We survey recent results about enumerating with constant delay the answers to a query over a database. More precisely, we focus on the case when enumeration can be achieved with a preprocessing running in time linear in the size of the database, followed by an enumeration process outputting the answers one by one with constant time between any consecutive outputs. We survey classes of databases and classes of queries for which this is possible. We also mention related problems such as computing the number of answers or sampling the set of answers.

/pdf/enumerating-with-constant-delay-the-answers-to-a-query-5bn2a6o88t.pdf

Enumerating with constant delay the answers to a query

String problems arise in various applications ranging from text mining to biological sequence analysis. Many string problems are NP-hard. This motivates the search for (fixed-parameter) tractable special cases of these problems. We survey parameterized and multivariate algorithmics results for NP-hard string problems and identify challenges for future research.

/pdf/multivariate-algorithmics-for-np-hard-string-problems-1a80pzjk10.pdf

Multivariate Algorithmics for NP-Hard String Problems

The aim of the paper is to examine the computational complexity and algorithmics of enumeration, the task to output all solutions of a given problem, from the point of view of parameterized complexity. First, we define formally different notions of efficient enumeration in the context of parameterized complexity: FPT-enumeration and delayFPT. Second, we show how different algorithmic paradigms can be used in order to get parameter-efficient enumeration algorithms in a number of examples. These paradigms use well-known principles from the design of parameterized decision as well as enumeration techniques, like for instance kernelization and self-reducibility. The concept of kernelization, in particular, leads to a characterization of fixed-parameter tractable enumeration problems. Furthermore, we study the parameterized complexity of enumerating all models of Boolean formulas having weight at least k, where k is the parameter, in the famous Schaefer's framework. We consider propositional formulas that are conjunctions of constraints taken from a fixed finite set Γ. Given such a formula and an integer k, we are interested in enumerating all the models of the formula that have weight at least k. We obtain a dichotomy classification and prove that, according to the properties of the constraint language Γ, either one can enumerate all such models in delayFPT, or no such delayFPT enumeration algorithm exists under some complexity-theoretic assumptions.

pdf/paradigms-for-parameterized-enumeration-50irqophc8.pdf

Paradigms for Parameterized Enumeration

We study fragments D(k∀) and D(k-dep) of dependence logic defined either by restricting the number k of universal quantifiers or the width of dependence atoms in formulas. We find the sublogics of existential second-order logic corresponding to these fragments of dependence logic. We also show that, for any fixed signature, the fragments D(k∀) give rise to an infinite hierarchy with respect to expressive power. On the other hand, for the fragments D(k-dep), a hierarchy theorem is otained only in the case the signature is also allowed to vary. For any fixed signature, this question is open and is related to the so-called Spectrum Arity Hierarchy Conjecture.

pdf/hierarchies-in-dependence-logic-4scdcrldpq.pdf

Hierarchies in Dependence Logic

In this paper, we enumerate enumeration problems and algorithms. This survey is under construction. If you know some results not in this survey or there is anything wrong, please let me know.

Enumeration of Enumeration Algorithms.

Recently, Creignou et al. (Theory Comput. Syst. 2017), introduced the class DelayFPT into parameterised complexity theory in order to capture the notion of efficiently solvable parameterised enumeration problems. In this paper, we propose a framework for parameterised ordered enumeration and will show how to obtain enumeration algorithms running with an FPT delay in the context of general modification problems. We study these problems considering two different orders of solutions, namely, lexicographic order and order by size. Furthermore, we present two generic algorithmic strategies. The first one is based on the well-known principle of self-reducibility and is used in the context of lexicographic order. The second one shows that the existence of a neighbourhood structure among the solutions implies the existence of an algorithm running with FPT delay which outputs all solutions ordered non-decreasingly by their size.

https://hal.archives-ouvertes.fr/hal-02439368/document

Parameterised Enumeration for Modification Problems

It is well known that monadic second-order logic with linear order captures exactly regular languages. On the other hand, if addition is allowed, then J.F.Lynch has proved that existential monadic secondorder logic captures at least all the languages in NTIME(n), and then expresses some NP-complete languages (e.g. knapsack problem).

Monadic Logical Definability of NP-Complete Problems

https://hal.archives-ouvertes.fr/hal-00786148/document

A logical approach to locality in pictures languages

Recently the class \(DelayFPT\) has been introduced into parameterized complexity in order to capture the notion of efficiently solvable parameterized enumeration problems. In this paper we propose a framework for parameterized ordered enumeration and will show how to obtain \(DelayFPT\) enumeration algorithms in the context of graph modification problems. We study these problems considering two different orders of solutions, lexicographic and by size. We present generic algorithmic strategies: The first one is based on the well-known principle of self-reducibility in the context of lexicographic order. The second one shows that the existence of some neighborhood structure among the solutions implies the existence of a \(DelayFPT\) algorithm which outputs all solutions ordered non-decreasingly by their size.

Parameterized Enumeration for Modification Problems

Monadic logical definability of NP-complete problems

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