

unified

framework;

moreover,

it

leads

us

to

believe

that

the

bootstrapping

technique

of

Theorem

may

have

other

applications.

We

evaluate

the

payment

bounds

introduced

here

with

respect

to

a

checklist

of

desirable

features.

In

particular,

we

note

that

the

payment

bound

ν

NTUmin

of

exhibits

some

counterintuitive

properties,

such

as

nonmonotonicity

with

respect

to

adding

a

new

feasible

set

(Proposition

and

is

NP-hard

to

compute

(Theorem

while

some

of

the

other

payment

bounds

do

not

suffer

from

these

problems.

This

can

be

seen

as

an

argument

in

favour

of

using

weaker

but

efficiently

computable

bounds

NTUmax

and

TUmax.

Related

work

Vertex-cover

auctions

have

been

studied

in

the

past

by

Talwar

and

Calinescu

Both

of

these

papers

are

based

on

the

definition

of

frugality

ratio

used

in

as

mentioned

before,

this

means

that

their

results

only

apply

to

bipartite

graphs.

Talwar

shows

that

the

frugality

ratio

of

VCG

is

at

most

Δ.

However,

since

finding

the

cheapest

vertex

cover

is

an

NP-hard

problem,

the

VCG

mechanism

is

computationally

infeasible.

The

first

(and,

to

the

best

of

our

knowledge,

only)

paper

to

investigate

polynomial-time

truthful

mechanisms

for

vertex

cover

is

This

paper

studies

an

auction

that

is

based

on

the

greedy

allocation

algorithm,

which

has

an

approximation

ratio

of

log

n.

While

the

main

focus

of

is

the

more

general

set

cover

problem,

the

results

of

imply

a

frugality

ratio

of

for

vertex

cover.

Our

results

improve

on

those

of

as

our

mechanism

is

polynomial-time

computable,

as

well

as

on

those

of

as

our

mechanism

has

a

better

approximation

ratio,

and

we

prove

a

stronger

bound

on

the

frugality

ratio;

moreover,

this

bound

also

applies

to

the

mechanism

of

PRELIMINARIES

In

most

of

this

paper,

we

discuss

auctions

for

set

systems.

A

set

system

is

a

pair

(E,

F),

where

E

is

the

ground

set,

|E|

n,

and

F

is

a

collection

of

feasible

sets,

which

are

subsets

of

E.

Two

particular

types

of

set

systems

are

of

interest

to

us

shortest

path

systems,

in

which

the

ground

set

consists

of

all

edges

of

a

network,

and

the

feasible

sets

are

paths

between

two

specified

vertices

s

and

t,

and

vertex

cover

systems,

in

which

the

elements

of

the

ground

set

are

the

vertices

of

a

graph,

and

the

feasible

sets

are

vertex

covers

of

this

graph.

In

set

system

auctions,

each

element

e

of

the

ground

set

is

owned

by

an

independent

agent

and

has

an

associated

non-negative

cost

ce.

The

goal

of

the

centre

is

to

select

(purchase)

a

feasible

set.

Each

element

e

in

the

selected

set

incurs

a

cost

of

ce.

The

elements

that

are

not

selected

incur

no

costs.

The

auction

proceeds

as

follows:

all

elements

of

the

ground

set

make

their

bids,

the

centre

selects

a

feasible

set

based

on

the

bids

and

makes

payments

to

the

agents.

Formally,

an

auction

is

defined

by

an

allocation

rule

A

Rn

F

and

a

payment

rule

P

Rn

Rn

The

allocation

rule

takes

as

input

a

vector

of

bids

and

decides

which

of

the

sets

in

F

should

be

selected.

The

payment

rule

also

takes

as

input

a

vector

of

bids

and

decides

how

much

to

pay

to

each

agent.

The

standard

requirements

are

individual

rationality,

i.e.,

the

payment

to

each

agent

should

be

at

least

as

high

as

his

incurred

cost

for

agents

not

in

the

selected

set

and

ce

for

agents

in

the

selected

set)

and

incentive

compatibility,

or

truthfulness,

i.e.,

each

agent"s

dominant

strategy

is

to

bid

his

true

cost.

An

allocation

rule

is

monotone

if

an

agent

cannot

increase

his

chance

of

getting

selected

by

raising

his

bid.

Formally,

for

any

bid

vector

b

and

any

e

E,

if

e

A(b)

then

e

be,

bn)

for

any

be

be.

Given

a

monotone

allocation

rule

A

and

a

bid

vector

b,

the

threshold

bid

te

of

an

agent

e

A(b)

is

the

highest

bid

of

this

agent

that

still

wins

the

auction,

given

that

the

bids

of

other

participants

remain

the

same.

Formally,

te

sup{be

R

e

be,

bn)}.

It

is

well

known

(see,

e.g.

that

any

auction

that

has

a

monotone

allocation

rule

and

pays

each

agent

his

threshold

bid

is

truthful;

conversely,

any

truthful

auction

has

a

monotone

allocation

rule.

The

VCG

mechanism

is

a

truthful

mechanism

that

maximises

the

social

welfare

and

pays

to

the

losing

agents.

For

set

system

auctions,

this

simply

means

picking

a

cheapest

feasible

set,

paying

each

agent

in

the

selected

set

his

threshold

bid,

and

paying

to

all

other

agents.

Note,

however,

that

the

VCG

mechanism

may

be

difficult

to

implement,

since

finding

a

cheapest

feasible

set

may

be

intractable.

If

U

is

a

set

of

agents,

c(U)

denotes

P

w∈U

cw.

Similarly,

b(U)

denotes

P

w∈U

bw.

FRUGALITY

RATIOS

We

start

by

reproducing

the

definition

of

the

quantity

ν

from

Definition

Let

(E,

F)

be

a

set

system

and

let

S

be

a

cheapest

feasible

set

with

respect

to

the

true

costs

ce.

Then

ν(c,

S)

is

the

solution

to

the

following

optimisation

problem.

Minimise

B

P

e∈S

be

subject

to

be

ce

for

all

e

E

P

e∈S\T

be

P

e∈T

\S

ce

for

all

T

F

for

every

e

S,

there

is

a

Te

F

such

that

e

Te

andP

e

∈S\Te

be

P

e

∈Te\S

ce

The

bound

ν(c,

S)

can

be

seen

as

an

outcome

of

a

two-stage

process,

where

first

each

agent

e

S

makes

a

bid

be

stating

how

much

it

wants

to

be

paid,

and

then

the

centre

decides

whether

to

accept

these

bids.

The

behaviour

of

both

parties

is

affected

by

the

following

considerations.

From

the

centre"s

point

of

view,

the

set

S

must

remain

the

most

attractive

choice,

i.e.,

it

must

be

among

the

cheapest

feasible

sets

under

the

new

costs

ce

ce

for

e

S,

ce

be

for

e

S

(condition

The

reason

for

that

is

that

if

is

violated

for

some

set

T,

the

centre

would

prefer

T

to

S.

On

the

other

hand,

no

agent

would

agree

to

a

payment

that

does

not

cover

his

costs

(condition

and

moreover,

each

agent

tries

to

maximise

his

profit

by

bidding

as

high

as

possible,

i.e.,

of

the

agents

can

increase

his

bid

without

violating

condition

(condition

The

centre

wants

to

minimise

the

total

payout,

so

ν(c,

S)

corresponds

to

the

best

possible

outcome

from

the

centre"s

point

of

view.

This

definition

captures

many

important

aspects

of

our

intuition

about

‘fair"

payments.

However,

it

can

be

modified

in

two

ways,

both

of

which

are

still

quite

natural,

but

result

in

different

payment

bounds.

First,

we

can

consider

the

worst

rather

than

the

best

possible

outcome

for

the

centre.

