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Constraint Satisfaction Problems: Probabilistic Approach and Applications to Social Choice Theory

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Constraint Satisfaction Problems: Probabilistic Approach and Applications to Social Choice Theory Constraint Satisfaction Problems: Probabilistic Approach and Applications to Social Choice Theory John Livieratos, Ph.D. December 4, 2020 Department of Mathematics, National and Kapodistrian University of Athens 2 Committee Members Josep D´ıaz, Computer Science Department, Universitat Polit`ecnica de Catalunya, Barcelona. Marcel Fernandez, Department of Network Engineering, Universitat Politecnica de Catalunya, Spain. Dimitris Fotakis, Division of Computer Science, School of Electrical and Computer Engineering, NTUA. Lefteris Kirousis (advisor), Department of Mathematics, NKUA. Phokion G. Kolaitis, Computer Science Department UC Santa Cruz and IBM Research - Almaden. Stavros Kolliopoulos, Department of Informatics and Telecommunications, NKUA. Dimitrios M. Thilikos, Department of Mathematics, NKUA. 3 Dedicated to Fani, Susana, John and Toula, who would have certainly appreciated all that and to Sara, for continuing to be around. 4 Abstract In this Ph.D thesis, we work in one of the most well studied class of problems in Computer Science, that of Constraint Satisfaction Problems (CSPs). In one of their usual formulations, CSPs consist of a set of variables that take values in a common domain set. Groups of variables are tied by constraints that restrict the possible combinations of values that the variables can have in a solution. In such a setting, there are many objectives that one might be interested in: checking if there is a solution, finding or approximating one, or considering how fast an algorithmic procedure can do all that. The framework of CSPs is broad enough to model a great number of in- teresting problems in computer science, like the satisfiability of propositional formulas and graph coloring problems. It is also a very developed field on its own accord, with a lot of interesting results that classify the computational complexity of classes of CSPs and delineate the bounds between tractability and NP-hardness. The machinery used to tackle such problems is broad, including polynomial-time algorithms that solve classes of CSPs, random- ized ones that find or approximate solutions given some conditions that the CSP in question must satisfy and algebraic manipulations that allow us to relate their computational complexity with structural properties of their sets of constraints. We begin with an overview of various approaches to CSPs: defining them in the language of Propositional, First or Second Order Logic and via homo- morphisms and we consider the subclass of multi-sorted CSPs, that is CSPs whose variables are divided in di↵erent sorts and take values in independent domains. Some of these variations are discussed to show the versatility of CSPs and provide some context to our work, while others are utilized to prove our results. The first part of our results concerns what is known as the probabilistic approach. Here, we devise randomized algorithms that (i) prove conditions 5 6 that guarantee the existence of solutions to a given instance of a CSP and (ii) in case a solution exists, find it efficiently. A solution in this setting is usually expressed as a point in a probability space such that no event, from a set of events that are deemed as “undesirable”, occurs. We work with the seminal Lov´aszLocal Lemma (LLL) and its variation, Shearer’s Lemma, which, given some bounds concerning the probabilities of undesirable events and the way these events depend on each other, provide conditions that imply all the events can be avoided with positive probability. A solution in this setting, is a point in a probability space such that none of the events occur. All our work is situated in the variable framework of Moser and Tardos, where the events are assumed to be defined upon independent random variables. Although this is a restriction of the general setting, it is a broad framework that easily translates to the language of CSPs and that is particularly handy for algorithmic purposes. Specifically, we define two new notions of dependency between the events, the variable-directed lopsidependency (VDL) and the directed dependency (d- dependency), which are specifically tailored to facilitate the algorithmic ma- nipulation of events that are negatively correlated. It is quite common in practice to depict dependencies between the events by a dependency graph, where the nodes correspond to the events and unconnected events are con- sidered independent. We thus discuss how the directed dependency graphs that our notions give rise to, relate with other such graphs in the bibliogra- phy. Furthermore, we show that the d-dependency condition gives rise to a sparser dependency graph than other known such conditions in the variable framework, thus allowing for stronger versions of the LLL to be proven. We then proceed to prove the simple version of the LLL of the VDL