Some Complete and Intermediate Polynomials in Algebraic Complexity Theory

Abstract

Abstract. We provide a list of new natural VNP-intermediate polynomial families, based on basic (combinatorial) NP-complete problems that are complete under parsimonious reductions. Over finite fields, these families are in VNP, and under the plausible hypothesis ModpP ⊆ P/poly, are neither VNP-hard (even under oracle-circuit reductions) nor in VP. Prior to this, only the Cut Enumerator polynomial was known to be VNP-intermediate, as shown by Bürgisser in 2000. We next show that over rationals and reals, two of our intermediate polynomials, based on satisfiability and Hamiltonian cycle, are not monotone affine polynomial-size projections of the permanent. This augments recent results along this line due to Grochow. Finally, we describe a (somewhat natural) polynomial defined independent of a computation model, and show that it is VP-complete under polynomial-size projections. This complements a recent result of Durand et al. (2014) which established VP-completeness of a related polynomial but under constant-depth oracle circuit reductions. Both polynomials are based on graph homomorphisms. A simple restriction yields a family similarly complete for VBP.

A note on VNP-completeness and border complexity

“…The notion of c-reductions in algebraic complexity theory is relatively new [7]. The difference between p-projections and c-reductions plays a prominent role in [19]. Proof of Theorem 1…”

“…The set of all VP-p-complete p-families is denoted by VPC(≤ p ). There exist specific graph homomorphism polynomial families that are in VPC(≤ p ), see the recent [18]. The search for natural problems in VPC(≤ p ) was in fact a long-standing open problem.…”

Accesibilidad Web. Un Aporte De Responsabilidad Social Universitaria

“…The complexity has an important academic acknowledgment and has been the subject of several studies in recent years in various areas of knowledge (Chae, 2014;Chettiparamb, 2016;Walton, 2016;Copelli et al, 2016;Degener;Berne, 2017;Olya;Mehran, 2017;Efremenko et al;Goldberg;Papadimitriou, 2018;Karatsanis et al, 2018;Mahajan;Saurabh, 2018).…”

On the relative power of reduction notions in arithmetic circuit complexity

“…Again the question comes up if there is a fundamental difference between these two notions of reductions. In fact, it was exactly this uncertainty about the relative power of p-projections and c-reductions that motivated the recent work Mahajan and Saurabh [MS16]: For the first time they prove a natural problem complete for the arithmetic circuit class VP under p-projections, where before there existed only such result under c-reductions.…”