That

is,

we

can

consider

the

maximum

total

payment

that

the

agents

can

extract

by

jointly

selecting

their

bids

subject

to

and

Such

a

bound

corresponds

to

maximising

B

subject

to

and

rather

than

minimising

it.

If

it

is

the

agents

who

make

the

original

bids

(rather

than

the

centre),

this

kind

of

bidding

behaviour

is

plausible.

On

the

other

hand,

in

a

game

in

which

the

centre

proposes

payments

to

the

agents

in

S

and

the

agents

accept

them

as

long

as

and

are

satisfied,

we

would

be

likely

to

observe

a

total

payment

of

ν(c,

S).

Hence,

the

difference

between

these

two

definitions

can

be

seen

as

the

price

of

initiative.

Second,

the

agents

may

be

able

to

make

payments

to

each

other.

In

this

case,

if

they

can

extract

more

money

from

the

centre

by

agreeing

on

a

vector

of

bids

that

violates

individual

rationality

(i.e.,

condition

for

some

bidders,

they

might

be

willing

to

do

so,

as

the

agents

who

are

paid

below

their

costs

will

be

compensated

by

other

members

of

the

group.

The

bids

must

still

be

realistic,

i.e.,

they

have

to

satisfy

be

The

resulting

change

in

payments

can

be

seen

as

the

price

of

co-operation

and

corresponds

to

replacing

condition

with

the

following

weaker

condition

be

for

all

e

E.

By

considering

all

possible

combinations

of

these

modifications,

we

obtain

four

different

payment

bounds,

namely

TUmin(c,

S),

which

is

the

solution

to

the

optimisation

problem

Minimise

B

subject

to

and

TUmax(c,

S),

which

is

the

solution

to

the

optimisation

problem

Maximise

B

subject

to

and

NTUmin(c,

S),

which

is

the

solution

to

the

optimisation

problem

Minimise

B

subject

to

and

NTUmax(c,

S),

which

is

the

solution

to

the

optimisation

problem

Maximise

B

subject

to

The

abbreviations

TU

and

NTU

correspond,

respectively,

to

transferable

utility

and

non-transferable

utility,

i.e.,

the

agents"

ability/inability

to

make

payments

to

each

other.

For

concreteness,

we

will

take

TUmin(c)

to

be

TUmin(c,

S)

where

S

is

the

lexicographically

least

amongst

the

cheapest

feasible

sets.

We

define

TUmax(c),

NTUmin(c),

NTUmax(c)

and

ν(c)

similarly,

though

we

will

see

in

Section

that,

in

fact,

NTUmin(c,

S)

and

NTUmax(c,

S)

are

independent

of

the

choice

of

S.

Note

that

the

quantity

ν(c)

from

is

NTUmin(c).

The

second

modification

(transferable

utility)

is

more

intuitively

appealing

in

the

context

of

the

maximisation

problem,

as

both

assume

some

degree

of

co-operation

between

the

agents.

While

the

second

modification

can

be

made

without

the

first,

the

resulting

payment

bound

TUmin(c,

S)

is

too

strong

to

be

a

realistic

benchmark,

at

least

for

general

set

systems.

In

particular,

it

can

be

smaller

than

the

total

cost

of

the

cheapest

feasible

set

S

(see

Section

Nevertheless,

we

provide

the

definition

as

well

as

some

results

about

TUmin(c,

S)

in

the

paper,

both

for

completeness

and

because

we

believe

that

it

may

help

to

understand

which

properties

of

the

payment

bounds

are

important

for

our

proofs.

Another

possibility

would

be

to

introduce

an

additional

constraint

P

e∈S

be

≥P

e∈S

ce

in

the

definition

of

TUmin(c,

S)

(note

that

this

condition

holds

automatically

for

TUmax(c,

S),

as

TUmax(c,

S)

NTUmax(c,

S));

however,

such

a

definition

would

have

no

direct

game-theoretic

interpretation,

and

some

of

our

results

(in

particular,

the

ones

in

Section

would

no

longer

be

true.

REMARK

For

the

payment

bounds

that

are

derived

from

maximisation

problems,

(i.e.,

TUmax(c,

S)

and

NTUmax(c,

S)),

constraints

of

type

are

redundant

and

can

be

dropped.

Hence,

TUmax(c,

S)

and

NTUmax(c,

S)

are

solutions

to

linear

programs,

and

therefore

can

be

computed

in

polynomial

time

as

long

as

we

have

a

separation

oracle

for

constraints

in

In

contrast,

NTUmin(c,

S)

can

be

NP-hard

to

compute

even

if

the

size

of

F

is

polynomial

(see

Section

The

first

and

third

inequalities

in

the

following

observation

follow

from

the

fact

that

condition

is

strictly

weaker

than

condition

PROPOSITION

TUmin(c,

S)

NTUmin(c,

S)

NTUmax(c,

S)

TUmax(c,

S).

Let

M

be

a

truthful

mechanism

for

(E,

F).

Let

pM(c)

denote

the

total

payments

of

M

when

the

actual

costs

are

c.

A

frugality

ratio

of

M

with

respect

to

a

payment

bound

is

the

ratio

between

the

payment

of

M

and

this

payment

bound.

In

particular,

φTUmin(M)

sup

c

pM(c)/TUmin(c),

φTUmax(M)

sup

c

pM(c)/TUmax(c),

φNTUmin(M)

sup

c

pM(c)/NTUmin(c),

φNTUmax(M)

sup

c

pM(c)/NTUmax(c).

We

conclude

this

section

by

showing

that

there

exist

set

systems

and

respective

cost

vectors

for

which

all

four

payment

bounds

are

different.

In

the

next

section,

we

quantify

this

difference,

both

for

general

set

systems,

and

for

specific

types

of

set

systems,

such

as

path

auctions

or

vertex

cover

auctions.

EXAMPLE

Consider

the

shortest-path

auction

on

the

graph

of

Figure

The

cheapest

feasible

sets

are

all

paths

from

A

to

D.

It

can

be

verified,

using

the

reasoning

of

Propositions

and

below,

that

for

the

cost

vector

cAB

cCD

cBC

cAC

cBD

we

have

TUmax(c)

(with

bAB

bCD

bBC

NTUmax(c)

(with

bAB

bCD

bBC

NTUmin(c)

(with

bAB

bCD

bBC

TUmin(c)

(with

bAB

bCD

bBC

COMPARING

PAYMENT

BOUNDS

Path

auctions

We

start

by

showing

that

for

path

auctions

any

two

consecutive

payment

bounds

can

differ

by

at

least

a

factor

of

PROPOSITION

There

is

an

instance

of

the

shortest-path

problem

for

which

we

have

NTUmax(c)/NTUmin(c)

PROOF.

This

construction

is

due

to

David

Kempe

Consider

the

graph

of

Figure

with

the

edge

costs

cAB

cBC

cCD

cAC

cBD

Under

these

costs,

ABCD

is

the

cheapest

path.

The

inequalities

in

are

bAB

bBC

cAC

bBC

bCD

cBD

By

condition

both

of

these

inequalities

must

be

tight

(the

former

one

is

the

only

inequality

involving

bAB,

and

the

latter

one

is

the

only

inequality

involving

bCD).

The

inequalities

in

are

bAB

bBC

bCD

Now,

if

the

goal

is

to

maximise

bAB

bBC

bCD,

the

best

choice

is

bAB

bCD

bBC

so

NTUmax(c)

On

the

other

hand,

if

the

goal

is

to

minimise

bAB

bBC

bCD,

one

should

set

bAB

bCD

bBC

so

NTUmin(c)

PROPOSITION

There

is

an

instance

of

the

shortest-path

problem

for

which

we

have

TUmax(c)/NTUmax(c)

PROOF.

