scholarly journals Lower Bounds and Faster Algorithms for Equality Constraints

2020 ◽  
Author(s):  
Peter Jonsson ◽  
Victor Lagerkvist
Keyword(s):  
Lower Bounds ◽  
Constraint Satisfaction Problems ◽  
Satisfiability Modulo Theories ◽  
Infinite Domain ◽  
First Order ◽  
Fine Grained ◽  
Background Theory ◽  
Definable Relations ◽  
Np Complete ◽  
Rule Out

We study the fine-grained complexity of NP-complete, infinite-domain constraint satisfaction problems (CSPs) parameterised by a set of first-order definable relations (with equality). Such CSPs are of central importance since they form a subclass of any infinite-domain CSP parameterised by a set of first-order definable relations. We prove that under the randomised exponential-time hypothesis it is not possible to find c > 1 such that a CSP over an arbitrary finite equality language is solvable in O(c^n) time (n is the number of variables). Stronger lower bounds are possible for infinite equality languages where we rule out the existence of 2^o(n log n) time algorithms; a lower bound which also extends to satisfiability modulo theories solving for an arbitrary background theory. Despite these lower bounds we prove that for each c > 1 there exists an NP-hard equality CSP solvable in O(c^n) time. Lower bounds like these immediately ask for closely matching upper bounds, and we prove that a CSP over a finite equality language is always solvable in O(c^n) time for a fixed c.

scholarly journals Fine-Grained Time Complexity of Constraint Satisfaction Problems

2021 ◽  
Vol 13 (1) ◽  
pp. 1-32
Author(s):  
Peter Jonsson ◽  
Victor Lagerkvist ◽  
Biman Roy
Keyword(s):  
Constraint Satisfaction ◽  
Time Complexity ◽  
Constraint Satisfaction Problem ◽  
Constraint Satisfaction Problems ◽  
Finite Domain ◽  
Infinite Domain ◽  
Worst Case ◽  
Fine Grained ◽  
Restricted Problems ◽  
Np Complete

We study the constraint satisfaction problem (CSP) parameterized by a constraint language Γ (CSPΓ) and how the choice of Γ affects its worst-case time complexity. Under the exponential-time hypothesis (ETH), we rule out the existence of subexponential algorithms for finite-domain NP-complete CSPΓ problems. This extends to certain infinite-domain CSPs and structurally restricted problems. For CSPs with finite domain D and where all unary relations are available, we identify a relation S D such that the time complexity of the NP-complete problem CSP({ S D }) is a lower bound for all NP-complete CSPs of this kind. We also prove that the time complexity of CSP({ S D }) strictly decreases when |D| increases (unless the ETH is false) and provide stronger complexity results in the special case when |D|=3.

The (Coarse) Fine-Grained Structure of NP-Hard SAT and CSP Problems

2022 ◽  
Vol 14 (1) ◽  
pp. 1-54
Author(s):  
Victor Lagerkvist ◽  
Magnus Wahlström
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Constraint Satisfaction Problems ◽  
Close Connection ◽  
Algebraic Characterization ◽  
Sat Problem ◽  
Fine Grained ◽  
Algebraic Framework ◽  
Fine Grained Structure ◽  
Np Complete

We study the fine-grained complexity of NP-complete satisfiability (SAT) problems and constraint satisfaction problems (CSPs) in the context of the strong exponential-time hypothesis (SETH) , showing non-trivial lower and upper bounds on the running time. Here, by a non-trivial lower bound for a problem SAT (Γ) (respectively CSP (Γ)) with constraint language Γ, we mean a value c 0 > 1 such that the problem cannot be solved in time O ( c n ) for any c < c 0 unless SETH is false, while a non-trivial upper bound is simply an algorithm for the problem running in time O ( c n ) for some c < 2. Such lower bounds have proven extremely elusive, and except for cases where c 0 =2 effectively no such previous bound was known. We achieve this by employing an algebraic framework, studying constraint languages Γ in terms of their algebraic properties. We uncover a powerful algebraic framework where a mild restriction on the allowed constraints offers a concise algebraic characterization. On the relational side we restrict ourselves to Boolean languages closed under variable negation and partial assignment, called sign-symmetric languages. On the algebraic side this results in a description via partial operations arising from system of identities, with a close connection to operations resulting in tractable CSPs, such as near unanimity operations and edge operations . Using this connection we construct improved algorithms for several interesting classes of sign-symmetric languages, and prove explicit lower bounds under SETH. Thus, we find the first example of an NP-complete SAT problem with a non-trivial algorithm which also admits a non-trivial lower bound under SETH. This suggests a dichotomy conjecture with a close connection to the CSP dichotomy theorem: an NP-complete SAT problem admits an improved algorithm if and only if it admits a non-trivial partial invariant of the above form.

scholarly journals A fine-grained arc-consistency algorithm for non-normalized constraint satisfaction problems

