Abstract
In the maximum solution equation problem a collection of equations are given over some algebraic structure. The objective is to find an assignment to the variables in the equations such that all equations are satisfied and the sum of the variables is maximised. We give tight approximability results for the maximum solution equation problem when the equations are given over groups of the form Z p , where p is prime. We also prove that the weighted and unweighted versions of this problem have equal approximability thresholds. Furthermore, we show that the problem is equally hard to solve even if each equation is restricted to contain at most three variables and solvable in polynomial time if the equations are restricted to contain at most two variables. All of our results also hold for a generalised version of maximum solution equation where the elements of the group are mapped arbitrarily to non-negative integers in the objective function.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arora, S.: Probabilistic checking of proofs and hardness of approximation problems. PhD thesis (1995)
Arora, S., Babai, L., Stern, J., Sweedyk, Z.: The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. Syst. Sci. 54(2), 317–331 (1997), doi:10.1006/jcss.1997.1472
Bovet, D.P., Crescenzi, P.: Introduction to the theory of complexity. Prentice-Hall, Englewood Cliffs (1994)
Bruck, J., Naor, M.: The hardness of decoding linear codes with preprocessing. IEEE Transactions on Information Theory 36(2), 381–385 (1990)
Crescenzi, P., Silvestri, R., Trevisan, L.: To weight or not to weight: Where is the question? In: ISTCS 1996: Proceedings of the 4th Israeli Symposium on Theory of Computing and Systems, pp. 68–77. IEEE Computer Society Press, Los Alamitos (1996)
Engebretsen, L., Holmerin, J., Russell, A.: Inapproximability results for equations over finite groups. Theor. Comput. Sci. 312(1), 17–45 (2004)
Feige, U., Micciancio, D.: The inapproximability of lattice and coding problems with preprocessing. J. Comput. Syst. Sci. 69(1), 45–67 (2004), doi:10.1016/j.jcss.2004.01.002
Goldmann, M., Russell, A.: The complexity of solving equations over finite groups. Inf. Comput. 178(1), 253–262 (2002)
Hastad, J.: Some optimal inapproximability results. J. ACM 48(4), 798–859 (2001)
Khanna, S., Sudan, M., Trevisan, L., Williamson, D.P.: The approximability of constraint satisfaction problems. SIAM J. Comput. 30(6), 1863–1920 (2000)
Klíma, O., Tesson, P., Thérien, D.: Dichotomies in the complexity of solving systems of equations over finite semigroups. Technical Report TR04-091, Electronic Colloq. on Computational Complexity (2004)
Larose, B., Zadori, L.: Taylor terms, constraint satisfaction and the complexity of polynomial equations over finite algebras. (Submitted for publication)
Moore, C., Tesson, P., Thérien, D.: Satisfiability of systems of equations over finite monoids. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 537–547. Springer, Heidelberg (2001)
Nordh, G.: The complexity of equivalence and isomorphism of systems of equations over finite groups. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 380–391. Springer, Heidelberg (2004)
Nordh, G., Jonsson, P.: The complexity of counting solutions to systems of equations over finite semigroups. In: Chwa, K.-Y., Munro, J.I.J. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 370–379. Springer, Heidelberg (2004)
Pascal Tesson. Computational Complexity Questions Related to Finite Monoids and Semigroups. PhD thesis (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kuivinen, F. (2005). Tight Approximability Results for the Maximum Solution Equation Problem over Z p . In: Jȩdrzejowicz, J., Szepietowski, A. (eds) Mathematical Foundations of Computer Science 2005. MFCS 2005. Lecture Notes in Computer Science, vol 3618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11549345_54
Download citation
DOI: https://doi.org/10.1007/11549345_54
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28702-5
Online ISBN: 978-3-540-31867-5
eBook Packages: Computer ScienceComputer Science (R0)