## The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory (1998)

Venue: | SIAM J. Comput |

Citations: | 253 - 31 self |

### BibTeX

@ARTICLE{Vardi98thecomputational,

author = {Moshe Y. Vardi},

title = {The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory},

journal = {SIAM J. Comput},

year = {1998},

volume = {28},

pages = {57--104}

}

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### Citations

944 | Temporal constraint networks
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Citation Context ...of the problems in CSP as polynomially solvable or NP-complete? 3s2 Preliminaries A large class of problems in AI and other areas of computer science can be viewed as constraint-satisfaction problems =-=[10, 34, 40, 41, 42, 43, 45]-=-. This includes problems in machine vision, belief maintenance, scheduling, temporal reasoning, type reconstruction, graph theory, and satisfiability. We start with some definitions. A vocabulary is a... |

270 |
Generalized first-order spectra and polynomial-time recognizable sets
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Citation Context ...e complete classification. An appendix explores the connection between constraint satisfaction and Tom Etter's link systems. 3 Monotone Monadic SNP without inequality The class SNP [32, 44] (see also =-=[14]-=-) consists of all problems expressible by an existential second-order sentence with a universal first-order part, namely, by a sentence of the form (9S0)(8x)\Phi (x; S; S0), where \Phisis a first-orde... |

229 | Easy problems for tree-decomposable graphs
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Citation Context ...instances, constraint-satisfaction problems are polynomially solvable, for l; k fixed. For more general results on polynomially solvable problems in the case of graphs of bounded tree-width, see e.g. =-=[1, 9]-=-. 6.1.2 Bounded Strict Width and the Helly Property The canonical algorithm for problems of width (l; k) involved inferring all possible constraints on l variables at a time by considering k variables... |

216 |
Duality of graph homomorphisms
- Hell, Neˇsetˇril, et al.
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Citation Context ..., when S is a graph and S is an (l; k)-tree, then it is said to have tree-width k \Gammas1 (see e.g. [38, 47]). Along the general lines of duality of graph homomorphisms (see Hell, Ne^set^ril and Zhu =-=[27]-=-), a constraint-satisfaction problem defined by a template T has (l; k)-tree duality if: A structure S can be mapped to T if and only if every (l; k)-tree that can be mapped to S can be mapped to T . ... |

156 |
Graph theory and probability
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Citation Context ...S0 of high girth. Furthermore, S0 can be constructed from S in randomized polynomial time. Proof. The transformation that enforces large girth is an adaptation of a randomized construction of Erd""os =-=[11]-=- of graphs with large girth and large chromatic number. Given a structure S on n elements, define S0 by making N = ns copies of each element, where 10ss is a large constant. If a relation R of arity r... |

140 |
On the complexity of H-coloring
- Hell, Neˇsetˇril
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Citation Context ...raph retract partial order retract may only contain problems in P (Datalog and Group Theory) or NP-complete (1-in-3 SAT) graph homomorphism boolean CSP in P (bipartite) or NP-complete (non-bipartite) =-=[24]-=- in P (Horn clauses, 2SAT, linear eq. mod 2) or NP-complete [48] ,, ,, ,, ,, ,, ,, XX XX XX XX XX XX ,, ,, ,, ,, ,, ,, XX XX XX XX XX XX Figure 1: Summary 2sWe examine MMSNP, and observe that it conta... |

94 | Two-prover one-round proof systems: Their power and their problems
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Citation Context ...chy collapses. The approach there was via interactive proofs for graph non-isomorphism; interactive proofs have already been shown to be relevant in the study of constraint satisfaction problems, see =-=[20, 7]-=-. It might be possible to answer the third conjecture with the theory of nearsubgroups as studied by Michael Aschbacher [3]. A basic questions remain open. Find a deterministic construction of small g... |

