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

Abstract

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 restrictions do not show this dichotomy. We then explore the structure of this class. We show that all problems in this class reduce to the seemingly simpler class CSP. We divide CSP into subclasses and try to unify the collection of all known polytime algorithms for CSP problems and extract properties that make CSP problems NP-hard. This is where the second part of the title, "a study through Datalog and group theory," comes in. We present conjectures about this class which would end in showing the dichotomy.

MSC codes

  1. 68Q15
  2. 68R99

Keywords

  1. satisfiability
  2. graph coloring
  3. datalog
  4. group theory
  5. linear equations

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References

1.
Stefan Arnborg, Jens Lagergren, Detlef Seese, Easy problems for tree‐decomposable graphs, J. Algorithms, 12 (1991), 308–340
2.
M. Aschbacher, Finite Group Theory, Cambridge Stud. Adv. Math. 10, Cambridge University Press, Cambridge, UK, 1986.
3.
Michael Aschbacher, Near subgroups of finite groups, J. Group Theory, 1 (1998), 113–129
4.
F. Afrati and S. S. Cosmadakis, Expressiveness of restricted recursive queries, in Proc. 21st ACM Symp. on Theory of Computing, ACM, New York, 1989, pp. 113–126.
5.
F. Afrati, S. S. Cosmadakis, and M. Yannakakis, On Datalog vs. polynomial time, in Proc. 10th ACM SIGACT‐SIGMOD‐SIGART Symp. on Principles of Database Systems, ACM, New York, 1991, pp. 13–25.
6.
L. Babai, Monte Carlo Algorithms in Graph Isomorphism Testing, manuscript, 1979.
7.
Roman Bačík, Sanjeev Mahajan, Semidefinite programming and its applications to NP problems, Lecture Notes in Comput. Sci., Vol. 959, Springer, Berlin, 1995, 566–575
8.
Jørgen Bang‐Jensen, Pavol Hell, The effect of two cycles on the complexity of colourings by directed graphs, Discrete Appl. Math., 26 (1990), 1–23
9.
Hans Bodlaender, Polynomial algorithms for graph isomorphism and chromatic index on partial k‐trees, J. Algorithms, 11 (1990), 631–643
10.
R. Dechter, Constraint networks, in Encyclopedia of Artificial Intelligence, 1992, pp. 276–285.
11.
P. Erdös, Graph theory and probability, Canad. J. Math., 11 (1959), 34–38
12.
T. Etter, Process, system, causality, and quantum mechanics – a psychoanalysis of animal faith, Internat. J. General Systems (Special Issue on General Systems and the Emergence of Physical Structure from Information Theory).
13.
T. Etter, Quantum mechanics as a branch of mereology, extended abstract,Phys. Comp., 116 (1995).
14.
Ronald Fagin, Generalized first‐order spectra and polynomial‐time recognizable sets, Amer. Math. Soc., Providence, R.I., 1974, 0–0, 43–73. SIAM‐AMS Proc., Vol. VII
15.
Tomás Feder, Stable networks and product graphs, Mem. Amer. Math. Soc., 116 (1995), 0–0xii+223
16.
T. Feder, Removing Inequalities and Negation for Homomorphism‐Closed Problems, manuscript.
17.
T. Feder, Classification of Homomorphisms to Oriented Cycles and of k‐Partite Satisfiability Problems, manuscript.
18.
T. Feder and P. Hell, List Problems for Reflexive Graphs, manuscript.
19.
T. Feder and P. Hell, Homomorphism Problems on Graphs with Some Self‐Loops, manuscript.
20.
U. Feige and L. Lovász, Two‐prover one‐round proof systems: Their power and their problems, 24th Annual ACM Symp. on Theory of Computing, ACM, New York, 1994, pp. 422–431.
21.
Merrick Furst, John Hopcroft, Eugene Luks, Polynomial‐time algorithms for permutation groups, IEEE, New York, 1980, 36–1
22.
David Goldschmidt, 2‐fusion in finite groups, Ann. of Math. (2), 99 (1974), 70–117
23.
Wolfgang Gutjahr, Emo Welzl, Gerhard Woeginger, Polynomial graph‐colorings, Discrete Appl. Math., 35 (1992), 29–45
24.
Pavol Hell, Jaroslav Nešetřil, On the complexity of H‐coloring, J. Combin. Theory Ser. B, 48 (1990), 92–110
25.
