The complexity of approximating conservative counting CSPs

We study the complexity of approximately solving the weighted counting constraint satisfaction problem #CSP(F). In the conservative case, where F contains all unary functions, there is a classification known for the case in which the domain of functions in F is Boolean. In this paper, we give a classification for the more general problem where functions in F have an arbitrary finite domain. We define the notions of weak log-modularity and weak log-supermodularity. We show that if F is weakly log-modular, then #CSP(F)is in FP. Otherwise, it is at least as difficult to approximate as #BIS, the problem of counting independent sets in bipartite graphs. #BIS is complete with respect to approximation-preserving reductions for a logically-defined complexity class #RHPi1, and is believed to be intractable. We further sub-divide the #BIS-hard case. If F is weakly log-supermodular, then we show that #CSP(F) is as easy as a (Boolean) log-supermodular weighted #CSP. Otherwise, we show that it is NP-hard to approximate. Finally, we give a full trichotomy for the arity-2 case, where #CSP(F) is in FP, or is #BIS-equivalent, or is equivalent in difficulty to #SAT, the problem of approximately counting the satisfying assignments of a Boolean formula in conjunctive normal form. We also discuss the algorithmic aspects of our classification.Comment: Minor revisio

Galois correspondence for counting quantifiers

We introduce a new type of closure operator on the set of relations, max-implementation, and its weaker analog max-quantification. Then we show that approximation preserving reductions between counting constraint satisfaction problems (#CSPs) are preserved by these two types of closure operators. Together with some previous results this means that the approximation complexity of counting CSPs is determined by partial clones of relations that additionally closed under these new types of closure operators. Galois correspondence of various kind have proved to be quite helpful in the study of the complexity of the CSP. While we were unable to identify a Galois correspondence for partial clones closed under max-implementation and max-quantification, we obtain such results for slightly different type of closure operators, k-existential quantification. This type of quantifiers are known as counting quantifiers in model theory, and often used to enhance first order logic languages. We characterize partial clones of relations closed under k-existential quantification as sets of relations invariant under a set of partial functions that satisfy the condition of k-subset surjectivity. Finally, we give a description of Boolean max-co-clones, that is, sets of relations on {0,1} closed under max-implementations.Comment: 28 pages, 2 figure

Counting Constraint Satisfaction Problems

This chapter surveys counting Constraint Satisfaction Problems (counting CSPs, or #CSPs) and their computational complexity. It aims to provide an introduction to the main concepts and techniques, and present a representative selection of results and open problems. It does not cover holants, which are the subject of a separate chapter

Boolean approximate counting CSPs with weak conservativity, and implications for ferromagnetic two-spin

We analyse the complexity of approximate counting constraint satisfactions problems $\mathrm{\#CSP}(\mathcal{F})$, where $\mathcal{F}$ is a set of nonnegative rational-valued functions of Boolean variables. A complete classification is known in the conservative case, where $\mathcal{F}$ is assumed to contain arbitrary unary functions. We strengthen this result by fixing any permissive strictly increasing unary function and any permissive strictly decreasing unary function, and adding only those to $\mathcal{F}$: this is weak conservativity. The resulting classification is employed to characterise the complexity of a wide range of two-spin problems, fully classifying the ferromagnetic case. In a further weakening of conservativity, we also consider what happens if only the pinning functions are assumed to be in $\mathcal{F}$ (instead of the two permissive unaries). We show that any set of functions for which pinning is not sufficient to recover the two kinds of permissive unaries must either have a very simple range, or must satisfy a certain monotonicity condition. We exhibit a non-trivial example of a set of functions satisfying the monotonicity condition.Comment: 37 page