Again,

consider

the

graph

of

Figure

Let

the

edge

costs

be

cAB

cCD

cBC

cAC

cBD

ABCD

is

the

lexicographically-least

cheapest

path,

so

we

can

assume

that

S

{AB,

BC,

CD}.

The

inequalities

in

are

the

same

as

in

the

previous

example,

and

by

the

same

argument

both

of

them

are,

in

fact,

equalities.

The

inequalities

in

are

bAB

bBC

bCD

Our

goal

is

to

maximise

bAB

bBC

bCD.

If

we

have

to

respect

the

inequalities

in

we

have

to

set

bAB

bCD

bBC

so

NTUmax(c)

Otherwise,

we

can

set

bAB

bCD

bBC

so

TUmax(c)

PROPOSITION

There

is

an

instance

of

the

shortest-path

problem

for

which

we

have

NTUmin(c)/TUmin(c)

PROOF.

This

construction

is

also

based

on

the

graph

of

Figure

The

edge

costs

are

cAB

cCD

cBC

cAC

cBD

ABCD

is

the

lexicographically

least

cheapest

path,

so

we

can

assume

that

S

{AB,

BC,

CD}.

Again,

the

inequalities

in

are

the

same,

and

both

are,

in

fact,

equalities.

The

inequalities

in

are

bAB

bBC

bCD

Our

goal

is

to

minimise

bAB

bBC

+bCD.

If

we

have

to

respect

the

inequalities

in

we

have

to

set

bAB

bCD

bBC

so

NTUmin(c)

Otherwise,

we

can

set

bAB

bCD

bBC

so

TUmin(c)

In

Section

(Theorem

we

show

that

the

separation

results

in

Propositions

and

are

optimal.

Connections

between

separation

results

The

separation

results

for

path

auctions

are

obtained

on

the

same

graph

using

very

similar

cost

vectors.

It

turns

out

that

this

is

not

coincidental.

Namely,

we

can

prove

the

following

theorem.

THEOREM

For

any

set

system

(E,

F),

and

any

feasible

set

S,

max

c

TUmax(c,

S)

NTUmax(c,

S)

max

c

NTUmax(c,

S)

NTUmin(c,

S)

max

c

NTUmax(c,

S)

NTUmin(c,

S)

max

c

NTUmin(c,

S)

TUmin(c,

S)

where

the

maximum

is

over

all

cost

vectors

c

for

which

S

is

a

cheapest

feasible

set.

The

proof

of

the

theorem

follows

directly

from

the

four

lemmas

proved

below;

more

precisely,

the

first

equality

in

Theorem

is

obtained

by

combining

Lemmas

and

and

the

second

equality

is

obtained

by

combining

Lemmas

and

We

prove

Lemma

here;

the

proofs

of

Lemmas

are

similar

and

can

be

found

in

the

full

version

of

this

paper

LEMMA

Suppose

that

c

is

a

cost

vector

for

(E,

F)

such

that

S

is

a

cheapest

feasible

set

and

TUmax(c,

S)/NTUmax(c,

S)

α.

Then

there

is

a

cost

vector

c

such

that

S

is

a

cheapest

feasible

set

and

NTUmax(c

S)/NTUmin(c

S)

α.

PROOF.

Suppose

that

TUmax(c,

S)

X

and

NTUmax(c,

S)

Y

where

X/Y

α.

Assume

without

loss

of

generality

that

S

consists

of

elements

k,

and

let

k)

and

k)

be

the

bid

vectors

that

correspond

to

TUmax(c,

S)

and

NTUmax(c,

S),

respectively.

Construct

the

cost

vector

c

by

setting

ci

ci

for

i

S,

ci

min{ci,

i

for

i

S.

Clearly,

S

is

a

cheapest

set

under

c

Moreover,

as

the

costs

of

elements

outside

of

S

remained

the

same,

the

right-hand

sides

of

all

constraints

in

did

not

change,

so

any

bid

vector

that

satisfies

and

with

respect

to

c,

also

satisfies

them

with

respect

to

c

We

will

construct

two

bid

vectors

and

that

satisfy

conditions

and

for

the

cost

vector

c

and

X

X

X

X

X

X

X

Figure

Graph

that

separates

payment

bounds

for

vertex

cover,

n

have

P

i∈S

i

X,

P

i∈S

i

Y

As

NTUmax(c

S)

X

and

NTUmin(c

S)

Y

this

implies

the

lemma.

We

can

set

i

i

this

bid

vector

satisfies

conditions

and

since

does,

and

we

have

i

min{ci,

i

ci,

which

means

that

satisfies

condition

Furthermore,

we

can

set

i

i

Again,

satisfies

conditions

and

since

does,

and

since

satisfies

condition

we

have

i

ci

ci,

which

means

that

satisfies

condition

LEMMA

Suppose

c

is

a

cost

vector

for

(E,

F)

such

that

S

is

a

cheapest

feasible

set

and

NTUmax(c,

S)/NTUmin(c,

S)

α.

Then

there

is

a

cost

vector

c

such

that

S

is

a

cheapest

feasible

set

and

TUmax(c

S)/NTUmax(c

S)

α.

LEMMA

Suppose

that

c

is

a

cost

vector

for

(E,

F)

such

that

S

is

a

cheapest

feasible

set

and

NTUmax(c,

S)/NTUmin(c,

S)

α.

Then

there

is

a

cost

vector

c

such

that

S

is

a

cheapest

feasible

set

and

NTUmin(c

S)/TUmin(c

S)

α.

LEMMA

Suppose

that

c

is

a

cost

vector

for

(E,

F)

such

that

S

is

a

cheapest

feasible

set

and

NTUmin(c,

S)/TUmin(c,

S)

α.

Then

there

is

a

cost

vector

c

such

that

S

is

a

cheapest

feasible

set

and

NTUmax(c

S)/NTUmin(c

S)

α.

Vertex-cover

auctions

In

contrast

to

the

case

of

path

auctions,

for

vertex-cover

auctions

the

gap

between

NTUmin(c)

and

NTUmax(c)

(and

hence

between

NTUmax(c)

and

TUmax(c),

and

between

TUmin(c)

and

NTUmin(c))

can

be

proportional

to

the

size

of

the

graph.

PROPOSITION

For

any

n

there

is

a

an

n-vertex

graph

and

a

cost

vector

c

for

which

TUmax(c)/NTUmax(c)

n

PROOF.

The

underlying

graph

consists

of

an

(n

on

the

vertices

and

an

extra

vertex

adjacent

to

The

costs

are

We

can

assume

that

S

(this

is

the

lexicographically

first

vertex

cover

of

cost

For

this

set

system,

the

constraints

in

are

bXi

for

i

n

Clearly,

we

can

satisfy

conditions

and

by

setting

bXi

for

i

n

Hence,

TUmax(c)

n

For

NTUmax(c),

there

is

an

additional

constraint

so

the

best

we

can

do

is

to

set

bXi

for

i

n

which

implies

NTUmax(c)

Combining

Proposition

with

Lemmas

and

we

derive

the

following

corollaries.

COROLLARY

For

any

n

we

can

construct

an

instance

of

the

vertex

cover

problem

on

a

graph

of

size

n

that

satisfies

NTUmax(c)/NTUmin(c)

n

COROLLARY

For

any

n

we

can

construct

an

instance

of

the

vertex

cover

problem

on

a

graph

of

size

n

that

satisfies

NTUmin(c)/TUmin(c)

n

ij

P

P

P

yijixix

j

i

y

y

i

i

j

e

e

P

P

Figure

Proof

of

Theorem

constraints

for

ˆPij

and

do

not

overlap

Upper

bounds

It

turns

out

that

the

lower

bound

proved

in

the

previous

subsection

is

almost

tight.

More

precisely,

the

following

theorem

shows

that

no

two

payment

bounds

can

differ

by

more

than

a

factor

of

n;

moreover,

this

is

the

case

not

just

for

the

vertex

cover

problem,

but

for

general

set

systems.

