16 research outputs found
The Complexity of Manipulative Attacks in Nearly Single-Peaked Electorates
Many electoral bribery, control, and manipulation problems (which we will
refer to in general as "manipulative actions" problems) are NP-hard in the
general case. It has recently been noted that many of these problems fall into
polynomial time if the electorate is single-peaked (i.e., is polarized along
some axis/issue). However, real-world electorates are not truly single-peaked.
There are usually some mavericks, and so real-world electorates tend to merely
be nearly single-peaked. This paper studies the complexity of
manipulative-action algorithms for elections over nearly single-peaked
electorates, for various notions of nearness and various election systems. We
provide instances where even one maverick jumps the manipulative-action
complexity up to \np-hardness, but we also provide many instances where a
reasonable number of mavericks can be tolerated without increasing the
manipulative-action complexity.Comment: 35 pages, also appears as URCS-TR-2011-96
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
An Atypical Survey of Typical-Case Heuristic Algorithms
Heuristic approaches often do so well that they seem to pretty much always
give the right answer. How close can heuristic algorithms get to always giving
the right answer, without inducing seismic complexity-theoretic consequences?
This article first discusses how a series of results by Berman, Buhrman,
Hartmanis, Homer, Longpr\'{e}, Ogiwara, Sch\"{o}ening, and Watanabe, from the
early 1970s through the early 1990s, explicitly or implicitly limited how well
heuristic algorithms can do on NP-hard problems. In particular, many desirable
levels of heuristic success cannot be obtained unless severe, highly unlikely
complexity class collapses occur. Second, we survey work initiated by Goldreich
and Wigderson, who showed how under plausible assumptions deterministic
heuristics for randomized computation can achieve a very high frequency of
correctness. Finally, we consider formal ways in which theory can help explain
the effectiveness of heuristics that solve NP-hard problems in practice.Comment: This article is currently scheduled to appear in the December 2012
issue of SIGACT New
Manipulating Districts to Win Elections: Fine-Grained Complexity
Gerrymandering is a practice of manipulating district boundaries and
locations in order to achieve a political advantage for a particular party.
Lewenberg, Lev, and Rosenschein [AAMAS 2017] initiated the algorithmic study of
a geographically-based manipulation problem, where voters must vote at the
ballot box closest to them. In this variant of gerrymandering, for a given set
of possible locations of ballot boxes and known political preferences of
voters, the task is to identify locations for boxes out of possible
locations to guarantee victory of a certain party in at least districts.
Here integers and are some selected parameter.
It is known that the problem is NP-complete already for 4 political parties
and prior to our work only heuristic algorithms for this problem were
developed. We initiate the rigorous study of the gerrymandering problem from
the perspectives of parameterized and fine-grained complexity and provide
asymptotically matching lower and upper bounds on its computational complexity.
We prove that the problem is W[1]-hard parameterized by and that it does
not admit an algorithm for any function of
and only, unless Exponential Time Hypothesis (ETH) fails. Our lower
bounds hold already for parties. On the other hand, we give an algorithm
that solves the problem for a constant number of parties in time
.Comment: Presented at AAAI-2
Parameterized Algorithmics for Computational Social Choice: Nine Research Challenges
Computational Social Choice is an interdisciplinary research area involving
Economics, Political Science, and Social Science on the one side, and
Mathematics and Computer Science (including Artificial Intelligence and
Multiagent Systems) on the other side. Typical computational problems studied
in this field include the vulnerability of voting procedures against attacks,
or preference aggregation in multi-agent systems. Parameterized Algorithmics is
a subfield of Theoretical Computer Science seeking to exploit meaningful
problem-specific parameters in order to identify tractable special cases of in
general computationally hard problems. In this paper, we propose nine of our
favorite research challenges concerning the parameterized complexity of
problems appearing in this context
Single-Peaked Consistency for Weak Orders Is Easy
In economics and social choice single-peakedness is one of the most important
and commonly studied models for preferences. It is well known that
single-peaked consistency for total orders is in P. However in practice a
preference profile is not always comprised of total orders. Often voters have
indifference between some of the candidates. In a weak preference order
indifference must be transitive. We show that single-peaked consistency for
weak orders is in P for three different variants of single-peakedness for weak
orders. Specifically, we consider Black's original definition of
single-peakedness for weak orders, Black's definition of single-plateaued
preferences, and the existential model recently introduced by Lackner. We
accomplish our results by transforming each of these single-peaked consistency
problems to the problem of determining if a 0-1 matrix has the consecutive ones
property.Comment: In Proceedings TARK 2015, arXiv:1606.0729