2011 ◽  
Vol 21 (4) ◽  
pp. 733-744 ◽  
Author(s):  
Marlene Arangú ◽  
Miguel Salido
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Programming ◽  
Real Life ◽  
Search Space ◽  
Constraint Satisfaction Problems ◽  
Fine Grained ◽  
Arc Consistency ◽  
Life Problems ◽  
Programming Techniques ◽  
Np Complete

A fine-grained arc-consistency algorithm for non-normalized constraint satisfaction problems Constraint programming is a powerful software technology for solving numerous real-life problems. Many of these problems can be modeled as Constraint Satisfaction Problems (CSPs) and solved using constraint programming techniques. However, solving a CSP is NP-complete so filtering techniques to reduce the search space are still necessary. Arc-consistency algorithms are widely used to prune the search space. The concept of arc-consistency is bidirectional, i.e., it must be ensured in both directions of the constraint (direct and inverse constraints). Two of the most well-known and frequently used arc-consistency algorithms for filtering CSPs are AC3 and AC4. These algorithms repeatedly carry out revisions and require support checks for identifying and deleting all unsupported values from the domains. Nevertheless, many revisions are ineffective, i.e., they cannot delete any value and consume a lot of checks and time. In this paper, we present AC4-OP, an optimized version of AC4 that manages the binary and non-normalized constraints in only one direction, storing the inverse founded supports for their later evaluation. Thus, it reduces the propagation phase avoiding unnecessary or ineffective checking. The use of AC4-OP reduces the number of constraint checks by 50% while pruning the same search space as AC4. The evaluation section shows the improvement of AC4-OP over AC4, AC6 and AC7 in random and non-normalized instances.

scholarly journals Model Theory of XPath on Data Trees. Part I: Bisimulation and Characterization

2015 ◽  
Vol 53 ◽  
pp. 271-314 ◽  
Author(s):  
Diego Figueira ◽  
Santiago Figueira ◽  
Carlos Areces
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Finite Alphabet ◽  
Infinite Domain ◽  
Complexity Measures ◽  
First Order ◽  
Fine Grained ◽  
Data Value ◽  
Formula Complexity ◽  
Data Trees

We investigate model theoretic properties of XPath with data (in)equality tests over the class of data trees, i.e., the class of trees where each node contains a label from a finite alphabet and a data value from an infinite domain. We provide notions of (bi)simulations for XPpath logics containing the child, parent, ancestor and descendant axes to navigate the tree. We show that these notions precisely characterize the equivalence relation associated with each logic. We study formula complexity measures consisting of the number of nested axes and nested subformulas in a formula; these notions are akin to the notion of quantifier rank in first-order logic. We show characterization results for fine grained notions of equivalence and (bi)simulation that take into account these complexity measures. We also prove that positive fragments of these logics correspond to the formulas preserved under (non-symmetric) simulations. We show that the logic including the child axis is equivalent to the fragment of first-order logic invariant under the corresponding notion of bisimulation. If upward navigation is allowed the characterization fails but a weaker result can still be established. These results hold both over the class of possibly infinite data trees and over the class of finite data trees. Besides their intrinsic theoretical value, we argue that bi-simulations are useful tools to prove (non)expressivity results for the logics studied here, and we substantiate this claim with examples.

scholarly journals Chapter 33. Satisfiability Modulo Theories

2021 ◽  
Author(s):  
Clark Barrett ◽  
Roberto Sebastiani ◽  
Sanjit A. Seshia ◽  
Cesare Tinelli
Keyword(s):  
Relevant Literature ◽  
Finite Size ◽  
Decision Procedures ◽  
Research Field ◽  
Satisfiability Modulo Theories ◽  
Practical Applications ◽  
First Order ◽  
Background Theory ◽  
Sat Solver ◽  
And Function

Applications in artificial intelligence, formal verification, and other areas have greatly benefited from the recent advances in SAT. It is often the case, however, that applications in these fields require determining the satisfiability of formulas in more expressive logics such as first-order logic. Also, these applications typically require not general first-order satisfiability, but rather satisfiability with respect to some background theory, which fixes the interpretations of certain predicate and function symbols. For many background theories, specialized methods yield decision procedures for the satisfiability of quantifier-free formulas or some subclass thereof. Specialized decision procedures have been discovered for a long and still growing list of theories with practical applications. These include the theory of equality, various theories of arithmetic, and certain theories of arrays, as well as theories of lists, tuples, records, and bit-vectors of a fixed or arbitrary finite size. The research field concerned with determining the satisfiability of formulas with respect to some background theory is called Satisfiability Modulo Theories (SMT). This chapter provides a brief overview of SMT together with references to the relevant literature for a deeper study. It begins with an overview of techniques for solving SMT problems by encodings to SAT. The rest of the chapter is mostly concerned with an alternative approach in which a SAT solver is integrated with a separate decision procedure (called a theory solver) for conjunctions of literals in the background theory. After presenting this approach as a whole, the chapter provides more details on how to construct and combine theory solvers, and discusses several extensions and enhancements.

