80 research outputs found
Approximation and Parameterized Complexity of Minimax Approval Voting
We present three results on the complexity of Minimax Approval Voting. First,
we study Minimax Approval Voting parameterized by the Hamming distance from
the solution to the votes. We show Minimax Approval Voting admits no algorithm
running in time , unless the Exponential
Time Hypothesis (ETH) fails. This means that the
algorithm of Misra et al. [AAMAS 2015] is essentially optimal. Motivated by
this, we then show a parameterized approximation scheme, running in time
, which is essentially
tight assuming ETH. Finally, we get a new polynomial-time randomized
approximation scheme for Minimax Approval Voting, which runs in time
,
almost matching the running time of the fastest known PTAS for Closest String
due to Ma and Sun [SIAM J. Comp. 2009].Comment: 14 pages, 3 figures, 2 pseudocode
Computational Aspects of Multi-Winner Approval Voting
We study computational aspects of three prominent voting rules that use
approval ballots to elect multiple winners. These rules are satisfaction
approval voting, proportional approval voting, and reweighted approval voting.
We first show that computing the winner for proportional approval voting is
NP-hard, closing a long standing open problem. As none of the rules are
strategyproof, even for dichotomous preferences, we study various strategic
aspects of the rules. In particular, we examine the computational complexity of
computing a best response for both a single agent and a group of agents. In
many settings, we show that it is NP-hard for an agent or agents to compute how
best to vote given a fixed set of approval ballots from the other agents
Mathematical Programming formulations for the efficient solution of the -sum approval voting problem
In this paper we address the problem of electing a committee among a set of
candidates and on the basis of the preferences of a set of voters. We
consider the approval voting method in which each voter can approve as many
candidates as she/he likes by expressing a preference profile (boolean
-vector). In order to elect a committee, a voting rule must be established
to `transform' the voters' profiles into a winning committee. The problem
is widely studied in voting theory; for a variety of voting rules the problem
was shown to be computationally difficult and approximation algorithms and
heuristic techniques were proposed in the literature. In this paper we follow
an Ordered Weighted Averaging approach and study the -sum approval voting
(optimization) problem in the general case . For this problem we
provide different mathematical programming formulations that allow us to solve
it in an exact solution framework. We provide computational results showing
that our approach is efficient for medium-size test problems ( up to 200,
up to 60) since in all tested cases it was able to find the exact optimal
solution in very short computational times
Parameterized Complexity of Multi-winner Determination: More Effort Towards Fixed-Parameter Tractability
We study the parameterized complexity of Winners Determination for three
prevalent -committee selection rules, namely the minimax approval voting
(MAV), the proportional approval voting (PAV), and the Chamberlin-Courant's
approval voting (CCAV). It is known that Winners Determination for these rules
is NP-hard. Moreover, these problems have been studied from the parameterized
complexity point of view with respect to some natural parameters recently.
However, many results turned out to be W[1]-hard or W[2]-hard. Aiming at
deriving more fixed-parameter algorithms, we revisit these problems by
considering more natural and important single parameters, combined parameters,
and structural parameters.Comment: 31 pages, 2 figures, AAMAS 201
Finding a Collective Set of Items: From Proportional Multirepresentation to Group Recommendation
We consider the following problem: There is a set of items (e.g., movies) and
a group of agents (e.g., passengers on a plane); each agent has some intrinsic
utility for each of the items. Our goal is to pick a set of items that
maximize the total derived utility of all the agents (i.e., in our example we
are to pick movies that we put on the plane's entertainment system).
However, the actual utility that an agent derives from a given item is only a
fraction of its intrinsic one, and this fraction depends on how the agent ranks
the item among the chosen, available, ones. We provide a formal specification
of the model and provide concrete examples and settings where it is applicable.
We show that the problem is hard in general, but we show a number of
tractability results for its natural special cases
Comparing Election Methods Where Each Voter Ranks Only Few Candidates
Election rules are formal processes that aggregate voters preferences,
typically to select a single candidate, called the winner. Most of the election
rules studied in the literature require the voters to rank the candidates from
the most to the least preferred one. This method of eliciting preferences is
impractical when the number of candidates to be ranked is large. We ask how
well certain election rules (focusing on positional scoring rules and the
Minimax rule) can be approximated from partial preferences collected through
one of the following procedures: (i) randomized-we ask each voter to rank a
random subset of candidates, and (ii) deterministic-we ask each voter to
provide a ranking of her most preferred candidates (the -truncated
ballot). We establish theoretical bounds on the approximation ratios and we
complement our theoretical analysis with computer simulations. We find that
mostly (apart from the cases when the preferences have no or very little
structure) it is better to use the randomized approach. While we obtain fairly
good approximation guarantees for the Borda rule already for , for
approximating the Minimax rule one needs to ask each voter to compare a larger
set of candidates in order to obtain good guarantees
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