We

bound

the

gap

between

TUmax(c)

and

TUmin(c).

Since

TUmin(c)

NTUmin(c)

NTUmax(c)

TUmax(c),

this

bound

applies

to

any

pair

of

payment

bounds.

THEOREM

For

any

set

system

(E,

F)

and

any

cost

vector

c,

we

have

TUmax(c)/TUmin(c)

n.

PROOF.

Assume

wlog

that

the

winning

set

S

consists

of

elements

k.

Let

ck

be

the

true

costs

of

elements

in

S,

let

bk

be

their

bids

that

correspond

to

TUmin(c),

and

let

bk

be

their

bids

that

correspond

to

TUmax(c).

Consider

the

conditions

and

for

S.

One

can

pick

a

subset

L

of

at

most

k

inequalities

in

so

that

for

each

i

k

there

is

at

least

one

inequality

in

L

that

is

tight

for

bi.

Suppose

that

the

jth

inequality

in

L

is

of

the

form

bit

c(Tj

S).

For

bi,

all

inequalities

in

L

are,

in

fact,

equalities.

Hence,

by

adding

up

all

of

them

we

obtain

k

P

bi

P

c(Tj

S).

On

the

other

hand,

all

these

inequalities

appear

in

condition

so

they

must

hold

for

bi

i.e.,

P

bi

P

c(Tj

S).

Combining

these

two

inequalities,

we

obtain

nTUmin(c)

kTUmin(c)

TUmax(c).

REMARK

The

final

line

of

the

proof

of

Theorem

shows

that,

in

fact,

the

upper

bound

on

TUmax(c)/TUmin(c)

can

be

strengthened

to

the

size

of

the

winning

set,

k.

Note

that

in

Proposition

as

well

as

in

Corollaries

and

k

so

these

results

do

not

contradict

each

other.

For

path

auctions,

this

upper

bound

can

be

improved

to

matching

the

lower

bounds

of

Section

THEOREM

For

any

instance

of

the

shortest

path

problem,

TUmax(c)

TUmin(c).

PROOF.

Given

a

network

(G,

s,

t),

assume

without

loss

of

generality

that

the

lexicographically-least

cheapest

s-t

path,

P,

in

G

is

ek},

where

(s,

ek

t).

Let

ck

be

the

true

costs

of

ek,

and

let

b

bk)

and

b

bk

be

bid

vectors

that

correspond

to

TUmin(c)

and

TUmax(c),

respectively.

For

any

i

k,

there

is

a

constraint

in

that

is

tight

for

bi

with

respect

to

the

bid

vector

b

i.e.,

an

s-t

path

Pi

that

avoids

ei

and

satisfies

b

(P

\Pi)

c(Pi

\P).

We

can

assume

without

loss

of

generality

that

Pi

coincides

with

P

up

to

some

vertex

xi,

then

deviates

from

P

to

avoid

ei,

and

finally

returns

to

P

at

a

vertex

yi

and

coincides

with

P

from

then

on

(clearly,

it

might

happen

that

s

xi

or

t

yi).

Indeed,

if

Pi

deviates

from

P

more

than

once,

one

of

these

deviations

is

not

necessary

to

avoid

ei

and

can

be

replaced

with

the

respective

segment

of

P

without

increasing

the

cost

of

Pi.

Among

all

paths

of

this

form,

let

ˆPi

be

the

one

with

the

largest

value

of

yi,

i.e.,

the

rightmost

one.

This

path

corresponds

to

an

inequality

Ii

of

the

form

byi

c(

ˆPi

P).

As

in

the

proof

of

Theorem

we

construct

a

set

of

tight

constraints

L

such

that

every

variable

bi

appears

in

at

least

one

of

these

constraints;

however,

now

we

have

to

be

more

careful

about

the

choice

of

constraints

in

L.

We

construct

L

inductively

as

follows.

Start

by

setting

L

At

the

jth

step,

suppose

that

all

variables

up

to

(but

not

including)

bij

appear

in

at

least

one

inequality

in

L.

Add

Iij

to

L.

Note

that

for

any

j

we

have

yij

This

is

because

the

inequalities

added

to

L

during

the

first

j

steps

did

not

cover

See

Figure

Since

we

must

also

have

yij

otherwise,

would

not

be

the

rightmost

constraint

for

Therefore,

the

variables

in

and

Iij

do

not

overlap,

and

hence

no

bi

can

appear

in

more

than

two

inequalities

in

L.

Now

we

follow

the

argument

of

the

proof

of

Theorem

to

finish.

By

adding

up

all

of

the

(tight)

inequalities

in

L

for

bi

we

obtain

P

bi

P

c(

ˆPj

P).

On

the

other

hand,

all

these

inequalities

appear

in

condition

so

they

must

hold

for

bi

i.e.,

P

bi

P

c(

ˆPj

P),

so

TUmax(c)

TRUTHFUL

MECHANISMS

FOR

VERTEX

COVER

Recall

that

for

a

vertex-cover

auction

on

a

graph

G

(V,

E),

an

allocation

rule

is

an

algorithm

that

takes

as

input

a

bid

bv

for

each

vertex

and

returns

a

vertex

cover

ˆS

of

G.

As

explained

in

Section

we

can

combine

a

monotone

allocation

rule

with

threshold

payments

to

obtain

a

truthful

auction.

Two

natural

examples

of

monotone

allocation

rules

are

Aopt,

i.e.,

the

algorithm

that

finds

an

optimal

vertex

cover,

and

the

greedy

algorithm

AGR.

However,

Aopt

cannot

be

guaranteed

to

run

in

polynomial

time

unless

P

NP

and

AGR

has

approximation

ratio

of

log

n.

Another

approximation

algorithm

for

vertex

cover,

which

has

approximation

ratio

is

the

local

ratio

algorithm

ALR

This

algorithm

considers

the

edges

of

G

one

by

one.

Given

an

edge

e

(u,

v),

it

computes

min{bu,

bv}

and

sets

bu

bu

bv

bv

After

all

edges

have

been

processed,

ALR

returns

the

set

of

vertices

{v

bv

It

is

not

hard

to

check

that

if

the

order

in

which

the

edges

are

considered

is

independent

of

the

bids,

then

this

algorithm

is

monotone

as

well.

Hence,

we

can

use

it

to

construct

a

truthful

auction

that

is

guaranteed

to

select

a

vertex

cover

whose

cost

is

within

a

factor

of

from

the

optimal.

However,

while

the

quality

of

the

solution

produced

by

ALR

is

much

better

than

that

of

AGR,

we

still

need

to

show

that

its

total

payment

is

not

too

high.

In

the

next

subsection,

we

bound

the

frugality

ratio

of

ALR

(and,

more

generally,

all

algorithms

that

satisfy

the

condition

of

local

optimality,

defined

later)

by

where

Δ

is

the

maximum

degree

of

G.

We

then

prove

a

matching

lower

bound

showing

that

for

some

graphs

the

frugality

ratio

of

any

truthful

auction

is

at

least

Upper

bound

We

say

that

an

allocation

rule

is

locally

optimal

if

whenever

bv

>P

w∼v

bw,

the

vertex

v

is

not

chosen.

Note

that

for

any

such

rule

the

threshold

bid

of

v

satisfies

tv

P

w∼v

bw.

CLAIM

The

algorithms

Aopt,

AGR,

and

ALR

are

locally

optimal.

THEOREM

Any

vertex

cover

auction

M

that

has

a

locally

optimal

and

monotone

allocation

rule

and

pays

each

agent

his

threshold

bid

has

frugality

ratio

φNTUmin(M)

To

prove

Theorem

we

first

show

that

the

total

payment

of

any

locally

optimal

mechanism

does

not

exceed

Δc(V

We

then

demonstrate

that

NTUmin(c)

c(V

By

combining

these

two

results,

the

theorem

follows.