60 |
Polynomial-Time Algorithms for Permutation Groups
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Citation Context ...neral subgroup problem for a finite group G is polynomially solvable. Proof. The result follows immediately from a known algorithm that finds generators for a group, obtain by Babai [6], Furst et al. =-=[21]-=-; see also Theorem II.12 in Hoffmann [31]. The main observation is that given a group H with known generators and a chain of subgroups H = H0 ? H1 ? \Deltas\Deltas\Deltas? Hr = f1g, one can obtain dis... |

53 | Zhu: Duality and polynomial testing of tree homomorphisms
- Hell, Neˇsetˇril, et al.
- 1996
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Citation Context ...nts are biconnected components. A tree is a structure whose biconnected components consist each of a single relation occurrence. Following and generalizing the terminology of Hell, Ne^set^ril and Zhu =-=[25]-=-, we say that a constraint-satisfaction problem defined by a template T has tree duality if: A structure S can be mapped to T if and only if every tree that can be mapped to S can be mapped to T . The... |

52 |
Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees
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Citation Context ...instances, constraint-satisfaction problems are polynomially solvable, for l; k fixed. For more general results on polynomially solvable problems in the case of graphs of bounded tree-width, see e.g. =-=[1, 9]-=-. 6.1.2 Bounded Strict Width and the Helly Property The canonical algorithm for problems of width (l; k) involved inferring all possible constraints on l variables at a time by considering k variables... |

38 |
Group-Theoretic Algorithms and Graph Isomorphism, volume 136 of LNCS
- Hoffmann
- 1982
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Citation Context ... G is polynomially solvable. Proof. The result follows immediately from a known algorithm that finds generators for a group, obtain by Babai [6], Furst et al. [21]; see also Theorem II.12 in Hoffmann =-=[31]-=-. The main observation is that given a group H with known generators and a chain of subgroups H = H0 ? H1 ? \Deltas\Deltas\Deltas? Hr = f1g, one can obtain distinct representatives from each coset of ... |

32 |
On Datalog vs polynomial time
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Citation Context ... l variables that violate a constraint. This tree is then precisely a derivation tree by which the canonical program can accept the instance. For related games, see Afrati, Cosmadakis, and Yannakakis =-=[5]-=-, Kolaitis and Vardi [33], and Lakshmanan and Mendelzon [36]. This notion of bounded width for constraint-satisfaction problems can also be extended to allow infinite Datalog programs (allowing infini... |

27 |
Monte Carlo algorithms in graph isomorphism testing. Universitat de Montreal
- Babai
- 1979
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Citation Context ... Theorem 33 The general subgroup problem for a finite group G is polynomially solvable. Proof. The result follows immediately from a known algorithm that finds generators for a group, obtain by Babai =-=[6]-=-, Furst et al. [21]; see also Theorem II.12 in Hoffmann [31]. The main observation is that given a group H with known generators and a chain of subgroups H = H0 ? H1 ? \Deltas\Deltas\Deltas? Hr = f1g,... |

24 |
Homomorphisms to oriented cycles and k-partite satisfiability
- Feder
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Citation Context ...ertices in the same side were given different colors. Also, 2SAT has strict width 2, and so does integer programming with two variables per inequality with variables ranging over a fixed range. Feder =-=[17]-=- showed that digraph-homomorphism for oriented cycles is either polynomially solvable or NP-complete. The proof is a good illustration of some of the techniques we have been using up to this point; we... |

19 | The Existence of Homomorphisms to Oriented Cycles
- Hell, Zhu
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Citation Context ...ction problem has (l; k)-tree duality if and only if it has width (l; k). Recall from section 3 the definition of the length of an oriented cycle and of balanced and unbalanced digraphs. Hell and Zhu =-=[29]-=- show that in the case where T is an 27sunbalanced cycle, it is sufficient to test oriented paths, which are (1; 2)-trees, and oriented cycles, which are (2; 3)-trees; therefore unbalanced cycles defi... |

