32 research outputs found
Recognising Multidimensional Euclidean Preferences
Euclidean preferences are a widely studied preference model, in which
decision makers and alternatives are embedded in d-dimensional Euclidean space.
Decision makers prefer those alternatives closer to them. This model, also
known as multidimensional unfolding, has applications in economics,
psychometrics, marketing, and many other fields. We study the problem of
deciding whether a given preference profile is d-Euclidean. For the
one-dimensional case, polynomial-time algorithms are known. We show that, in
contrast, for every other fixed dimension d > 1, the recognition problem is
equivalent to the existential theory of the reals (ETR), and so in particular
NP-hard. We further show that some Euclidean preference profiles require
exponentially many bits in order to specify any Euclidean embedding, and prove
that the domain of d-Euclidean preferences does not admit a finite forbidden
minor characterisation for any d > 1. We also study dichotomous preferencesand
the behaviour of other metrics, and survey a variety of related work.Comment: 17 page
The Complexity of Fully Proportional Representation for Single-Crossing Electorates
We study the complexity of winner determination in single-crossing elections
under two classic fully proportional representation
rules---Chamberlin--Courant's rule and Monroe's rule. Winner determination for
these rules is known to be NP-hard for unrestricted preferences. We show that
for single-crossing preferences this problem admits a polynomial-time algorithm
for Chamberlin--Courant's rule, but remains NP-hard for Monroe's rule. Our
algorithm for Chamberlin--Courant's rule can be modified to work for elections
with bounded single-crossing width. To circumvent the hardness result for
Monroe's rule, we consider single-crossing elections that satisfy an additional
constraint, namely, ones where each candidate is ranked first by at least one
voter (such elections are called narcissistic). For single-crossing
narcissistic elections, we provide an efficient algorithm for the egalitarian
version of Monroe's rule.Comment: 23 page
Fair Knapsack
We study the following multiagent variant of the knapsack problem. We are
given a set of items, a set of voters, and a value of the budget; each item is
endowed with a cost and each voter assigns to each item a certain value. The
goal is to select a subset of items with the total cost not exceeding the
budget, in a way that is consistent with the voters' preferences. Since the
preferences of the voters over the items can vary significantly, we need a way
of aggregating these preferences, in order to select the socially best valid
knapsack. We study three approaches to aggregating voters' preferences, which
are motivated by the literature on multiwinner elections and fair allocation.
This way we introduce the concepts of individually best, diverse, and fair
knapsack. We study the computational complexity (including parameterized
complexity, and complexity under restricted domains) of the aforementioned
multiagent variants of knapsack.Comment: Extended abstract will appear in Proc. of 33rd AAAI 201
Reallocating Multiple Facilities on the Line
We study the multistage -facility reallocation problem on the real line,
where we maintain facility locations over stages, based on the
stage-dependent locations of agents. Each agent is connected to the nearest
facility at each stage, and the facilities may move from one stage to another,
to accommodate different agent locations. The objective is to minimize the
connection cost of the agents plus the total moving cost of the facilities,
over all stages. -facility reallocation was introduced by de Keijzer and
Wojtczak, where they mostly focused on the special case of a single facility.
Using an LP-based approach, we present a polynomial time algorithm that
computes the optimal solution for any number of facilities. We also consider
online -facility reallocation, where the algorithm becomes aware of agent
locations in a stage-by-stage fashion. By exploiting an interesting connection
to the classical -server problem, we present a constant-competitive
algorithm for facilities
Iterative Delegations in Liquid Democracy with Restricted Preferences
In this paper, we study liquid democracy, a collective decision making
paradigm which lies between direct and representative democracy. One main
feature of liquid democracy is that voters can delegate their votes in a
transitive manner so that: A delegates to B and B delegates to C leads to A
delegates to C. Unfortunately, this process may not converge as there may not
even exist a stable state (also called equilibrium). In this paper, we
investigate the stability of the delegation process in liquid democracy when
voters have restricted types of preference on the agent representing them
(e.g., single-peaked preferences). We show that various natural structures of
preferences guarantee the existence of an equilibrium and we obtain both
tractability and hardness results for the problem of computing several
equilibria with some desirable properties
Approximating the MaxCover Problem with Bounded Frequencies in FPT Time
We study approximation algorithms for several variants of the MaxCover
problem, with the focus on algorithms that run in FPT time. In the MaxCover
problem we are given a set N of elements, a family S of subsets of N, and an
integer K. The goal is to find up to K sets from S that jointly cover (i.e.,
include) as many elements as possible. This problem is well-known to be NP-hard
and, under standard complexity-theoretic assumptions, the best possible
polynomial-time approximation algorithm has approximation ratio (1 - 1/e). We
first consider a variant of MaxCover with bounded element frequencies, i.e., a
variant where there is a constant p such that each element belongs to at most p
sets in S. For this case we show that there is an FPT approximation scheme
(i.e., for each B there is a B-approximation algorithm running in FPT time) for
the problem of maximizing the number of covered elements, and a randomized FPT
approximation scheme for the problem of minimizing the number of elements left
uncovered (we take K to be the parameter). Then, for the case where there is a
constant p such that each element belongs to at least p sets from S, we show
that the standard greedy approximation algorithm achieves approximation ratio
exactly (1-e^{-max(pK/|S|, 1)}). We conclude by considering an unrestricted
variant of MaxCover, and show approximation algorithms that run in exponential
time and combine an exact algorithm with a greedy approximation. Some of our
results improve currently known results for MaxVertexCover