LEMMA

Consider

a

graph

G

(V,

E)

with

maximum

degree

Δ.

Let

M

be

a

vertex-cover

auction

on

G

that

satisfies

the

conditions

of

Theorem

Then

for

any

cost

vector

c,

the

total

payment

of

M

satisfies

pM(c)

Δc(V

PROOF.

First

note

that

any

such

auction

is

truthful,

so

we

can

assume

that

each

agent"s

bid

is

equal

to

his

cost.

Let

ˆS

be

the

vertex

cover

selected

by

M.

Then

by

local

optimality

pM(c)

X

v∈

ˆS

tv

X

v∈

ˆS

X

w∼v

cw

X

w∈V

Δcw

Δc(V

We

now

derive

a

lower

bound

on

TUmax(c);

while

not

essential

for

the

proof

of

Theorem

it

helps

us

build

the

intuition

necessary

for

that

proof.

LEMMA

For

a

vertex

cover

instance

G

(V,

E)

in

which

S

is

a

minimum

vertex

cover,

TUmax(c,

S)

c(V

S)

PROOF.

For

a

vertex

w

with

at

least

one

neighbour

in

S,

let

d(w)

denote

the

number

of

neighbours

that

w

has

in

S.

Consider

the

bid

vector

b

in

which,

for

each

v

S,

bv

P

w∼v,w∈S

cw

d(w)

Then

P

v∈S

bv

P

v∈S

P

w∼v,w∈S

cw/d(w)

P

w

/∈S

cw

c(V

S).

To

finish

we

want

to

show

that

b

is

feasible

in

the

sense

that

it

satisfies

Consider

a

vertex

cover

T,

and

extend

the

bid

vector

b

by

assigning

bv

cv

for

v

S.

Then

b(T)

c(T

\S)+b(S∩T)

c(T

\S)+

X

v∈S∩T

X

w∈S∩T

:w∼v

cw

d(w)

and

since

all

edges

between

S

T

and

S

go

to

S

T,

the

righthand-side

is

equal

to

c(T

\S)+

X

w∈S∩T

cw

c(T

\S)+c(S

∩T)

c(V

\S)

b(S).

Next,

we

prove

a

lower

bound

on

NTUmax(c,

S);

we

will

then

use

it

to

obtain

a

lower

bound

on

NTUmin(c).

LEMMA

For

a

vertex

cover

instance

G

(V,

E)

in

which

S

is

a

minimum

vertex

cover,

NTUmax(c,

S)

c(V

S)

PROOF.

If

c(S)

c(V

S),

by

condition

we

are

done.

Therefore,

for

the

rest

of

the

proof

we

assume

that

c(S)

c(V

S).

We

show

how

to

construct

a

bid

vector

(be)e∈S

that

satisfies

conditions

and

such

that

b(S)

c(V

S);

clearly,

this

implies

NTUmax(c,

S)

c(V

S).

Recall

that

a

network

flow

problem

is

described

by

a

directed

graph

Γ

(VΓ,

EΓ),

a

source

node

s

VΓ,

a

sink

node

t

VΓ,

and

a

vector

of

capacity

constraints

ae,

e

EΓ.

Consider

a

network

(VΓ,

EΓ)

such

that

VΓ

V

∪{s,

t},

EΓ

where

{(s,

v)

v

S},

{(v,

w)

v

S,

w

V

S,

(v,

w)

E},

{(w,

t)

w

V

S}.

Since

S

is

a

vertex

cover

for

G,

no

edge

of

E

can

have

both

of

its

endpoints

in

V

S,

and

by

construction,

contains

no

edges

with

both

endpoints

in

S.

Therefore,

the

graph

(V,

is

bipartite

with

parts

(S,

V

S).

Set

the

capacity

constraints

for

e

EΓ

as

follows:

a(s,v)

cv,

a(w,t)

cw,

a(v,w)

for

all

v

S,

w

V

S.

Recall

that

a

cut

is

a

partition

of

the

vertices

in

VΓ

into

two

sets

and

so

that

s

t

we

denote

such

a

cut

by

C

Abusing

notation,

we

write

e

(u,

v)

C

if

u

v

or

u

v

and

say

that

such

an

edge

e

(u,

v)

crosses

the

cut

C.

The

capacity

of

a

cut

C

is

computed

as

cap(C)

P

(v,w)∈C

a(v,w).

We

have

cap(s,

V

∪{t})

c(S),

cap({s}

V,

t)

c(V

S).

Let

Cmin

({s}

S

W

{t}

S

W

be

a

minimum

cut

in

Γ,

where

S

S

S,

W

W

V

S.

See

Figure

As

cap(Cmin)

cap(s,

V

{t})

c(S)

and

any

edge

in

has

infinite

capacity,

no

edge

(u,

v)

crosses

Cmin.

Consider

the

network

Γ

(VΓ

EΓ

where

VΓ

{s}

S

W

{t},

EΓ

{(u,

v)

EΓ

u,

v

VΓ

Clearly,

C

({s}

S

W

{t})

is

a

minimum

cut

in

Γ

(otherwise,

there

would

exist

a

smaller

cut

for

Γ).

As

cap(C

c(W

we

have

c(S

c(W

Now,

consider

the

network

Γ

(VΓ

EΓ

where

VΓ

{s}

S

W

{t},

EΓ

{(u,

v)

EΓ

u,

v

VΓ

Similarly,

C

({s},

S

W

{t})

is

a

minimum

cut

in

Γ

cap(C

c(S

As

the

size

of

a

maximum

flow

from

s

to

t

is

equal

to

the

capacity

of

a

minimum

cut

separating

s

and

t,

there

exists

a

flow

F

(fe)e∈EΓ

of

size

c(S

This

flow

has

to

saturate

all

edges

between

s

and

S

i.e.,

f(s,v)

cv

for

all

v

S

Now,

increase

the

capacities

of

all

edges

between

s

and

S

to

In

the

modified

network,

the

capacity

of

a

minimum

cut

(and

hence

the

size

of

a

maximum

flow)

is

c(W

and

a

maximum

flow

F

(fe)e∈EΓ

can

be

constructed

by

greedily

augmenting

F.

Set

bv

cv

for

all

v

S

bv

f(s,v)

for

all

v

S

As

F

is

constructed

by

augmenting

F,

we

have

bv

cv

for

all

v

S,

i.e.,

condition

is

satisfied.

Now,

let

us

check

that

no

vertex

cover

T

V

can

violate

condition

Set

T

S

T

S

T

W

T

W

our

goal

is

to

show

that

b(S

b(S

Consider

all

edges

(u,

v)

E

such

that

u

S

If

(u,

v)

then

v

(no

edge

in

can

cross

the

cut),

and

if

u,

v

S

then

v

Hence,

is

a

vertex

cover

for

G,

and

therefore

c(S

c(S)

c(S

c(S

Consequently,

c(S

b(S

Now,

consider

the

vertices

in

S

Any

edge

in

that

starts

in

one

of

these

vertices

has

to

end

in

(this

edge

has

to

be

covered

by

T,

and

it

cannot

go

across

the

cut).

Therefore,

the

total

flow

out

of

S

is

at

most

the

total

flow

out

of

i.e.,

b(S

Hence,

b(S

b(S

Finally,

we

derive

a

lower

bound

on

the

payment

bound

that

is

of

interest

to

us,

namely,

NTUmin(c).

LEMMA

For

a

vertex

cover

instance

G

(V,

E)

in

which

S

is

a

minimum

vertex

cover,

NTUmin(c,

S)

c(V

S)

PROOF.

Suppose

for

contradiction

that

c

is

a

cost

vector

with

minimum-cost

vertex

cover

S

and

NTUmin(c,

S)

c(V

\S).

