A simple algorithmic proof of the symmetric lopsided Lovász local lemma

We provide a simple algorithmic proof for the symmetric Lopsided Lovász Local Lemma, a variant of the classic Lovász Local Lemma, where, roughly, only the degree of the negatively correlated undesirable events counts. Our analysis refers to the algorithm by Moser (2009), however it is based on a simple application of the probabilistic method, rather than a counting argument, as are most of the analyses of algorithms for variants of the Lovász Local Lemma. © 2019, Springer Nature Switzerland AG

Aggregation of votes with multiple positions on each issue

We consider the problem of aggregating votes cast by a society on a fixed set of issues, where each member of the society may vote for one of several positions on each issue, but the combination of votes on the various issues is restricted to a set of feasible voting patterns. We follow the aggregation framework used by Dokow and Holzman [Aggregation of non-binary evaluations, Advances in Applied Mathematics, 45:4, 487-504, 2010], in which both preference aggregation and judgment aggregation can be cast. We require the aggregation to be independent on each issue, and also supportive, i.e., for every issue, the corresponding component of every aggregator, when applied to a tuple of votes, must take as value one of the votes in that tuple.We prove that, in such a setup, non-dictatorial aggregation of votes in a society of an arbitrary size is possible if and only if either there is a non-dictatorial aggregator for two voters or there is an aggregator for three voters such that, for each issue, the corresponding component of the aggregator, when restricted to two-element sets of votes, is a majority operation or a minority operation. We then introduce a notion of a uniform non-dictatorial aggregator, which is an aggregator such that on every issue, and when restricted to arbitrary two-element subsets of the votes for that issue, it differs from all projection functions. We first give a characterization of sets of feasible voting patterns that admit a uniform non-dictatorial aggregator. After this and by making use of Bulatov's dichotomy theorem for conservative constraint satisfaction problems, we connect social choice theory with the computational complexity of constraint satisfaction by proving that if a set of feasible voting patterns has a uniform non-dictatorial aggregator of some arity, then themulti-sorted conservative constraint satisfaction problem on that set (with each issue representing a different sort) is solvable in polynomial time; otherwise, it is NP-complete. © 2019 Association for Computing Machinery

On the Computational Complexity of Non-Dictatorial Aggregation

We investigate when non-dictatorial aggregation is possible from an algorithmic perspective, where non-dictatorial aggregation means that the votes cast by the members of a society can be aggregated in such a way that there is no single member of the society that always dictates the collective outcome. We consider the setting in which the members of a society take a position on a fixed collection of issues, where for each issue several different alternatives are possible, but the combination of choices must belong to a given set X of allowable voting patterns. Such a set X is called a possibility domain if there is an aggregator that is non-dictatorial, operates separately on each issue, and returns values among those cast by the society on each issue. We design a polynomial-time algorithm that decides, given a set X of voting patterns, whether or not X is a possibility domain. Furthermore, if X is a possibility domain, then the algorithm constructs in polynomial time a non-dictatorial aggregator for X. Furthermore, we show that the question of whether a Boolean domain X is a possibility domain is in NLOGSPACE. We also design a polynomial-time algorithm that decides whether X is a uniform possibility domain, that is, whether X admits an aggregator that is non-dictatorial even when restricted to any two positions for each issue. As in the case of possibility domains, the algorithm also constructs in polynomial time a uniform non-dictatorial aggregator, if one exists. Then, we turn our attention to the case where X is given implicitly, either as the set of assignments satisfying a propositional formula, or as a set of consistent evaluations of a sequence of propositional formulas. In both cases, we provide bounds to the complexity of deciding if X is a (uniform) possibility domain. Finally, we extend our results to four types of aggregators that have appeared in the literature: generalized dictatorships, whose outcome is always an element of their input, anonymous aggregators, whose outcome is not affected by permutations of their input, monotone, whose outcome does not change if more individuals agree with it and systematic, which aggregate every issue in the same way. ©2021 AI Access Foundation. All rights reserved

Aggregation of votes with multiple positions on each issue

We consider the problem of aggregating votes cast by a society on a fixed set of issues, where each member of the society may vote for one of several positions on each issue, but the combination of votes on the various issues is restricted to a set of feasible voting patterns. We require the aggregation to be supportive, i.e., for every issue, the corresponding component of every aggregator, when applied to a tuple of votes, must take as value one of the votes in that tuple. We prove that, in such a set-up, non-dictatorial aggregation of votes in a society of an arbitrary size is possible if and only if a non-dictatorial binary aggregator exists or a non-dictatorial ternary aggregator exists such that, for each issue, the corresponding component of the aggregator, when restricted to twoelement sets of votes, is a majority operation or a minority operation. We then introduce a notion of a uniform non-dictatorial aggregator, which is an aggregator such that on every issue, and when restricted to arbitrary two-element subsets of the votes for that issue, differs from all projection functions. We first give a characterization of sets of feasible voting patterns that admit a uniform non-dictatorial aggregator. After this and by making use of Bulatov’s dichotomy theorem for conservative constraint satisfaction problems, we connect social choice theory with the computational complexity of constraint satisfaction by proving that if a set of feasible voting patterns has a uniform non-dictatorial aggregator of some arity, then the multi-sorted conservative constraint satisfaction problem on that set (with each issue representing a different sort) is solvable in polynomial time; otherwise, it is NP-complete. © Springer International Publishing AG 2017

Correction to: Directed Lovász Local Lemma and Shearer’s Lemma (Annals of Mathematics and Artificial Intelligence, (2020), 88, 1-3, (133-155), 10.1007/s10472-019-09671-5)

The proof of Theorem 1a of our article that appears in Annals of Mathematics and Artificial Intelligence 88(1–3):133-155 (2020) has a mistake. We give here the corrected proof, together with a new version of Definition 3 in that article that the correction necessitated. © 2020, Springer Nature Switzerland AG

A dual modality approach to quantitative quality control in emission tomography

Routine quality control (QC) and optimization of image quality of reconstructed images in single photon emission computed tomography (SPECT) and positron emission tomography (PET) remains a relatively qualitative exercise. With the advent of combined SPECT/CT and PET/CT devices, and accurate post hoc co-registration algorithms, the potential exists to utilize high resolution structural information for QC evaluation in addition to their use for anatomical correlation in clinical studies. The aim of this work was to explore, in principle, the uses of x-ray CT data of QC phantoms used in SPECT and PET to develop more objective assessments of performance of the emission tomographic (ET) devices and reconstructed data. A CT reconstruction of a novel ET QC phantom was segmented into the various compartments it contained. Using software, the voxel values in the different compartments were then altered to correspond to the concentration of the radioactivity in the actual scan of the same phantom on the SPECT system. This produces a high resolution version of a ‘perfect’ ET scan. Image co-registration techniques were then used to spatially align the synthetic high resolution SPECT scan to the measured SPECT scan. Various parameters can then be objectively derived from the registered data, for example, image contrast, spatial resolution, spatial non-uniformity, etc. In this study, we have used this approach to estimate spatial resolution (full width at half maximum, FWHM) and recovered contrast in reconstructed images of a SPECT phantom. Two independent methods were used to measure spatial resolution, obtaining excellent agreement. In conclusion, the ability to produce high resolution synthetic phantoms in emission tomography QC affords an objective approach to assessing system performance and optimizing protocols which is readily automated and quantifiable