Constraint Network Satisfaction for Finite Relation Algebras

Network satisfaction problems (NSPs) for finite relation algebras are computational decision problems, studied intensively since the 1990s. The major open research challenge in this field is to understand which of these problems are solvable by polynomial-time algorithms. Since there are known examples of undecidable NSPs of finite relation algebras it is advisable to restrict the scope of such a classification attempt to well-behaved subclasses of relation algebras. The class of relation algebras with a normal representation is such a well-behaved subclass. Many well-known examples of relation algebras, such as the Point Algebra, RCC5, and Allen’s Interval Algebra admit a normal representation. The great advantage of finite relation algebras with normal representations is that their NSP is essentially the same as a constraint satisfaction problem (CSP). For a relational structure B the problem CSP(B) is the computational problem to decide whether a given finite relational structure C has a homomorphism to B. The study of CSPs has a long and rich history, culminating for the time being in the celebrated proofs of the Feder-Vardi dichotomy conjecture. Bulatov and Zhuk independently proved that for every finite structure B the problem CSP(B) is in P or NP-complete. Both proofs rely on the universal-algebraic approach, a powerful theory that connects algebraic properties of structures B with complexity results for the decision problems CSP(B). Our contributions to the field are divided into three parts. Firstly, we provide two algebraic criteria for NP-hardness of NSPs. Our second result is a complete classification of the complexity of NSPs for symmetric relation algebras with a flexible atom; these problems are in P or NP-complete. Our result is obtained via a decidable condition on the relation algebra which implies polynomial-time tractability of the NSP. As a third contribution we prove that for a large class of NSPs, non-hardness implies that the problems can even be solved by Datalog programs, unless P = NP. This result can be used to strengthen the dichotomy result for NSPs of symmetric relation algebras with a flexible atom: every such problem can be solved by a Datalog program or is NP-complete. Our proof relies equally on known results and new observations in the algebraic analysis of finite structures. The CSPs that emerge from NSPs are typically of the form CSP(B) for an infinite structure B and therefore do not fall into the scope of the dichotomy result for finite structures. In this thesis we study NSPs of finite relation algebras with normal representations by the universal algebraic methods which were developed for the study of finite and infinite-domain CSPs. We additionally make use of model theory and a Ramsey-type result of Nešetril and Rödl. Our contributions to the field are divided into three parts. Firstly, we provide two algebraic criteria for NP-hardness of NSPs. Our second result is a complete classification of the complexity of NSPs for symmetric relation algebras with a flexible atom; these problems are in P or NP-complete. Our result is obtained via a decidable condition on the relation algebra which implies polynomial-time tractability of the NSP. As a third contribution we prove that for a large class of NSPs the containment in P implies that the problems can even be solved by Datalog programs, unless P = NP. As a third contribution we prove that for a large class of NSPs, non-hardness implies that the problems can even be solved by Datalog programs, unless P = NP. This result can be used to strengthen the dichotomy result for NSPs of symmetric relation algebras with a flexible atom: every such problem can be solved by a Datalog program or is NP-complete. Our proof relies equally on known results and new observations in the algebraic analysis of finite structures

Network satisfaction for symmetric relation algebras with a flexible atom

Robin Hirsch posed in 1996 the Really Big Complexity Problem: classify the computational complexity of the network satisfaction problem for all finite relation algebras $\bf A$. We provide a complete classification for the case that $\bf A$ is symmetric and has a flexible atom; the problem is in this case NP-complete or in P. If a finite integral relation algebra has a flexible atom, then it has a normal representation $\mathfrak{B}$. We can then study the computational complexity of the network satisfaction problem of ${\bf A}$ using the universal-algebraic approach, via an analysis of the polymorphisms of $\mathfrak{B}$. We also use a Ramsey-type result of Ne\v{s}et\v{r}il and R\"odl and a complexity dichotomy result of Bulatov for conservative finite-domain constraint satisfaction problems.Comment: 32 pages, 2 figure

