231 research outputs found
Pareto Optimal Allocation under Uncertain Preferences
The assignment problem is one of the most well-studied settings in social
choice, matching, and discrete allocation. We consider the problem with the
additional feature that agents' preferences involve uncertainty. The setting
with uncertainty leads to a number of interesting questions including the
following ones. How to compute an assignment with the highest probability of
being Pareto optimal? What is the complexity of computing the probability that
a given assignment is Pareto optimal? Does there exist an assignment that is
Pareto optimal with probability one? We consider these problems under two
natural uncertainty models: (1) the lottery model in which each agent has an
independent probability distribution over linear orders and (2) the joint
probability model that involves a joint probability distribution over
preference profiles. For both of the models, we present a number of algorithmic
and complexity results.Comment: Preliminary Draft; new results & new author
Hunting for Tractable Languages for Judgment Aggregation
Judgment aggregation is a general framework for collective decision making
that can be used to model many different settings. Due to its general nature,
the worst case complexity of essentially all relevant problems in this
framework is very high. However, these intractability results are mainly due to
the fact that the language to represent the aggregation domain is overly
expressive. We initiate an investigation of representation languages for
judgment aggregation that strike a balance between (1) being limited enough to
yield computational tractability results and (2) being expressive enough to
model relevant applications. In particular, we consider the languages of Krom
formulas, (definite) Horn formulas, and Boolean circuits in decomposable
negation normal form (DNNF). We illustrate the use of the positive complexity
results that we obtain for these languages with a concrete application: voting
on how to spend a budget (i.e., participatory budgeting).Comment: To appear in the Proceedings of the 16th International Conference on
Principles of Knowledge Representation and Reasoning (KR 2018
A Belief Model for Conflicting and Uncertain Evidence -- Connecting Dempster-Shafer Theory and the Topology of Evidence
One problem to solve in the context of information fusion, decision-making,
and other artificial intelligence challenges is to compute justified beliefs
based on evidence. In real-life examples, this evidence may be inconsistent,
incomplete, or uncertain, making the problem of evidence fusion highly
non-trivial. In this paper, we propose a new model for measuring degrees of
beliefs based on possibly inconsistent, incomplete, and uncertain evidence, by
combining tools from Dempster-Shafer Theory and Topological Models of Evidence.
Our belief model is more general than the aforementioned approaches in two
important ways: (1) it can reproduce them when appropriate constraints are
imposed, and, more notably, (2) it is flexible enough to compute beliefs
according to various standards that represent agents' evidential demands. The
latter novelty allows the users of our model to employ it to compute an agent's
(possibly) distinct degrees of belief, based on the same evidence, in
situations when, e.g, the agent prioritizes avoiding false negatives and when
it prioritizes avoiding false positives. Finally, we show that computing
degrees of belief with this model is #P-complete in general.Comment: To appear in the proceedings of KR 202
On Existential MSO and its Relation to ETH
Impagliazzo et al. proposed a framework, based on the logic fragment defining the complexity class SNP, to identify problems that are equivalent to k-CNF-Sat modulo subexponential-time reducibility (serf-reducibility). The subexponential-time solvability of any of these problems implies the failure of the Exponential Time Hypothesis (ETH). In this paper, we extend the framework of Impagliazzo et al., and identify a larger set of problems that are equivalent to k-CNF-Sat modulo serf-reducibility. We propose a complexity class, referred to as Linear Monadic NP, that consists of all problems expressible in existential monadic second order logic whose expressions have a linear measure in terms of a complexity parameter, which is usually the universe size of the problem.
This research direction can be traced back to Fagin\u27s celebrated theorem stating that NP coincides with the class of problems expressible in existential second order logic. Monadic NP, a well-studied class in the literature, is the restriction of the aforementioned logic fragment to existential monadic second order logic. The proposed class Linear Monadic NP is then the restriction of Monadic NP to problems whose expressions have linear measure in the complexity parameter.
We show that Linear Monadic NP includes many natural complete problems such as the satisfiability of linear-size circuits, dominating set, independent dominating set, and perfect code. Therefore, for any of these problems, its subexponential-time solvability is equivalent to the failure of ETH. We prove, using logic games, that the aforementioned problems are inexpressible in the monadic fragment of SNP, and hence, are not captured by the framework of Impagliazzo et al. Finally, we show that Feedback Vertex Set is inexpressible in existential monadic second order logic, and hence is not in Linear Monadic NP, and investigate the existence of certain reductions between Feedback Vertex Set (and variants of it) and 3-CNF-Sat
Egalitarian Judgment Aggregation
Egalitarian considerations play a central role in many areas of social choice
theory. Applications of egalitarian principles range from ensuring everyone
gets an equal share of a cake when deciding how to divide it, to guaranteeing
balance with respect to gender or ethnicity in committee elections. Yet, the
egalitarian approach has received little attention in judgment aggregation -- a
powerful framework for aggregating logically interconnected issues. We make the
first steps towards filling that gap. We introduce axioms capturing two
classical interpretations of egalitarianism in judgment aggregation and situate
these within the context of existing axioms in the pertinent framework of
belief merging. We then explore the relationship between these axioms and
several notions of strategyproofness from social choice theory at large.
Finally, a novel egalitarian judgment aggregation rule stems from our analysis;
we present complexity results concerning both outcome determination and
strategic manipulation for that rule.Comment: Extended version of paper in proceedings of the 20th International
Conference on Autonomous Agents and Multiagent Systems (AAMAS), 202
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