17 |
Complexity of tree homomorphisms
- Hell, Neˇsetˇril, et al.
- 1996
(Show Context)
Citation Context ...e of A to the image of B, as required. A more general case is the case of oriented trees, which defines both polynomial and NP-complete problems, as well as problems that have not yet been classified =-=[26]-=-. We say that a constraint-satisfaction problem defined by a template T with T a core has extended tree duality if: A connected structure S with one element s preassigned a value t in T can be mapped ... |

13 | Finite Group Theory. Cambridge Studies in Advanced Mathematics - Aschbacher - 2000 |

12 |
Tools for Datalog boundedness
- Hillebrand, Kanellakis, et al.
- 1991
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Citation Context ...taneously imposed. Theorem 1 Every problem in NP has an equivalent (under polynomial time reductions) problem in monotone monadic SNP with inequality. Proof. Hillebrand, Kanellakis, Mairson and Vardi =-=[30]-=- showed that monadic Datalog with inequality (but without negation) can verify a polynomial time encoding of a Turing machine computation; the machine can be nondeterministic. A Datalog program is a f... |

11 |
Homomorphisms to oriented paths
- Hell, Zhu
- 1994
(Show Context)
Citation Context ...n procedure. Consider the special case where the template T is an oriented path (a path each of whose edges may be oriented in either direction). This T was shown to have tree duality by Hell and Zhu =-=[28]-=-; in fact, they showed that it satisfies the stronger path duality property that S maps to T if and only if every oriented path that maps to S maps to T . We give a simple proof of tree duality via th... |

10 | 2-fusion in finite groups - Goldschmidt - 1974 |

9 |
The effect of two cycles on the complexity of colourings by directed graphs, Discrete
- Bang-Jensen, Hell
- 1990
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Citation Context ...at contains all triples except (0; 0; 0) and (1; 1; 1). Hell and Ne^set^ril [24] showed that the H-coloring problem is in P if H is bipartite and NP-complete for H non-bipartite. Bang-Jensen and Hell =-=[8]-=- conjecture that this result extends to the digraph case when every vertex in the template has at least one incoming and at least one incoming and at least one outgoing edge: if the template is equiva... |

4 |
Stable networks and product graphs, doctoral dissertation
- Feder
- 1990
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Citation Context ...bounded-width case and the subgroup case. Remark: A similar situation of only three polynomially solvable cases was observed for Boolean network stability problems by Mayr and Subramanian [35], Feder =-=[12]-=-; the three cases there are monotone networks, linear networks, and nonexpansive networks, in close correspondence with Horn clauses, linear equations modulo 2, and 2SAT respectively; it is the genera... |

2 |
Stable Networks and Product Graphs," doctoral dissertation
- Feder
- 1991
(Show Context)
Citation Context ...bounded-width case and the subgroup case. Remark: A similar situation of only three polynomially solvable cases was observed for Boolean network stability problems by Mayr and Subramanian [39], Feder =-=[15]-=-; the three cases there are monotone networks, linear networks, and nonexpansive networks, in close correspondence with Horn clauses, linear equations modulo 2, and 2SAT respectively; it is the genera... |

2 |
Removing Inequalities and Negation for Homomorphism-Closed Problems. manuscript
- Feder
(Show Context)
Citation Context ...onadic SNP with inequality is no more powerful than monotone monadic SNP without inequality, and the same holds for (binary) SNP with inequality compared with monotone (binary) SNP without inequality =-=[16]-=-. 6.1.1 Width 1 and Tree Duality Horn clauses have width (1; k), where k is the maximum number of variables per Horn clause. To see this, express Horn clauses as implications with a conjunction of pos... |

2 |
Graph theory and probability, Canadian
- Erdős
- 1959
(Show Context)
Citation Context ...n can be derandomized using quasi-random hypergraphs. The general transformation from monotone monadic SNP to constraint satisfaction problems is an adaptation of a randomized construction of Erd""os =-=[10]-=- of graphs with large girth and large chromatic number. Consider a monotone monadic SNP problem that asks for a input structure S whether there exists a monadic structure S0 such that for all x, \Phi ... |

