Preferences Single-Peaked on Nice Trees
Dominik Peters and Edith Elkind
Department of Computer Science, University of Oxford, UK
Finding
Nice
Trees
(if
possible)
Single-Peaked on a Tree
Preferences Single-Peaked on Nice Trees
introduced by Demange (1982) • Question: Can we modify Trick’s algorithm to returnEdith
a nice
tree if that’s possible?
Dominik
Peters
and
Elkind
more general than single-peaked on a line
Department of •Computer
Science
Answer: Yes, by keeping track of the decisions that
Condorcet winner guaranteed to exist University of Oxford, UK
the algorithm made.
{dominik.peters, edith.elkind}@cs.ox.ac.uk
allows faster algorithms for voting problems • In fact, we can concisely represent all trees on which a given
profile is single-peaked using the “attachment digraph”:
efficiently recognisable
j
Abstract
Preference
profiles
that
are
single-peaked
on
trees
enDefinitions. Let A be an m-element set of alternatives, and
joy desirable properties: they admit a Condorcet winner
let V be an n-element
set1982),
(or profile)
votes,
strict
(Demange
and thereofare
hard i.e.,
voting
problems
thatover
become
tractable
this)domain
(Yu,over
Chan,A.and
preference orders
A. Let
T =on
(A,E
be a tree
Elkind 2013). Trick (1989) proposed a polynomial-time
algorithm that finds some tree with respect to which
preference profile ison
single-peaked.
V aisgiven
single-peaked
T iff However,
some voting problems are only known to be easy for profiles that are single-peaked
on “nice”
trees,
V is single-peaked
on every
path
ofand
T.Trick’s
algorithm provides no guarantees on the properties of
the tree that it outputs. To overcome this issue, we build
on is
thesingle-peaked
work of Trick andon
YuTet if
al.for
to develop
structural
Equivalently, V
every avoter
i
approach that enables us to compactly represent all trees
and all numbers
k,
the
set
of
i’s
k
most
preferred
alternawith respect to which a given profile is single-peaked.
tives induces aWe(connected)
subtree
of T.
show how to use
this representation
to efficiently find
the “best” tree for a given profile, according to a number
of criteria; for other criteria, we obtain NP-hardness results. In particular, we show that it is NP-hard to decide
whether an inputmosprofile is single-peaked with respect
t-pref
erred
alterdemonstrate
to a given tree. To
the
applicability
of
our
native
s
framework,
we use it to identify a new
class
of
profiles
a
b
that admit an efficient algorithm for a popular variant of
the Chamberlin–Courant (1983) rule.
w
b
1 Introduction
a
Preference aggregation is a difficult task when voters’ preferleast-p
refe
ences
may
be arbitrary:
one has tost-deal
z
dwith voting paradoxes
e
r
r
e
f
alterna rred
e
r
tive
lea p ative
w
altern hard problems (Brandt,
(Arrow 1951) and computationally
z
Conitzer, and Endriss 2013). This observation motivates the
study of domain restrictions, i.e., special classes of voters’
preferences that rule out paradoxical outcomes and/or allow
one to circumvent computational hardness results. Perhaps
Trick’stheRecognition
Algorithm
most-studied restricted domain is that of single-peaked
preferences
1948).2) This
domain for
captures
profiles
gives(Black
an O(nm
algorithm
checking
• Trick (1987)
where voters’ preferences are determined by candidates’ poif a preference
single-peaked
a tree.
sitions on profile
a single is
issue,
and has many on
desirable
properties:
forreturns
instance, some
single-peaked
elections
always have a Condorcet
tree that
works.
• If yes, it
winner (a candidate that is preferred to every other candidate
by a majority of voters), admit a non-manipulable voting
Key insightrule
behind
algorithm:
if an
alternative
appears
in lastdeter(Moulin
1991), and
allow
for an efficient
winner
algorithm
for a popular
committee
selection
rule
position inmination
some voter’s
preference,
then the
alternative
must be
(Betzler,
Slinko, and
Uhlmann 2013).
a leaf (similar
to standard
single-peaked).
Identify such a leaf alCopyright
forchosen
the Advancement
of Artificial
ternative and
attachcit2016,
to anAssociation
arbitrarily
other alternative,
Intelligence (www.aaai.org). All rights reserved.
but without violating single-peakedness. Repeat.
