Abstract
We present polynomial families complete for the well-studied algebraic complexity classes \(\textsf {VF}\), \(\textsf {VBP}\), \(\textsf {VP}\), and \(\textsf {VNP}\). The polynomial families are based on the homomorphism polynomials studied in the recent works of Durand et al. (2014) and Mahajan et al. [10]. We consider three different variants of graph homomorphisms, namely injective homomorphisms, directed homomorphisms and injective directed homomorphisms and obtain polynomial families complete for \(\textsf {VF}\), \(\textsf {VBP}\), \(\textsf {VP}\), and \(\textsf {VNP}\) under each one of these. The polynomial families have the following properties:
-
The polynomial families complete for \(\textsf {VF}\), \(\textsf {VBP}\), and \(\textsf {VP}\) are model independent, i.e. they do not use a particular instance of a formula, ABP or circuit for characterising \(\textsf {VF}\), \(\textsf {VBP}\), or \(\textsf {VP}\), respectively.
-
All the polynomial families are hard under p-projections.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
The hardness is shown with respect to p-projection reductions. We will define them formally in Sect. 2.
- 2.
Valiant [1] raised the question of whether the Permanent is computable in \(\textsf {VP}\). This question is equivalent to asking whether \(\textsf {VP}\) \(=\) \(\textsf {VNP}\), which is the algebraic analogue of the P vs. NP question.
- 3.
Note that if we set all Y variables to 1, then this polynomial essentially counts the number of homomorphisms from G to H.
- 4.
We do not use c-reductions in this work. They are more general than p-reductions. The formal definition can be found in [6].
- 5.
It is a layer preserving isomorphic copy which maps the root node of \(\mathsf{MAT}_{{i}}\) to the root of \(H_n\).
- 6.
Recall that \(m(n) = 2c \lceil \log n\rceil + 1\), which is odd. Also this is without loss of generality.
References
Valiant, L.G.: Completeness classes in algebra. In: Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, STOC 1979, pp. 249–261 (1979)
Bürgisser, P.: Completeness and Reduction in Algebraic Complexity Theory, vol. 7. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-662-04179-6
Raz, R.: Elusive functions and lower bounds for arithmetic circuits. Theory Comput. 6(1), 135–177 (2010)
Mengel, S.: Characterizing arithmetic circuit classes by constraint satisfaction problems. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 700–711. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22006-7_59
Capelli, F., Durand, A., Mengel, S.: The arithmetic complexity of tensor contraction. Theory Comput. Syst. 58(4), 506–527 (2016)
Durand, A., Mahajan, M., Malod, G., de Rugy-Altherre, N., Saurabh, N.: Homomorphism polynomials complete for VP. In: LIPIcs-Leibniz International Proceedings in Informatics, vol. 29 (2014)
Cook, S.A.: The complexity of theorem-proving procedures. STOC 71, 151–158 (1971)
Levin, L.A.: Universal search problems. Probl. Inf. Transm. 9(3), 115–116 (1973). (in Russian)
Greenlaw, R., Hoover, H.J., Ruzzo, W.: A compendium of problems complete for P, vol. 11 (1992)
Mahajan, M., Saurabh, N.: Some complete and intermediate polynomials in algebraic complexity theory. Theory Comput. Syst. 62(3), 622–652 (2018)
Rugy-Altherre, N.: A Dichotomy theorem for homomorphism polynomials. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 308–322. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32589-2_29
Engels, C.: Dichotomy theorems for homomorphism polynomials of graph classes. J. Graph Algorithms Appl. 20(1), 3–22 (2016)
Saurabh, N.: Analysis of algebraic complexity classes and boolean functions. Ph.D. thesis (2017)
Malod, G., Portier, N.: Characterizing Valiant’s algebraic complexity classes. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 704–716. Springer, Heidelberg (2006). https://doi.org/10.1007/11821069_61
Chaugule, P., Limaye, N., Varre, A.: Variants of homomorphism polynomials complete for algebraic complexity classes, 31 July 2018
Shpilka, A., Yehudayoff, A., et al.: Arithmetic circuits: a survey of recent results and open questions. Found. Trends® Theor. Comput. Sci. 5(3–4), 207–388 (2010)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Chaugule, P., Limaye, N., Varre, A. (2019). Variants of Homomorphism Polynomials Complete for Algebraic Complexity Classes. In: Du, DZ., Duan, Z., Tian, C. (eds) Computing and Combinatorics. COCOON 2019. Lecture Notes in Computer Science(), vol 11653. Springer, Cham. https://doi.org/10.1007/978-3-030-26176-4_8
Download citation
DOI: https://doi.org/10.1007/978-3-030-26176-4_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-26175-7
Online ISBN: 978-3-030-26176-4
eBook Packages: Computer ScienceComputer Science (R0)