1 | Finite group theory," Cambridge studies in advanced mathematics 10 - Aschbacher |

1 |
Near subgroups of finite groups," manuscript. [4] F. Afrati and
- Aschbacher
- 1989
(Show Context)
Citation Context ...ditional nearsubgroup constraints cannot simulate one-in-three SAT (and is thus unlikely to be NP-complete). Proof. Since the intersection of nearsubgroups is a nearsubgroup by a result of Aschbacher =-=[3]-=-, it suffices to show that a single one-nearsubgroup K of G3 cannot represent the one-in-three SAT relation via some cK. Suppose that it does, so that cK contains three elements (a; b; b), (b; a; b), ... |

1 |
Quantum mechanics as a branch of mereology," extended abstract for PhysComp paper
- Etter
(Show Context)
Citation Context ...ing two variables x and x0 corresponding to coin tosses, where the velocity is the probability of heads minus the probability of tails. In a second paper, "Quantum Mechanics as a Branch of Mereology" =-=[13]-=- proposes an approach to link systems from relational databases instead of probability theory. The word mereology means the mathematical theory of parts and wholes. The idea is to use the record count... |

1 |
List problems for reflexive graphs," manuscript. [19
- Feder, Hell
(Show Context)
Citation Context ... vertices in the template. The reflexive graph-connected list problem allows only subsets that induce a connected subset of the vertices in the template. The following results are from Feder and Hell =-=[18, 19]-=-. Theorem 14 Every constraint-satisfaction problem is polynomially equivalent to a reflexive graph-retract problem. The reflexive graph-retract problem is NP-complete for graphs without triangles othe... |

1 |
Near subgroups of finite groups," manuscript. [4] F. Afrati and
- Aschbacher
- 1989
(Show Context)
Citation Context ...ditional nearsubgroup constraints cannot simulate one-in-three SAT (and is thus unlikely to be NP-complete). Proof. Since the intersection of nearsubgroups is a nearsubgroup by a result of Aschbacher =-=[3]-=-, it suffices to show that a single one-nearsubgroup K of G3 cannot represent the one-in-three SAT relation via some cK. Suppose that it does, so that cK contains three elements (a, b, b), (b, a, b), ... |

1 |
List problems for reflexive graphs," manuscript. [16
- Feder, Hell
(Show Context)
Citation Context ... vertices in the template. The reflexive graph-connected list problem allows only subsets that induce a connected subset of the vertices in the template. The following results are from Feder and Hell =-=[15, 16]-=-. Theorem 13 Every constraint-satisfaction problem is polynomially equivalent to a reflexive graph-retract problem. The reflexive graph-retract problem is NP-complete for graphs without triangles othe... |

1 |
Complexity of tree homomorphisms," manuscript
- Hell, Nesetril, et al.
(Show Context)
Citation Context ...e of A to the image of B, as required. A more general case is the case of oriented trees, which defines both polynomial and NP-complete problems, as well as problems that have not yet been classified =-=[22]-=-. We say that a constraint-satisfaction problem defined by a template T with T a core has extended tree duality if: A connected structure S with one element s preassigned a value t in T can be mapped ... |

1 |
Duality of graph homomorphisms," manuscript. [24] P. Hell and X. Zhu, "Homomorphisms to oriented paths," manuscript. [25
- Hell, Nesetril, et al.
(Show Context)
Citation Context ...ature, when S is a graph and S is an (l, k)-tree, then it is said to have tree-width k - 1 (see e.g. [34, 43]). Along the general lines of duality of graph homomorphisms (see Hell, Ne^set^ril and Zhu =-=[23]-=-), a constraint-satisfaction problem defined by a template T has (l, k)-tree duality if: A structure S can be mapped to T if and only if every (l, k)-tree that can be mapped to S can be mapped to T . ... |