• Some algorithms for voting problems that exploit
the tree structure of preferences work better if the
tree is “nice”
• For example, nice could mean having few leaves, or
low degrees, or small pathwidth.
i
a
h
b
c
d
e
f
g
k
Instructions
one outgoing
arc computed
for each alternative,
forget
Figure for
1: use:
The select
‘attachment
digraph’
for the profile {kfedghcijba,
dcbeafghijk,
which
is singleorientations,
and we obtain
a tree on gfhiedcbajk}
which the profile
{kfedghcijba,
peakedgfhiedcbajk}
on 336 different
trees. All ofAllthem
as subtrees
dcbeafghijk,
is single-peaked.
suchappear
trees can
be obtained
2
the above
digraph,
which
canare
be forced,
computed
O(|V
|C| are
)
in thisofway.
Note that
the black
arcs
andinthe
grey| ·arcs
time.
The
black
edges
must
appear
in
any
tree;
for
each
free
‘free’. Crucially, among the grey arcs, the digraph is transitive.
vertex with gray outgoing arcs, we choose exactly one of
them. The transitivity among the gray edges will be crucial
to
our
approach.
Recognition Algorithms
Theorem. Given a profile V that is single-peaked on a
tree, we
can find
in polynomial
a suitable
tree that
Demange
(1982)
introduced atime
weaker
domain restriction,
namely,
single-peakedness
on a tree.number
Briefly, of
a profile
among
suitable
tree has a minimum
leaves,is a
single-peaked on a tree if candidates can be mapped to the
minimum
internal
vertices,
minimum
diameter,
verticesnumber
of some of
tree
so that the
restriction
of this profile
to
every path
in this tree minimum
is single-peaked.
This is a considerminimum
max-degree,
path-width.
ably broader domain than that of the single-peaked elections,
Further,
wenevertheless,
can decideretains
in polynomial
whether
a given
which,
some of thetime
desirable
properties
of the
latter: profiles that
single-peaked
on a tree always
profile
is single-peaked
on are
a line,
star, caterpillar,
lobster,
have a Condorcet winner (Demange 1982), and there are
or oncommittee
a subdivision
of
a
star.
selection problems that become easier when preferences are single-peaked on a tree (Yu, Chan, and Elkind
Theorem.
However,some
it is of
NP-complete
decide,
given a
2013). However,
the algorithmstofor
this domain
require
the
input
profile
to
be
single-peaked
on
a
“nice”
tree,
profile V and an unlabelled tree T with |A| vertices,
such as a tree with a small number of leaves or a star (Yu,
whether
is single-peaked
on some
labelling
Chan,Vand
Elkind 2013); indeed,
positive
results of
forT.
singlepeaked preferences
can also be
viewed in this
as they
Also NP-hard
to decide whether
single-peaked
on light,
a regular
tree.
require the preferences to be single-peaked on a specific tree,
namely, a line (path).
Application:
Committee
Selection
Now, forcing Trick’s algorithm to output a “nice” tree is
a trivial task: indeed, this
algorithm
may output
a complex
• ThenotChamberlin-Courant
rule
is a popular
voting
rule for
tree even
the input
profile isofsingle-peaked
line.
selecting
anwhen
“optimal”
committee
size k from on
theam
canFortunately, there are efficient algorithms for recognizing
didates
in
A.
when a given profile is single-peaked on a line (Bartholdi and
• Optimal
committee
minimises
a misrepresentation
function.
Trick 1986;
Doignon
and Falmagne
1994; Escoffier, Lang,
Öztürk
2008), and
Chan, and
Elkind (2013) explain
• Theand
rule
is NP-hard
to Yu,
evaluate
in general.
how
to
modify
Trick’s
algorithm
so
that
it
outputs
a
tree
• Yu, Chan, and Elkind (2013) show this problem becomes easier for preferences single-peaked on a tree with few leaves.
• This paper: also easy for trees with few internal vertices.
Theorem. Given a profile that is single-peaked on a tree
with r internal vertices, we can find a winning committee of
size k under the Chamberlin-Courant rule with Borda misrepresentation function in time poly(n, m, (k + 1)r).
Presented at AAAI 2016, Phoenix, Arizona