Constraint Network Satisfaction for Finite Relation Algebras

Network satisfaction problems (NSPs) for finite relation algebras are computational decision problems, studied intensively since the 1990s. The major open research challenge in this field is to understand which of these problems are solvable by polynomial-time algorithms. Since there are known examples of undecidable NSPs of finite relation algebras it is advisable to restrict the scope of such a classification attempt to well-behaved subclasses of relation algebras. The class of relation algebras with a normal representation is such a well-behaved subclass. Many well-known examples of relation algebras, such as the Point Algebra, RCC5, and Allen’s Interval Algebra admit a normal representation. The great advantage of finite relation algebras with normal representations is that their NSP is essentially the same as a constraint satisfaction problem (CSP). For a relational structure B the problem CSP(B) is the computational problem to decide whether a given finite relational structure C has a homomorphism to B. The study of CSPs has a long and rich history, culminating for the time being in the celebrated proofs of the Feder-Vardi dichotomy conjecture. Bulatov and Zhuk independently proved that for every finite structure B the problem CSP(B) is in P or NP-complete. Both proofs rely on the universal-algebraic approach, a powerful theory that connects algebraic properties of structures B with complexity results for the decision problems CSP(B). Our contributions to the field are divided into three parts. Firstly, we provide two algebraic criteria for NP-hardness of NSPs. Our second result is a complete classification of the complexity of NSPs for symmetric relation algebras with a flexible atom; these problems are in P or NP-complete. Our result is obtained via a decidable condition on the relation algebra which implies polynomial-time tractability of the NSP. As a third contribution we prove that for a large class of NSPs, non-hardness implies that the problems can even be solved by Datalog programs, unless P = NP. This result can be used to strengthen the dichotomy result for NSPs of symmetric relation algebras with a flexible atom: every such problem can be solved by a Datalog program or is NP-complete. Our proof relies equally on known results and new observations in the algebraic analysis of finite structures. The CSPs that emerge from NSPs are typically of the form CSP(B) for an infinite structure B and therefore do not fall into the scope of the dichotomy result for finite structures. In this thesis we study NSPs of finite relation algebras with normal representations by the universal algebraic methods which were developed for the study of finite and infinite-domain CSPs. We additionally make use of model theory and a Ramsey-type result of Nešetril and Rödl. Our contributions to the field are divided into three parts. Firstly, we provide two algebraic criteria for NP-hardness of NSPs. Our second result is a complete classification of the complexity of NSPs for symmetric relation algebras with a flexible atom; these problems are in P or NP-complete. Our result is obtained via a decidable condition on the relation algebra which implies polynomial-time tractability of the NSP. As a third contribution we prove that for a large class of NSPs the containment in P implies that the problems can even be solved by Datalog programs, unless P = NP. As a third contribution we prove that for a large class of NSPs, non-hardness implies that the problems can even be solved by Datalog programs, unless P = NP. This result can be used to strengthen the dichotomy result for NSPs of symmetric relation algebras with a flexible atom: every such problem can be solved by a Datalog program or is NP-complete. Our proof relies equally on known results and new observations in the algebraic analysis of finite structures

Constraint Network Satisfaction for Finite Relation Algebras

Network satisfaction problems (NSPs) for finite relation algebras are computational decision problems, studied intensively since the 1990s. The major open research challenge in this field is to understand which of these problems are solvable by polynomial-time algorithms. Since there are known examples of undecidable NSPs of finite relation algebras it is advisable to restrict the scope of such a classification attempt to well-behaved subclasses of relation algebras. The class of relation algebras with a normal representation is such a well-behaved subclass. Many well-known examples of relation algebras, such as the Point Algebra, RCC5, and Allen’s Interval Algebra admit a normal representation. The great advantage of finite relation algebras with normal representations is that their NSP is essentially the same as a constraint satisfaction problem (CSP). For a relational structure B the problem CSP(B) is the computational problem to decide whether a given finite relational structure C has a homomorphism to B. The study of CSPs has a long and rich history, culminating for the time being in the celebrated proofs of the Feder-Vardi dichotomy conjecture. Bulatov and Zhuk independently proved that for every finite structure B the problem CSP(B) is in P or NP-complete. Both proofs rely on the universal-algebraic approach, a powerful theory that connects algebraic properties of structures B with complexity results for the decision problems CSP(B). Our contributions to the field are divided into three parts. Firstly, we provide two algebraic criteria for NP-hardness of NSPs. Our second result is a complete classification of the complexity of NSPs for symmetric relation algebras with a flexible atom; these problems are in P or NP-complete. Our result is obtained via a decidable condition on the relation algebra which implies polynomial-time tractability of the NSP. As a third contribution we prove that for a large class of NSPs, non-hardness implies that the problems can even be solved by Datalog programs, unless P = NP. This result can be used to strengthen the dichotomy result for NSPs of symmetric relation algebras with a flexible atom: every such problem can be solved by a Datalog program or is NP-complete. Our proof relies equally on known results and new observations in the algebraic analysis of finite structures. The CSPs that emerge from NSPs are typically of the form CSP(B) for an infinite structure B and therefore do not fall into the scope of the dichotomy result for finite structures. In this thesis we study NSPs of finite relation algebras with normal representations by the universal algebraic methods which were developed for the study of finite and infinite-domain CSPs. We additionally make use of model theory and a Ramsey-type result of Nešetril and Rödl. Our contributions to the field are divided into three parts. Firstly, we provide two algebraic criteria for NP-hardness of NSPs. Our second result is a complete classification of the complexity of NSPs for symmetric relation algebras with a flexible atom; these problems are in P or NP-complete. Our result is obtained via a decidable condition on the relation algebra which implies polynomial-time tractability of the NSP. As a third contribution we prove that for a large class of NSPs the containment in P implies that the problems can even be solved by Datalog programs, unless P = NP. As a third contribution we prove that for a large class of NSPs, non-hardness implies that the problems can even be solved by Datalog programs, unless P = NP. This result can be used to strengthen the dichotomy result for NSPs of symmetric relation algebras with a flexible atom: every such problem can be solved by a Datalog program or is NP-complete. Our proof relies equally on known results and new observations in the algebraic analysis of finite structures

Network Satisfaction for Symmetric Relation Algebras with a Flexible Atom

Robin Hirsch posed in 1996 the Really Big Complexity Problem: classify the computational complexity of the network satisfaction problem for all finite relation algebras A. We provide a complete classification for the case that A is symmetric and has a flexible atom; the problem is in this case NP-complete or in P. If a finite integral relation algebra has a flexible atom, then it has a normal representation B. We can then study the computational complexity of the network satisfaction problem of A using the universal-algebraic approach, via an analysis of the polymorphisms of B. We also use a Ramsey-type result of Nešetřil and Rödl and a complexity dichotomy result of Bulatov for conservative finite-domain constraint satisfaction problems