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

Tractability in Constraint Satisfaction Problems: A Survey

International audienceEven though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP

Backdoor Sets for CSP

A backdoor set of a CSP instance is a set of variables whose instantiation moves the instance into a fixed class of tractable instances (an island of tractability). An interesting algorithmic task is to find a small backdoor set efficiently: once it is found we can solve the instance by solving a number of tractable instances. Parameterized complexity provides an adequate framework for studying and solving this algorithmic task, where the size of the backdoor set provides a natural parameter. In this survey we present some recent parameterized complexity results on CSP backdoor sets, focusing on backdoor sets into islands of tractability that are defined in terms of constraint languages

Combining Treewidth and Backdoors for CSP

We show that CSP is fixed-parameter tractable when parameterized by the treewidth of a backdoor into any tractable CSP problem over a finite constraint language. This result combines the two prominent approaches for achieving tractability for CSP: (i) structural restrictions on the interaction between the variables and the constraints and (ii) language restrictions on the relations that can be used inside the constraints. Apart from defining the notion of backdoor-treewidth and showing how backdoors of small treewidth can be used to efficiently solve CSP, our main technical contribution is a fixed-parameter algorithm that finds a backdoor of small treewidth

A Graph Algorithmic Approach to Separate Direct from Indirect Neural Interactions

Network graphs have become a popular tool to represent complex systems composed of many interacting subunits; especially in neuroscience, network graphs are increasingly used to represent and analyze functional interactions between neural sources. Interactions are often reconstructed using pairwise bivariate analyses, overlooking their multivariate nature: it is neglected that investigating the effect of one source on a target necessitates to take all other sources as potential nuisance variables into account; also combinations of sources may act jointly on a given target. Bivariate analyses produce networks that may contain spurious interactions, which reduce the interpretability of the network and its graph metrics. A truly multivariate reconstruction, however, is computationally intractable due to combinatorial explosion in the number of potential interactions. Thus, we have to resort to approximative methods to handle the intractability of multivariate interaction reconstruction, and thereby enable the use of networks in neuroscience. Here, we suggest such an approximative approach in the form of an algorithm that extends fast bivariate interaction reconstruction by identifying potentially spurious interactions post-hoc: the algorithm flags potentially spurious edges, which may then be pruned from the network. This produces a statistically conservative network approximation that is guaranteed to contain non-spurious interactions only. We describe the algorithm and present a reference implementation to test its performance. We discuss the algorithm in relation to other approximative multivariate methods and highlight suitable application scenarios. Our approach is a tractable and data-efficient way of reconstructing approximative networks of multivariate interactions. It is preferable if available data are limited or if fully multivariate approaches are computationally infeasible.Comment: 24 pages, 8 figures, published in PLOS On

Guarantees and Limits of Preprocessing in Constraint Satisfaction and Reasoning

We present a first theoretical analysis of the power of polynomial-time preprocessing for important combinatorial problems from various areas in AI. We consider problems from Constraint Satisfaction, Global Constraints, Satisfiability, Nonmonotonic and Bayesian Reasoning under structural restrictions. All these problems involve two tasks: (i) identifying the structure in the input as required by the restriction, and (ii) using the identified structure to solve the reasoning task efficiently. We show that for most of the considered problems, task (i) admits a polynomial-time preprocessing to a problem kernel whose size is polynomial in a structural problem parameter of the input, in contrast to task (ii) which does not admit such a reduction to a problem kernel of polynomial size, subject to a complexity theoretic assumption. As a notable exception we show that the consistency problem for the AtMost-NValue constraint admits a polynomial kernel consisting of a quadratic number of variables and domain values. Our results provide a firm worst-case guarantees and theoretical boundaries for the performance of polynomial-time preprocessing algorithms for the considered problems.Comment: arXiv admin note: substantial text overlap with arXiv:1104.2541, arXiv:1104.556