condition. That is, we design an algorithm which, if the probabilities of the events are upper bounded by a common number p [0, 1), the VDL graph has maximum degree d and epd 1, efficiently finds2 a point in the probability space such that none of the events occur, thus showing at the same time that such a point must exist in the first place. We also prove the more general asymmetric version of the LLL for the d-dependency graph, where the probability of each event is bounded by a number relating to the probabilities of the events depending on it. We then prove the even stronger Shearer’s lemma for the underlying undirected graph of the d-dependency one, which bounds the probabilities of the events by polynomials defined over the independent sets of the graph. The proofs for these versions of the LLL and Shearer’s lemma employ a 7 direct probabilistic approach, in which we show that the probability that our algorithms last for at least n steps is inverse exponential to n, by express- ing it by a recurrence relation which we subsequently solve using advanced analytic tools, such as Bender and Richmond’s Lagrange Inversion Formula and Gelfand’s Formula for the spectral radius of matrices. In contrast, most extant work bounds only the expectation of the steps performed by such algorithms. We believe that this fact is interesting in each own accord. Nev- ertheless, we have applied our method in two interesting combinatorial prob- lems. First, we show that 2∆ 1colorssuffice to acyclicaly color the edges of a graph with maximum degree− ∆, that is, we want the resulting coloring to contain neither incident edges with the same color, nor bichromatic cy- cles. We thus match the best possible bound for Moser-like algorithms, as observed by Cai et al. [Acyclic edge colourings of graphs with large girth. Random Structures & Algorithms, 50(4):511–533, 2017]. We also show how to explicitly construct binary c-separating codes whose rate matches the op- timal known one. c-separating codes are M n matrices over some alphabet , where, in any two sets U and V of at most⇥ c rows, there is at least one columnQ such that the set of elements in U is disjoint with that in V . Al- though such codes are very useful for applications, explicit constructions are scarce. The second part of our results lies in Social Choice Theory and, specifi- cally, in Judgment Aggregation, where a group of agents collectively decides a set of issues and where, both the individual positions of each agent and the aggregated positions (the social outcome)needstoadheretosomere- strictions that reflect logical consistency requirements. The aim is to find aggregating procedures that preserve these requirements and do not degen- erate to dictatorships, that is aggregators that always output the positions of a specific agent. Firstly, we consider the case where these restrictions are expressed by a set of m-ary vectors X over some finite domain , where m is the number of issues to be decided. That is, m contains the allowedD combinations of votes over the issues and a vector not in X is deemed “irrational”. In this setting, we characterize possibility domains, that is sets X where non-dictatorial ag- gregation is possible, via the types of aggregators they admit. Furthermore, we provide an analogous characterization for a subclass of possibility do- mains we named uniform possibility domains, which are domains that admit aggregators that are not dictatorial even when restricted to any issue and any binary subset of allowed positions. We also show that uniform possi- 8 bility domains give rise to tractable multi-sorted CSPs, while any domain that is not uniform, gives rise to NP-complete multi-sorted CSPs, thus tying the possibility of non-dictatorial aggregation with a dichotomy result in the complexity of multi-sorted CSPs. We then proceed to consider Boolean such domains, that are given as the sets of models of propositional formulas, which, in the bibliography, are called integrity constraints. We provide syntactic characterizations for integrity constraints that give rise to (uniform) possibility domains and also to domains admitting a variety of non-dictatorial aggregators with specific properties that have appeared in the bibliography. We also show how to efficiently identify integrity constraints of these types and how to efficiently construct such constraints given a Boolean domain X of the corresponding type. Finally, we turn our attention to the problem of recognizing if a domain admits a (uniform) non-dictatorial aggregator. In case X is provided explic- itly, as a set of m-ary vectors, we design polynomial-time algorithms that solve this problem. In case X is Boolean and provided either via an integrity constraint, or, as in the original framework of Judgment Aggregation, as the set of consistent evaluations of a set of propositional formulas, called an agenda, we provide upper and lower complexity bounds in the polynomial hierarchy. We extend these results to include the cases where X admits non-dictatorial aggregators with desirable properties.
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