P. Hell, J. Nešetřil, X. Zhu, Duality and polynomial testing of tree homomorphisms, Trans. Amer. Math. Soc., 348 (1996), 1281–1297
26.
P. Hell, J. Nešetřil, X. Zhu, Complexity of tree homomorphisms, Discrete Appl. Math., 70 (1996), 23–36
27.
P. Hell, J. Nešetřil, X. Zhu, Duality of graph homomorphisms, Bolyai Soc. Math. Stud., Vol. 2, János Bolyai Math. Soc., Budapest, 1996, 271–282
28.
Pavol Hell, Xu Zhu, Homomorphisms to oriented paths, Discrete Math., 132 (1994), 107–114
29.
Pavol Hell, Xu Zhu, The existence of homomorphisms to oriented cycles, SIAM J. Discrete Math., 8 (1995), 208–222
30.
G. G. Hillebrand, P. C. Kanellakis, H. G. Mairson, and M. Y. Vardi, Tools for Datalog boundedness, in Proc. 10th ACM SIGACT‐SIGMOD‐SIGART Symp. on Principles of Database Systems, ACM, New York, 1991, pp. 1–12.
31.
Christoph Hoffmann, Group‐theoretic algorithms and graph isomorphism, Lecture Notes in Computer Science, Vol. 136, Springer‐Verlag, 1982viii+311
32.
P. G. Kolaitis and M. Y. Vardi, The decision problem for the probabilities of higher‐order properties, in Proc. 19th ACM Symp. on Theory of Computing, 1987, pp. 425–435.
33.
P. G. Kolaitis and M. Y. Vardi, On the expressive power of Datalog: Tools and a case study, in Proc. 9th ACM SIGACT‐SIGMOD‐SIGART Symp. on Principles of Database Systems, ACM, New York, 1990, pp. 61–71.
34.
V. Kumar, Algorithms for constraint‐satisfaction problems, AI Magazine, 13 (1992), pp. 32–44.
35.
R. E. Ladner, On the structure of polynomial time reducibility, J. Assoc. Comput. Mach., 22 (1975), pp. 155–171.
36.
V. S. Lakshmanan and A. O. Mendelzon, Inductive pebble games and the expressive power of Datalog, in Proc. 8th ACM SIGACT‐SIGMOD‐SIGART Symp. on Principles of Database Systems, ACM, New York, 1989, pp. 301–310.
37.
Anna Lubiw, Some NP‐complete problems similar to graph isomorphism, SIAM J. Comput., 10 (1981), 11–21
38.
Jiří Matoušek, Robin Thomas, Algorithms finding tree‐decompositions of graphs, J. Algorithms, 12 (1991), 1–22
39.
Ernst Mayr, Ashok Subramanian, The complexity of circuit value and network stability, J. Comput. System Sci., 44 (1992), 302–323
40.
P. Meseguer, Constraint satisfaction problem: An overview, AICOM, 2 (1989), pp. 3–16.
41.
J. C. Mitchell, Coercion and type inference (summary), in Conf. Rec. 11th ACM Symp. on Principles of Programming Languages, ACM, New York, 1984, pp. 175–185.
42.
P. Lincoln and J. C. Mitchell, Algorithmic aspects of type inference with subtypes, in Conf. Rec. 19th ACM Symp. on Principles of Programming Languages, ACM, New York, 1992, pp. 293–304.
43.
M. Wand and P. M. O’Keefe, On the complexity of type inference with coercion, in Conf. on Functional Programming Languages and Computer Architecture, 1989.
44.
Christos Papadimitriou, Mihalis Yannakakis, Optimization, approximation, and complexity classes, J. Comput. System Sci., 43 (1991), 425–440
45.
Vaughan Pratt, Jerzy Tiuryn, Satisfiability of inequalities in a poset, Fund. Inform., 28 (1996), 165–182
46.
A. A. Razborov, Lower bounds on monotone complexity of the logical permanent, Math. Notes Acad. Sci. USSR, 37 (1985), pp. 485–493.
47.
N. Robertson and P. Seymour, Graph minors. II. Algorithmic aspects of tree‐width, J. Algorithms, 7 (1985), pp. 309–322.
48.
Thomas Schaefer, The complexity of satisfiability problems, ACM, New York, 1978, 216–226
49.
E. Tardos, The gap between monotone and non‐monotone circuit complexity is exponential, Combinatorica, 7–4 (1987), pp. 141–142.
50.
J. D. Ullman, Principles of Database and Knowledge‐Base Systems, Vol. I, Computer Science Press, Rockville, MD, 1989.

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cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 57 - 104
ISSN (online): 1095-7111

History

Published online: 28 July 2006

MSC codes

  1. 68Q15
  2. 68R99

Keywords

  1. satisfiability
  2. graph coloring
  3. datalog
  4. group theory
  5. linear equations

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