Abstract
We present polynomial families complete for the well-studied algebraic complexity classes VF, VBP, VP, and VNP. The polynomial families are based on the homomorphism polynomials studied in the recent works of Durand et al. (2014) and Mahajan et al. (2018). We consider three different variants of graph homomorphisms, namely injective homomorphisms, directed homomorphisms, and injective directed homomorphisms, and obtain polynomial families complete for VF, VBP, VP, and VNP under each one of these. The polynomial families have the following properties:
• | The polynomial families complete for VF, VBP, and VP are model independent, i.e., they do not use a particular instance of a formula, algebraic branching programs, or circuit for characterising VF, VBP, or VP, respectively. | ||||
• | All the polynomial families are hard under p-projections. | ||||
- Michael Ben-Or and Richard Cleve. 1992. Computing algebraic formulas using a constant number of registers. SIAM J. Comput. 21, 1 (1992), 54–58. Google Scholar
Digital Library
- Peter Bürgisser. 2013. Completeness and Reduction in Algebraic Complexity Theory. Vol. 7. Springer Science & Business Media.Google Scholar
Cross Ref
- Florent Capelli, Arnaud Durand, and Stefan Mengel. 2016. The arithmetic complexity of tensor contraction. Theory Comput. Syst. 58, 4 (2016), 506–527. Google Scholar
Digital Library
- Stephen A. Cook. 1971. The complexity of theorem-proving procedures. In Proceedings of the 3rd Annual ACM Symposium on Theory of Computing (STOC'71). 151–158. Google Scholar
Digital Library
- Nicolas de Rugy-Altherre. 2012. A dichotomy theorem for homomorphism polynomials. In Proceedings of the International Symposium on Mathematical Foundations of Computer Science. Springer, 308–322. Google Scholar
Digital Library
- Arnaud Durand, Meena Mahajan, Guillaume Malod, Nicolas de Rugy-Altherre, and Nitin Saurabh. 2014. Homomorphism polynomials complete for VP. In Leibniz International Proceedings in Informatics, Vol. 29.Google Scholar
- Christian Engels. 2016. Dichotomy theorems for homomorphism polynomials of graph classes. J. Graph Algor. Appl. 20, 1 (2016), 3–22.Google Scholar
Cross Ref
- Raymond Greenlaw, H. James Hoover, and Walter Ruzzo. 1992. A compendium of problems complete for P. Citeseer.Google Scholar
- Leonid A. Levin. 1973. Universal search problems. Probl. Inf. Transmiss. 9, 3 (1973). [in Russian]Google Scholar
- Meena Mahajan. 2013. Algebraic complexity classes. arXiv:cs.CC/1307.3863. Retrieved from https://arxiv.org/abs/1307.3863.Google Scholar
- Meena Mahajan and Nitin Saurabh. 2018. Some complete and intermediate polynomials in algebraic complexity theory. Theory Comput. Syst. 62, 3 (2018), 622–652. Google Scholar
Digital Library
- Guillaume Malod and Natacha Portier. 2006. Characterizing valiant's algebraic complexity classes. In Proceedings of the International Symposium on Mathematical Foundations of Computer Science (MFCS'06). Springer, 704–716. Google Scholar
Digital Library
- Stefan Mengel. 2011. Characterizing arithmetic circuit classes by constraint satisfaction problems. In Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP'11). Springer, 700–711. Google Scholar
Digital Library
- Ran Raz. 2008. Elusive functions and lower bounds for arithmetic circuits. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing. ACM, 711–720. Google Scholar
Digital Library
- Nitin Saurabh. 2016. Analysis of Algebraic Complexity Classes and Boolean Functions. Ph.D. Dissertation. Institute of Mathematical sciences.Google Scholar
- L. G. Valiant. 1979. Completeness classes in algebra. In Proceedings of the 11th Annual ACM Symposium on Theory of Computing (STOC'79). 249–261. Google Scholar
Digital Library
Index Terms
Variants of Homomorphism Polynomials Complete for Algebraic Complexity Classes
Recommendations
Circuit Complexity, Proof Complexity, and Polynomial Identity Testing
FOCS '14: Proceedings of the 2014 IEEE 55th Annual Symposium on Foundations of Computer ScienceWe introduce a new and natural algebraic proof system, which has tight connections to (algebraic) circuit complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent ...
Some Complete and Intermediate Polynomials in Algebraic Complexity Theory
CSR 2016: Proceedings of the 11th International Computer Science Symposium on Computer Science --- Theory and Applications - Volume 9691We provide a list of new natural $$\mathsf {VNP}$$-Intermediate polynomial families, based on basic combinatorial $$\mathsf {NP}$$-Complete problems that are complete under parsimonious reductions. Over finite fields, these families are in $$\mathsf {...
Succinct hitting sets and barriers to proving algebraic circuits lower bounds
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of ComputingWe formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich for boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all ...






Comments