Let

b

be

the

corresponding

bid

vector

and

let

c

be

a

new

cost

vector

with

cv

bv

for

v

S

and

cv

cv

for

v

S.

Condition

guarantees

that

S

is

an

optimal

solution

to

the

cost

vector

c

Now

compute

a

bid

vector

b

corresponding

to

NTUmax(c

S).

We

S"

W""

S""

W"

s

t

Figure

Proof

of

Lemma

Dashed

lines

correspond

to

edges

in

E

claim

that

bv

cv

for

any

v

S.

Indeed,

suppose

that

bv

cv

for

some

v

S

(bv

cv

for

v

S

by

construction).

As

b

satisfies

conditions

among

the

inequalities

in

there

is

one

that

is

tight

for

v

and

the

bid

vector

b.

That

is,

b(S

T)

c(T

S).

By

the

construction

of

c

c

(S

T)

c

(T

S).

Now

since

bw

cw

for

all

w

S,

bv

cv

implies

b

(S

\T)

c

(S

\T)

c

(T

\S).

But

this

violates

So

we

now

know

b

c

Hence,

we

have

NTUmax(c

S)

P

v∈S

bv

NTUmin(c,

S)

c(V

S),

giving

a

contradiction

to

the

fact

that

NTUmax(c

S)

c

(V

\S)

which

we

proved

in

Lemma

As

NTUmin(c,

S)

satisfies

condition

it

follows

that

we

have

NTUmin(c,

S)

c(S).

Together

will

Lemma

this

implies

NTUmin(c,

S)

max{c(V

S),

c(S)}

c(V

Combined

with

Lemma

this

completes

the

proof

of

Theorem

REMARK

As

NTUmin(c)

NTUmax(c)

TUmax(c),

our

bound

of

2Δ

extends

to

the

smaller

frugality

ratios

that

we

consider,

i.e.,

φNTUmax(M)

and

φTUmax(M).

It

is

not

clear

whether

it

extends

to

the

larger

frugality

ratio

φTUmin(M).

However,

the

frugality

ratio

φTUmin(M)

is

not

realistic

because

the

payment

bound

TUmin(c)

is

inappropriately

low

we

show

in

Section

that

TUmin(c)

can

be

significantly

smaller

than

the

total

cost

of

a

cheapest

vertex

cover.

Extensions

We

can

also

apply

our

results

to

monotone

vertex-cover

algorithms

that

do

not

necessarily

output

locally-optimal

solutions.

To

do

so,

we

simply

take

the

vertex

cover

produced

by

any

such

algorithm

and

transform

it

into

a

locally-optimal

one,

considering

the

vertices

in

lexicographic

order

and

replacing

a

vertex

v

with

its

neighbours

whenever

bv

P

u∼v

bu.

Note

that

if

a

vertex

u

has

been

added

to

the

vertex

cover

during

this

process,

it

means

that

it

has

a

neighbour

whose

bid

is

higher

than

bu,

so

after

one

pass

all

vertices

in

the

vertex

cover

satisfy

bv

P

u∼v

bu.

This

procedure

is

monotone

in

bids,

and

it

can

only

decrease

the

cost

of

the

vertex

cover.

Therefore,

using

it

on

top

of

a

monotone

allocation

rule

with

imation

ratio

α,

we

obtain

a

monotone

locally-optimal

allocation

rule

with

approximation

ratio

α.

Combining

it

with

threshold

payments,

we

get

an

auction

with

φNTUmin

Since

any

truthful

auction

has

a

monotone

allocation

rule,

this

procedure

transforms

any

truthful

mechanism

for

the

vertex-cover

problem

into

a

frugal

one

while

preserving

the

approximation

ratio.

Lower

bound

In

this

subsection,

we

prove

that

the

upper

bound

of

Theorem

is

essentially

optimal.

Our

proof

uses

the

techniques

of

where

the

authors

prove

a

similar

result

for

shortest-path

auctions.

THEOREM

For

any

Δ

and

any

n,

there

exist

a

graph

G

of

maximum

degree

Δ

and

size

N

n

such

that

for

any

truthful

mechanism

M

on

G

we

have

φNTUmin(M)

PROOF.

Given

n

and

Δ,

set

k

Let

G

be

the

graph

that

consists

of

k

blocks

Bk

of

size

2Δ

each,

where

each

Bi

is

a

complete

bipartite

graph

with

parts

Li

and

Ri,

|Li|

|Ri|

Δ.

We

will

consider

two

families

of

cost

vectors

for

G.

Under

a

cost

vector

x

X,

each

block

Bi

has

one

vertex

of

cost

all

other

vertices

cost

Under

a

cost

vector

y

Y

there

is

one

block

that

has

two

vertices

of

cost

one

in

each

part,

all

other

blocks

have

one

vertex

of

cost

and

all

other

vertices

cost

Clearly,

|X|

|Y

Δ2

We

will

now

construct

a

bipartite

graph

W

with

the

vertex

set

X

Y

as

follows.

Consider

a

cost

vector

y

Y

that

has

two

vertices

of

cost

in

Bi;

let

these

vertices

be

vl

Li

and

vr

Ri.

By

changing

the

cost

of

either

of

these

vertices

to

we

obtain

a

cost

vector

in

X.

Let

xl

and

xr

be

the

cost

vectors

obtained

by

changing

the

cost

of

vl

and

vr,

respectively.

The

vertex

cover

chosen

by

M(y)

must

either

contain

all

vertices

in

Li

or

it

must

contain

all

vertices

in

Ri.

In

the

former

case,

we

put

in

W

an

edge

from

y

to

xl

and

in

the

latter

case

we

put

in

W

an

edge

from

y

to

xr

(if

the

vertex

cover

includes

all

of

Bi,

W

contains

both

of

these

edges).

The

graph

W

has

at

least

Δ2

edges,

so

there

must

exist

an

x

X

of

degree

at

least

Let

be

the

other

endpoints

of

the

edges

incident

to

x,

and

for

each

i

let

vi

be

the

vertex

whose

cost

is

different

under

x

and

yi;

note

that

all

vi

are

distinct.

It

is

not

hard

to

see

that

NTUmin(x)

k:

the

cheapest

vertex

cover

contains

the

part

of

each

block,

and

we

can

satisfy

conditions

by

letting

one

of

the

vertices

in

the

part

of

each

block

to

bid

while

all

other

the

vertices

in

the

cheapest

set

bid

On

the

other

hand,

by

monotonicity

of

M

we

have

vi

M(x)

for

i

(vi

is

in

the

winning

set

under

yi,

and

x

is

obtained

from

yi

by

decreasing

the

cost

of

vi),

and

moreover,

the

threshold

bid

of

each

vi

is

at

least

so

the

total

payment

of

M

on

x

is

at

least

Hence,

φNTUmin(M)

M(x)/NTUmin(x)

REMARK

The

lower

bound

of

Theorem

can

be

generalised

to

randomised

mechanisms,

where

a

randomised

mechanism

is

considered

to

be

truthful

if

it

can

be

represented

as

a

probability

distribution

over

truthful

mechanisms.

In

this

case,

instead

of

choosing

the

vertex

x

X

with

the

highest

degree,

we

put

both

(y,

xl)

and

(y,

xr)

into

W

label

each

edge

with

the

probability

that

the

respective

part

of

the

block

is

chosen,

and

pick

x

X

with

the

highest

weighted

degree.

The

argument

can

be

further

extended

to

a

more

permissive

definition

of

truthfulness

for

randomised

mechanisms,

but

this

discussion

is

beyond

the

scope

of

this

paper.

PROPERTIES

OF

PAYMENT

BOUNDS

In

this

section

we

consider

several

desirable

properties

of

payment

bounds

and

evaluate

the

four

payment

bounds

proposed

in

this

paper

with

respect

to

them.

