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 the collective outcome is not simply the choices made by a single member of the society. 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 such a non-dictatorial aggregator for $X$. We then 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 an sequence of propositional formulas. In both cases, we provide bounds to the complexity of deciding if $X$ is a (uniform) possibility domain.Comment: 21 page

A new upper bound for 3-SAT

We show that a randomly chosen 3-CNF formula over n variables with clauses-to-variables ratio at least 4.4898 is, as n grows large, asymptotically almost surely unsatisfiable. The previous best such bound, due to Dubois in 1999, was 4.506. The first such bound, independently discovered by many groups of researchers since 1983, was 5.19. Several decreasing values between 5.19 and 4.506 were published in the years between. The probabilistic techniques we use for the proof are, we believe, of independent interest.Comment: 20 page

Solving order constraints in logarithmic space.

We combine methods of order theory, finite model theory, and universal algebra to study, within the constraint satisfaction framework, the complexity of some well-known combinatorial problems connected with a finite poset. We identify some conditions on a poset which guarantee solvability of the problems in (deterministic, symmetric, or non-deterministic) logarithmic space. On the example of order constraints we study how a certain algebraic invariance property is related to solvability of a constraint satisfaction problem in non-deterministic logarithmic space

GRS 1915+105 : High-energy Insights with SPI/INTEGRAL

We report on results of two years of INTEGRAL/SPI monitoring of the Galactic microquasar GRS 1915+105. From September 2004 to May 2006, the source has been observed twenty times with long (approx 100 ks) exposures. We present an analysis of the SPI data and focus on the description of the high-energy (> 20 keV) output of the source. We found that the 20 - 500 keV spectral emission of GRS 1915+105 was bound between two states. It seems that these high-energy states are not correlated with the temporal behavior of the source, suggesting that there is no direct link between the macroscopic characteristics of the coronal plasma and the the variability of the accretion flow. All spectra are well fitted by a thermal comptonization component plus an extra high-energy powerlaw. This confirms the presence of thermal and non-thermal electrons around the black hole.Comment: 7 pages, 8 figures, 2 tables; accepted (09/11/2008) for publication in A&

A Graph Based Backtracking Algorithm for Solving General CSPs

Many AI tasks can be formalized as constraint satisfaction problems (CSPs), which involve finding values for variables subject to constraints. While solving a CSP is an NP-complete task in general, tractable classes of CSPs have been identified based on the structure of the underlying constraint graphs. Much effort has been spent on exploiting structural properties of the constraint graph to improve the efficiency of finding a solution. These efforts contributed to development of a class of CSP solving algorithms called decomposition algorithms. The strength of CSP decomposition is that its worst-case complexity depends on the structural properties of the constraint graph and is usually better than the worst-case complexity of search methods. Its practical application is limited, however, since it cannot be applied if the CSP is not decomposable. In this paper, we propose a graph based backtracking algorithm called omega-CDBT, which shares merits and overcomes the weaknesses of both decomposition and search approaches

Minimum Energy Broadcast and Disk Cover in Grid Wireless Networks

Abstract. The Minimum Energy Broadcast problem consists in finding the minimum-energy range assignment for a given set S of n stations of an ad hoc wireless network that allows a source station to perform broadcast operations over S. We prove a nearly tight asymptotical bound on the optimal cost for the Minimum Energy Broadcast problem on square grids. We emphasize that finding tight bounds for this problem restriction is far to be easy: it involves the Gauss’s Circle problem and the Apollonian Circle Packing. We also derive near-tight bounds for the Bounded-Hop version of this problem. Our results imply that the best-known heuristic, the MST-based one, for the Minimum Energy Broadcast problem is far to achieve optimal solutions (even) on very regular, well-spread instances: its worst-case approximation ratio is about pi and it yields Ω( n) hops. As a by product, we get nearly tight bounds for the Minimum Disk Cover problem and for its restriction in which the allowed disks must have non-constant radius. Finally, we emphasize that our upper bounds are obtained via polynomial time constructions.

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