Abstract
Forbidden Patterns problem (FPP) is a proper generalisation of Constraint Satisfaction Problem (CSP). FPP was introduced in [1] as a combinatorial counterpart of MMSNP, a logic which was in turn introduced in relation to CSP by Feder and Vardi [2]. We prove that Forbidden Patterns Problems are Constraint Satisfaction Problems when restricted to graphs of bounded degree. This is a generalisation of a result by Häggkvist and Hell who showed that F-moteness of bounded-degree graphs is a CSP (that is, for a given graph F there exists a graph H so that the class of bounded-degree graphs that do not admit a homomorphism from F is exactly the same as the class of bounded-degree graphs that are homomorphic to H). Forbidden-pattern property is a strict generalisation of F-moteness (in fact of F-moteness combined with a CSP) as it involves both vertex- and edge-colourings of the graph F, and thus allows to express \(\mathcal{N}p\)-complete problems (while F-moteness is always in \(\mathcal{P}\)). We finally extend our result to arbitrary relational structures, and prove that every problem in MMSNP, restricted to connected inputs of bounded (hyper-graph) degree, is in fact in CSP.
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References
Madelaine, F., Stewart, I.A.: Constraint satisfaction, logic and forbidden patterns (submitted, 2005)
Feder, T., Vardi, M.Y.: The computational structure of monotone monadic SNP and constraint satisfaction: a study through datalog and group theory. SIAM J. Comput. 28, 57–104 (1999)
Hell, P., Nešetřil, J.: Graphs and homomorphisms. Oxford University Press, Oxford (2004)
Nešetřil, J., Tardif, C.: Duality theorems for finite structures (characterising gaps and good characterisations). Journal of Combin. Theory Ser. B 80, 80–97 (2000)
Madelaine, F., Stewart, I.A.: Some problems not definable using structures homomorphisms. Ars Combinatoria LXVII (2003)
Bordisky, M.: Constraint Satisfaction with Infinite Domains. PhD thesis, Humboldt-Universität zu Berlin (2004)
Madelaine, F.: Constraint satisfaction problems and related logic. PhD thesis, University of Leicester, Department of Mathematics and Computer Science (2003)
Häggkvist, R., Hell, P.: Universality of A-mote graphs. European Journal of Combinatorics 14, 23–27 (1993)
Hell, P., Galluccio, A., Nešetřil, J.: The complexity of H-coloring bounded degree graphs. Discrete Mathematics 222, 101–109 (2000)
Dreyer Jr., P.A., Malon, C., Nešetřil, J.: Universal H-colourable graphs without a given configuration. Discrete Mathematics 250(1-3), 245–252 (2002)
Courcelle, B.: The monadic second order logic of graphs VI: On several representations of graphs by relational structures. Discrete Applied Mathematics 54, 117–149 (1994)
Achlioptas, D.: The complexity of G-free colourability. Discrete Mathematics 165-166, 21–30 (1997)
Garey, M., Johnson, D.: Computers and intractability: a guide to NP-completeness. Freeman, San Francisco (1979)
Madelaine, F., Dantchev, S.: Online appendix of this paper (2006), Electronic form available from: www.dur.ac.uk/f.r.madelaine
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Dantchev, S., Madelaine, F. (2006). Bounded-Degree Forbidden Patterns Problems Are Constraint Satisfaction Problems. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds) Computer Science – Theory and Applications. CSR 2006. Lecture Notes in Computer Science, vol 3967. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11753728_18
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DOI: https://doi.org/10.1007/11753728_18
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