The

particular

properties

that

we

are

interested

in

are

independence

of

the

choice

of

S

(Section

monotonicity

(Section

computational

hardness

(Section

and

the

relationship

with

other

reasonable

bounds,

such

as

the

total

cost

of

the

cheapest

set

(Section

or

the

total

VCG

payment

(Section

Comparison

with

total

cost

Our

first

requirement

is

that

a

payment

bound

should

not

be

less

than

the

total

cost

of

the

selected

set.

Payment

bounds

are

used

to

evaluate

the

performance

of

set-system

auctions.

The

latter

have

to

satisfy

individual

rationality,

i.e.,

the

payment

to

each

agent

must

be

at

least

as

large

as

his

incurred

costs;

it

is

only

reasonable

to

require

the

payment

bound

to

satisfy

the

same

requirement.

Clearly,

NTUmax(c)

and

NTUmin(c)

satisfy

this

requirement

due

to

condition

and

so

does

TUmax(c),

since

TUmax(c)

NTUmax(c).

However,

TUmin(c)

fails

this

test.

The

example

of

Proposition

shows

that

for

path

auctions,

TUmin(c)

can

be

smaller

than

the

total

cost

by

a

factor

of

Moreover,

there

are

set

systems

and

cost

vectors

for

which

TUmin(c)

is

smaller

than

the

cost

of

the

cheapest

set

S

by

a

factor

of

Ω(n).

Consider,

for

example,

the

vertex-cover

auction

for

the

graph

of

Proposition

with

the

costs

The

cost

of

a

cheapest

vertex

cover

is

n

and

the

lexicographically

first

vertex

cover

of

cost

is

The

constraints

in

are

bXi

Clearly,

we

can

satisfy

conditions

and

by

setting

which

means

that

TUmin(c)

This

observation

suggests

that

the

payment

bound

TUmin(c)

is

too

strong

to

be

realistic,

since

it

can

be

substantially

lower

than

the

cost

of

the

cheapest

feasible

set.

Nevertheless,

some

of

the

positive

results

that

were

proved

in

for

NTUmin(c)

go

through

for

TUmin(c)

as

well.

In

particular,

one

can

show

that

if

the

feasible

sets

are

the

bases

of

a

monopolyfree

matroid,

then

φTUmin(VCG)

To

show

that

φTUmin(VCG)

is

at

most

one

must

prove

that

the

VCG

payment

is

at

most

TUmin(c).

This

is

shown

for

NTUmin(c)

in

the

first

paragraph

of

the

proof

of

Theorem

in

Their

argument

does

not

use

condition

at

all,

so

it

also

applies

to

TUmin(c).

On

the

other

hand,

φTUmin(VCG)

since

φTUmin(VCG)

φNTUmin(VCG)

and

φNTUmin(VCG)

by

Proposition

of

(and

also

by

Proposition

below).

Comparison

with

VCG

payments

Another

measure

of

suitability

for

payment

bounds

is

that

they

should

not

result

in

frugality

ratios

that

are

less

then

for

wellknown

truthful

mechanisms.

If

this

is

indeed

the

case,

the

payment

bound

may

be

too

weak,

as

it

becomes

too

easy

to

design

mechanisms

that

perform

well

with

respect

to

it.

It

particular,

a

reasonable

requirement

is

that

a

payment

bound

should

not

exceed

the

total

payment

of

the

classical

VCG

mechanism.

The

following

proposition

shows

that

NTUmax(c),

and

therefore

also

NTUmin(c)

and

TUmin(c),

do

not

exceed

the

VCG

payment

pVCG(c).

The

proof

essentially

follows

the

argument

of

Proposition

of

and

can

be

found

in

the

full

version

of

this

paper

PROPOSITION

φNTUmax(VCG)

Proposition

shows

that

of

the

payment

bounds

TUmin(c),

NTUmin(c)

and

NTUmax(c)

exceeds

the

payment

of

VCG.

However,

the

payment

bound

TUmax(c)

can

be

larger

that

the

total

VCG

payment.

In

particular,

for

the

instance

in

Proposition

the

VCG

payment

is

smaller

than

TUmax(c)

by

a

factor

of

n

We

have

already

seen

that

TUmax(c)

n

On

the

other

hand,

under

VCG,

the

threshold

bid

of

any

Xi,

i

n

is

if

any

such

vertex

bids

above

it

is

deleted

from

the

winning

set

together

with

and

replaced

with

Similarly,

the

threshold

bid

of

is

because

if

bids

above

it

can

be

replaced

with

So

the

VCG

payment

is

This

result

is

not

surprising:

the

definition

of

TUmax(c)

implicitly

assumes

there

is

co-operation

between

the

agents,

while

the

computation

of

VCG

payments

does

not

take

into

account

any

interaction

between

them.

Indeed,

co-operation

enables

the

agents

to

extract

higher

payments

under

VCG.

That

is,

VCG

is

not

groupstrategyproof.

This

suggests

that

as

a

payment

bound,

TUmax(c)

may

be

too

liberal,

at

least

in

a

context

where

there

is

little

or

no

co-operation

between

agents.

Perhaps

TUmax(c)

can

be

a

good

benchmark

for

measuring

the

performance

of

mechanisms

designed

for

agents

that

can

form

coalitions

or

make

side

payments

to

each

other,

in

particular,

group-strategyproof

mechanisms.

Another

setting

in

which

bounding

φTUmax

is

still

of

some

interest

is

when,

for

the

underlying

problem,

the

optimal

allocation

and

VCG

payments

are

NP-hard

to

compute.

In

this

case,

finding

a

polynomial-time

computable

mechanism

with

good

frugality

ratio

with

respect

to

TUmax(c)

is

a

non-trivial

task,

while

bounding

the

frugality

ratio

with

respect

to

more

challenging

payment

bounds

could

be

too

difficult.

To

illustrate

this

point,

compare

the

proofs

of

Lemma

and

Lemma

both

require

some

effort,

but

the

latter

is

much

more

difficult

than

the

former.

The

choice

of

S

All

payment

bounds

defined

in

this

paper

correspond

to

the

total

bid

of

all

elements

in

the

cheapest

feasible

set,

where

ties

are

broken

lexicographically.

While

this

definition

ensures

that

our

payment

bounds

are

well-defined,

the

particular

choice

of

the

drawresolution

rule

appears

arbitrary,

and

one

might

wonder

if

our

payment

bounds

are

sufficiently

robust

to

be

independent

of

this

choice.

It

turns

out

that

is

indeed

the

case

for

NTUmin(c)

and

NTUmax(c),

i.e.,

these

bounds

do

not

depend

on

the

draw-resolution

rule.

To

see

this,

suppose

that

two

feasible

sets

and

have

the

same

cost.

In

the

computation

of

NTUmin(c,

all

vertices

in

would

have

to

bid

their

true

cost,

since

otherwise

would

become

cheaper

than

Hence,

any

bid

vector

for

can

only

have

be

ce

for

e

and

hence

constitutes

a

valid

bid

vector

for

and

vice

versa.

A

similar

argument

applies

to

NTUmax(c).

However,

for

TUmin(c)

and

TUmax(c)

this

is

not

the

case.

For

example,

consider

the

set

system

E

F

with

the

costs

The

cheapest

sets

are

and

Now

TUmax(c,

as

the

total

bid

of

the

elements

in

cannot

exceed

the

total

cost

of

On

the

other

hand,

TUmax(c,

as

we

can

set

Similarly,

TUmin(c,

because

the

inequalities

in

are

and

But

TUmin(c,

as

we

can

set

Negative

results

for

NTUmin(c)

and

TUmin(c)

The

results

in

and

our

vertex

cover

results

are

proved

for

the

frugality

ratio

φNTUmin.

