The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory

@article{Feder1998TheCS,
  title={The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory},
  author={Tom{\'a}s Feder and Moshe Y. Vardi},
  journal={SIAM J. Comput.},
  year={1998},
  volume={28},
  pages={57-104}
}
This paper starts with the project of finding a large subclass of NP which exhibits a dichotomy. The approach is to find this subclass via syntactic prescriptions. While the paper does not achieve this goal, it does isolate a class (of problems specified by) "monotone monadic SNP without inequality" which may exhibit this dichotomy. We justify the placing of all these restrictions by showing, essentially using Ladner's theorem, that classes obtained by using only two of the above three… 
A Dichotomy Theorem for Typed Constraint Satisfaction Problems
TLDR
A dichotomy theorem is proved for these monotone function CSPs, and characterize those monot one functions such that the corresponding problem resides in P.
Towards a dichotomy theorem for the counting constraint satisfaction problem
TLDR
It is proved that if a subclass of the #CSP is tractable, then constraints allowed by the class satisfy some very restrictive condition: it has to have a Mal'tsev polymorphism, that is a ternary operation m(x, y, z) such that x = x.
Constraint Satisfaction Problems over semilattice block Mal'tsev algebras
  • A. Bulatov
  • Computer Science, Mathematics
    Inf. Comput.
  • 2019
TLDR
A new method of ‘hybrid’ algorithms that allows us to solve certain CSPs that has been out of reach for a quite a while, and eventually leads to resolving the Dichotomy Conjecture.
A universal-algebraic proof of the complexity dichotomy for Monotone Monadic SNP
TLDR
This work presents a new proof of the reduction to finite-domain CSPs that does not rely on the results of Kun, and uses the universal-algebraic approach to study the computational complexity of MMSNP.
Topology is relevant (in a dichotomy conjecture for infinite-domain constraint satisfaction problems)
TLDR
It is shown that local satisfaction and global satisfaction of nontrivial height 1 identities differ for $\omega$ -categorical structures with less than double exponential orbit growth, thereby resolving one of the main open problems in the algebraic theory of such structures.
Topology is relevant (in the infinite-domain dichotomy conjecture for constraint satisfaction problems)
TLDR
It is shown that local satisfaction and global satisfaction of non-trivial height 1 identities differ for $\omega$-categorical structures with less than double exponential orbit growth, thereby resolving one of the main open problems in the algebraic theory of such structures.
On the complexity of #CSP
TLDR
An elementary proof of Bulatov's dichotomy, based on succinct representations, of a class of highly structured relations, which are precisely the relations that are invariant under a Mal'tsev polymorphism, is given.
A combinatorial constraint satisfaction problem dichotomy classification conjecture
TLDR
The Feder-Hell-Huang conjecture is reduced to the CSP dichotomy classification conjecture, and the Kostochka-Nesetril-Smolikova conjecture is proved, although these results were proved independently by Jonsson et al. and Kun respectively.
A NEW COMBINATORIAL APPROACH TO THE CONSTRAINT SATISFACTION PROBLEM DICHOTOMY CLASSIFICATION
We introduce a new general polynomial-time constructionthe fibre constructionwhich reduces any constraint satisfaction problem CSP(H) to the constraint satisfaction problem CSP(P ), where P is any
Full Constraint Satisfaction Problems
TLDR
This work deduces the fact that all three-inclusive constraint satisfaction problems restricted to W-full input structures are NP-complete or "quasi-polynomial" (of order $n^{O(\log n)}$).
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 50 REFERENCES
Inductive pebble games and the expressive power of datalog
TLDR
This paper sketches a proof that the query “find all pairs of nodes connected by a directed simple path of even length” cannot be expressed in DATALOG, the language of function-free Horn clauses.
The decision problem for the probabilities of higher-order properties
TLDR
Logics which on the one hand go beyond fixpoint in terms of expressive power and on the other possess the 0-1 law are investigated, which establishes that the associated decision problem is NEXPTIME-complete and proofs of the decidability and complexity results require certain combinatorial machinery.
The complexity of satisfiability problems
TLDR
An infinite class of satisfiability problems is considered which contains these two particular problems as special cases, and it is shown that every member of this class is either polynomial-time decidable or NP-complete.
Algorithmic aspects of type inference with subtypes
TLDR
It is NP-hard to decide whether a lambda term has a type with respect to a fixed subtype hierarchy (involving only atomic type names), and PSPACE upper bounds for deciding polymorphic typability are given.
Easy Problems for Tree-Decomposable Graphs
Abstract Using a variation of the interpretability concept we show that all graph properties definable in monadic second-order logic (MS properties) with quantification over vertex and edge sets can
Duality and Polynomial Testing of Tree Homomorphisms
Let H be a fixed digraph. We consider the H-colouring problem, i.e., the problem of deciding which digraphs G admit a homomorphism to H. We are interested in a characterization in terms of the
Generalized first-order spectra, and polynomial. time recognizable sets
The spectrum of a first-order sentence σ is the set of cardinalities of its finite models. Jones and Selman showed that a set C of numbers (written in binary) is a spectrum if and only if C is in the
Optimization, Approximation, and Complexity Classes
TLDR
It follows that such a complete problem has a polynomial-time approximation scheme iff the whole class does, and that a number of common optimization problems are complete for MAX SNP under a kind of careful transformation that preserves approximability.
The gap between monotone and non-monotone circuit complexity is exponential
TLDR
The proof is immediate by combining the Alon—Boppana version of another argument of Razborov with results of Grötschel—Lovász—Schrijver on the Lovász — capacity of a graph.
On the Structure of Polynomial Time Reducibility
  • R. Ladner
  • Mathematics, Computer Science
    JACM
  • 1975
TLDR
The method of showing density ymlds the result that if P ~ NP then there are members of NP -P that are not polynomml complete is shown, which means there is a strictly ascending sequence with a minimal pair of upper bounds to the sequence.
...
1
2
3
4
5
...