scholarly journals An exact expression of positive periodic solution for a first-order singular equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yun Xin ◽  
Xiaoxiao Cui ◽  
Jie Liu
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Lower Bounds ◽  
Topological Degree ◽  
Order Differential Equation ◽  
Positive Periodic Solution ◽  
Exact Expression ◽  
Upper And Lower Bounds ◽  
Singular Equation ◽  
First Order

Abstract The main purpose of this paper is to obtain an exact expression of the positive periodic solution for a first-order differential equation with attractive and repulsive singularities. Moreover, we prove the existence of at least one positive periodic solution for this equation with an indefinite singularity by applications of topological degree theorem, and give the upper and lower bounds of the positive periodic solution.

scholarly journals Ontology-Mediated Querying with the Description Logic EL: Trichotomy and Linear Datalog Rewritability

2017 ◽  
Author(s):  
Carsten Lutz ◽  
Leif Sabellek
Keyword(s):  
Description Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Data Complexity ◽  
First Order ◽  
Fine Grained ◽  
Linear Recursion

We consider ontology-mediated queries (OMQs) based on an EL ontology and an atomic query (AQ), provide an ultimately fine-grained analysis of data complexity and study rewritability into linear Datalog-aiming to capture linear recursion in SQL. Our main results are that every such OMQ is in AC0, NL-complete or PTime-complete, and that containment in NL coincides with rewritability into linear Datalog (whereas containment in AC0 coincides with rewritability into first-order logic). We establish natural characterizations of the three cases, show that deciding linear Datalog rewritability (as well as the mentioned complexities) is ExpTime-complete, give a way to construct linear Datalog rewritings when they exist, and prove that there is no constant bound on the arity of IDB relations in linear Datalog rewritings.

scholarly journals Optimal Bounds for the k -cut Problem

2022 ◽  
Vol 69 (1) ◽  
pp. 1-18
Author(s):  
Anupam Gupta ◽  
David G. Harris ◽  
Euiwoong Lee ◽  
Jason Li
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Connected Components ◽  
Small Integer ◽  
Main Ingredient ◽  
Fine Grained ◽  
Optimal Bounds ◽  
Sunflower Lemma ◽  
General Graphs ◽  
New Algorithms

In the k -cut problem, we want to find the lowest-weight set of edges whose deletion breaks a given (multi)graph into k connected components. Algorithms of Karger and Stein can solve this in roughly O ( n 2k ) time. However, lower bounds from conjectures about the k -clique problem imply that Ω ( n (1- o (1)) k ) time is likely needed. Recent results of Gupta, Lee, and Li have given new algorithms for general k -cut in n 1.98k + O(1) time, as well as specialized algorithms with better performance for certain classes of graphs (e.g., for small integer edge weights). In this work, we resolve the problem for general graphs. We show that the Contraction Algorithm of Karger outputs any fixed k -cut of weight α λ k with probability Ω k ( n - α k ), where λ k denotes the minimum k -cut weight. This also gives an extremal bound of O k ( n k ) on the number of minimum k -cuts and an algorithm to compute λ k with roughly n k polylog( n ) runtime. Both are tight up to lower-order factors, with the algorithmic lower bound assuming hardness of max-weight k -clique. The first main ingredient in our result is an extremal bound on the number of cuts of weight less than 2 λ k / k , using the Sunflower lemma. The second ingredient is a fine-grained analysis of how the graph shrinks—and how the average degree evolves—in the Karger process.

scholarly journals Probabilistic Inference for Predicate Constraint Satisfaction

2020 ◽  
Vol 34 (02) ◽  
pp. 1644-1651
Author(s):  
Yuki Satake ◽  
Hiroshi Unno ◽  
Hinata Yanagi
Keyword(s):  
Graphical Models ◽  
Constraint Satisfaction ◽  
Linear Time ◽  
Predicate Logic ◽  
Probabilistic Inference ◽  
Search Space ◽  
Constraint Satisfaction Problems ◽  
Safety Properties ◽  
First Order ◽  
Guided Inductive

In this paper, we present a novel constraint solving method for a class of predicate Constraint Satisfaction Problems (pCSP) where each constraint is represented by an arbitrary clause of first-order predicate logic over predicate variables. The class of pCSP properly subsumes the well-studied class of Constrained Horn Clauses (CHCs) where each constraint is restricted to a Horn clause. The class of CHCs has been widely applied to verification of linear-time safety properties of programs in different paradigms. In this paper, we show that pCSP further widens the applicability to verification of branching-time safety properties of programs that exhibit finitely-branching non-determinism. Solving pCSP (and CHCs) however is challenging because the search space of solutions is often very large (or unbounded), high-dimensional, and non-smooth. To address these challenges, our method naturally combines techniques studied separately in different literatures: counterexample guided inductive synthesis (CEGIS) and probabilistic inference in graphical models. We have implemented the presented method and obtained promising results on existing benchmarks as well as new ones that are beyond the scope of existing CHC solvers.

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