Indeed,

it

can

be

argued

that

φNTUmin

is

the

best

definition

of

frugality

ratio,

because

among

all

reasonable

payment

bounds

(i.e.,

ones

that

are

at

least

as

large

as

the

cost

of

the

cheapest

feasible

set),

it

is

most

demanding

of

the

algorithm.

However,

NTUmin(c)

is

not

always

the

easiest

or

the

most

natural

payment

bound

to

work

with.

In

this

subsection,

we

discuss

several

disadvantages

of

NTUmin(c)

(and

also

TUmin(c))

as

compared

to

NTUmax(c)

and

TUmax(c).

Nonmonotonicity

The

first

problem

with

NTUmin(c)

is

that

it

is

not

monotone

with

respect

to

F,

i.e.,

it

may

increase

when

one

adds

a

feasible

set

to

F.

(It

is,

however,

monotone

in

the

sense

that

a

losing

agent

cannot

become

a

winner

by

raising

his

cost.)

Intuitively,

a

good

payment

bound

should

satisfy

this

monotonicity

requirement,

as

adding

a

feasible

set

increases

the

competition,

so

it

should

drive

the

prices

down.

Note

that

this

indeed

the

case

for

NTUmax(c)

and

TUmax(c)

since

a

new

feasible

set

adds

a

constraint

in

thus

limiting

the

solution

space

for

the

respective

linear

program.

PROPOSITION

Adding

a

feasible

set

to

F

can

increase

the

value

of

NTUmin(c)

by

a

factor

of

Ω(n).

PROOF.

Let

E

{x,

xx,

yn,

zn}.

Set

Y

yn},

S

Y

{x},

Ti

Y

{yi}

{zi},

i

n,

and

suppose

that

F

{S,

Tn}.

The

costs

are

cx

cxx

cyi

czi

for

i

n.

Note

that

S

is

the

cheapest

feasible

set.

Let

F

F

where

Y

{xx}.

For

F,

the

bid

vector

byn

bx

satisfies

and

so

NTUmin(c)

For

F

S

is

still

the

lexicographically-least

cheapest

set.

Any

optimal

solution

has

bx

(by

constraint

in

with

Condition

for

yi

implies

bx

byi

czi

so

byi

and

NTUmin(c)

n.

For

path

auctions,

it

has

been

shown

that

NTUmin(c)

is

non-monotone

in

a

slightly

different

sense,

i.e.,

with

respect

to

adding

a

new

edge

(agent)

rather

than

a

new

feasible

set

(a

team

of

existing

agents).

REMARK

We

can

also

show

that

NTUmin(c)

is

non-monotone

for

vertex

cover.

In

this

case,

adding

a

new

feasible

set

corresponds

to

deleting

edges

from

the

graph.

It

turns

out

that

deleting

a

single

edge

can

increase

NTUmin(c)

by

a

factor

of

n

the

construction

is

similar

to

that

of

Proposition

NP-Hardness

Another

problem

with

NTUmin(c,

S)

is

that

it

is

NP-hard

to

compute

even

if

the

number

of

feasible

sets

is

polynomial

in

n.

Again,

this

puts

it

at

a

disadvantage

compared

to

NTUmax(c,

S)

and

TUmax(c,

S)

(see

Remark

THEOREM

Computing

NTUmin(c)

is

NP-hard,

even

when

the

lexicographically-least

feasible

set

S

is

given

in

the

input.

PROOF.

We

reduce

EXACT

COVER

BY

to

our

problem.

An

instance

of

is

given

by

a

universe

G

gn}

and

a

collection

of

subsets

Cm,

Ci

G,

|Ci|

where

the

goal

is

to

decide

whether

one

can

cover

G

by

of

these

sets.

Observe

that

if

this

is

indeed

the

case,

each

element

of

G

is

contained

in

exactly

one

set

of

the

cover.

LEMMA

Consider

a

minimisation

problem

P

of

the

following

form:

Minimise

P

bi

under

conditions

bi

for

all

i

n;

for

any

j

k

we

have

P

bi∈Sj

bi

aj,

where

Sj

bn};

for

each

bj

one

of

the

constraints

in

involving

it

is

tight.

For

any

such

P,

one

can

construct

a

set

system

S

and

a

vector

of

costs

c

such

that

NTUmin(c)

is

the

optimal

solution

to

P.

PROOF.

The

construction

is

straightforward:

there

is

an

element

of

cost

for

each

bi,

an

element

of

cost

aj

for

each

aj,

the

feasible

solutions

are

bn},

or

any

set

obtained

from

bn}

by

replacing

the

elements

in

Sj

by

aj.

By

this

lemma,

all

we

have

to

do

to

prove

Theorem

is

to

show

how

to

solve

by

using

the

solution

to

a

minimisation

problem

of

the

form

given

in

Lemma

We

do

this

as

follows.

For

each

Ci,

we

introduce

variables

xi,

¯xi,

ai,

and

bi.

Also,

for

each

element

gj

of

G

there

is

a

variable

dj.

We

use

the

following

set

of

constraints:

In

we

have

constraints

xi

¯xi

ai

bi

dj

for

all

i

m,

j

n.

In

for

all

i

m,

we

have

the

following

constraints:

xi

¯xi

xi

+ai

¯xi

+ai

xi

+bi

¯xi

bi

Also,

for

all

j

n

we

have

a

constraint

of

the

form

xik

dj

where

Cik

are

the

sets

that

contain

gj.

The

goal

is

to

minimize

z

P

i(xi

¯xi

ai

bi)

P

j

dj.

Observe

that

for

each

j,

there

is

only

one

constraint

involving

dj

so

by

condition

it

must

be

tight.

Consider

the

two

constraints

involving

ai.

One

of

them

must

be

tight,

and

therefore

xi

+¯xi

+ai

+bi

xi

+¯xi

+ai

Hence,

for

any

feasible

solution

to

we

have

z

m.

Now,

suppose

that

there

is

an

exact

set

cover.

Set

dj

for

j

n.

Also,

if

Ci

is

included

in

this

cover,

set

xi

¯xi

ai

bi

otherwise

set

¯xi

xi

ai

bi

Clearly,

all

inequalities

in

are

satisfied

(we

use

the

fact

that

each

element

is

covered

exactly

once),

and

for

each

variable,

one

of

the

constraints

involving

it

is

tight.

This

assignment

results

in

z

m.

Conversely,

suppose

there

is

a

feasible

solution

with

z

m.

As

each

addend

of

the

form

xi

¯xi

ai

bi

contributes

at

least

we

have

xi

¯xi

ai

bi

for

all

i,

dj

for

all

j.

We

will

now

show

that

for

each

i,

either

xi

and

¯xi

or

xi

and

¯xi

For

the

sake

of

contradiction,

suppose

that

xi

δ

¯xi

δ

As

one

of

the

constraints

involving

ai

must

be

tight,

we

have

ai

δ,

δ

Similarly,

bi

δ,

δ

Hence,

xi

¯xi

ai

bi

δ

+δ

1−δ

To

finish

the

proof,

note

that

for

each

j

m

we

have

xik

dj

and

dj

so

the

subsets

that

correspond

to

xi

constitute

a

set

cover.

REMARK

In

the

proofs

of

Proposition

and

Theorem

all

constraints

in

are

of

the

form

be

Hence,

the

same

results

are

true

for

TUmin(c).

REMARK

For

shortest-path

auctions,

the

size

of

F

can

be

superpolynomial.

However,

there

is

a

polynomial-time

separation

oracle

for

constraints

in

(to

construct

one,

use

any

algorithm

for

finding

shortest

paths),

so

one

can

compute

NTUmax(c)

and

TUmax(c)

in

polynomial

time.

On

the

other

hand,

recently

and

independently

it

was

shown

that

computing

NTUmin(c)

for

shortest-path

auctions

is

NP-hard.

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auction
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