Complexity dichotomies for approximations of counting problems

Αυτή η διπλωματική αποτελεί μια επισκόπηση θεωρημάτων διχοτομίας για υπολογιστικά προβλήματα, και ειδικότερα προβλήματα μέτρησης. Θεώρημα διχοτομίας στην υπολογιστική πολυπλοκότητα είναι ένας πλήρης χαρασκτηρισμός των μελών μιας κλάσης προβλημάτων, σε υπολογιστικά δύσκολα και υπολογιστικά εύκολα, χωρίς να υπάρχουν προβλήματα ενδιάμεσης πολυπλοκότητας στην κλάση αυτή. Λόγω του θεωρήματος του Ladner, δεν μπορούμε να έχουμε διχοτομία για ολόκληρες τις κλάσεις NP και #P, παρόλα αυτά υπάρχουν μεγάλες υποκλάσεις της NP (#P) για τις οποίες ισχύουν θεωρήματα διχοτομίας. Συνεχίζουμε με την εκδοχή απόφασης του προβλήματος ικανοποίησης περιορισμών (CSP), μία κλάση προβλήμάτων της NP στην οποία δεν εφαρμόζεται το θεώρημα του Ladner. Δείχνουμε τα θεωρήματα διχοτομίας που υπάρχουν για ειδικές περιπτώσεις του CSP. Στη συνέχεια επικεντρωνόμαστε στα προβλήματα μέτρησης παρουσιάζοντας τα παρακάτω μοντέλα: Ομομορφισμοί γράφων, μετρητικό πρόβλημα ικανοποίησης περιορισμών (#CSP), και προβλήματα Holant. Αναφέρουμε τα θεωρήματα διχοτομίας που γνωρίζουμε γι' αυτά. Στο τελευταίο και κύριο κεφάλαιο, χαλαρώνουμε την απαίτηση ακριβών υπολογισμών, και αρκούμαστε στην προσέγγιση των προβλημάτων. Παρουσιάζουμε τα μέχρι σήμερα γνωστά θεωρήματα κατάταξης για το #CSP. Πολλά ερωτήματα στην περιοχή παραμένουν ανοιχτά. Το παράρτημα είναι μια εισαγωγή στους ολογραφικούς αλγορίθμους, μία πρόσφατη αλγοριθμική τεχνική για την εύρεση πολυωνυμικών αλγορίθμων (ακριβείς υπολογισμοί) σε προβλήματα μέτρησης.This thesis is a survey of dichotomy theorems for computational problems, focusing in counting problems. A dichotomy theorem in computational complexity, is a complete classification of the members of a class of problems, in computationally easy and computationally hard, with the set of problems of intermediate complexity being empty. Due to Ladner's theorem we cannot find a dichotomy theorem for the whole classes NP and #P, however there are large subclasses of NP (#P), that model many "natural" problems, for which dichotomy theorems exist. We continue with the decision version of constraint satisfaction problems (CSP), a class of problems in NP, for which Ladner's theorem doesn't apply. We obtain a dichotomy theorem for some special cases of CSP. We then focus on counting problems presenting the following frameworks: graph homomorphisms, counting constraint satisfaction (#CSP) and Holant problems; we provide the known dichotomies for these frameworks. In the last and main chapter of this thesis we relax the requirement of exact computation, and settle in approximating the problems. We present the known cassification theorems for cases of #CSP. Many questions in terms of approximate counting problems remain open. The appendix introduces a recent technique for obtaining exact polynomial-time algorithms for counting problems, namely the holographic algorithms

The computational complexity of approximation of partition functions

This thesis studies the computational complexity of approximately evaluating partition functions. For various classes of partition functions, we investigate whether there is an FPRAS: a fully polynomial randomised approximation scheme. In many of these settings we also study “expressibility”, a simple notion of defining a constraint by combining other constraints, and we show that the results cannot be extended by expressibility reductions alone. The main contributions are: -� We show that there is no FPRAS for evaluating the partition function of the hard-core gas model on planar graphs at fugacity 312, unless RP = NP. -� We generalise an argument of Jerrum and Sinclair to give FPRASes for a large class of degree-two Boolean #CSPs. -� We initiate the classification of degree-two Boolean #CSPs where the constraint language consists of a single arity 3 relation. -� We show that the complexity of approximately counting downsets in directed acyclic graphs is not affected by restricting to graphs of maximum degree three. -� We classify the complexity of degree-two #CSPs with Boolean relations and weights on variables. -� We classify the complexity of the problem #CSP(F) for arbitrary finite domains when enough non-negative-valued arity 1 functions are in the constraint language. -� We show that not all log-supermodular functions can be expressed by binary logsupermodular functions in the context of #CSPs