New Algorithms and Hard Instances for Non-Commutative Computation

TL;DR

This paper investigates the computational complexity of non-commutative permanent and determinant, establishing hardness results for certain graph classes and providing algorithms for specific cases, along with polynomial equivalences.

Contribution

It characterizes hard instances of non-commutative permanent, proves exponential lower bounds, and offers efficient algorithms for graphs with bounded component size.

Findings

Computing Cayley permanent/determinant on graphs with component size six is P-complete.

Lower bound of 2^{(n)} on branching program size for graphs with component size two.

Polynomial equivalences that are easy to compute over commutative domains.

Abstract

Motivated by the recent developments on the complexity of non-com\-mu\-ta\-tive determinant and permanent [Chien et al.\ STOC 2011, Bl\"aser ICALP 2013, Gentry CCC 2014] we attempt at obtaining a tight characterization of hard instances of non-commutative permanent. We show that computing Cayley permanent and determinant on weight\-ed adjacency matrices of graphs of component size six is $\#{\sf P}$ complete on algebras that contain $2\times 2$ matrices and the permutation group $S_3$. Also, we prove a lower bound of $2^{\Omega(n)}$ on the size of branching programs computing the Cayley permanent on adjacency matrices of graphs with component size bounded by two. Further, we observe that the lower bound holds for almost all graphs of component size two. On the positive side, we show that the Cayley permanent on graphs of component size $c$ can be computed in time $n^{c{\sf poly}(t)}$, where $t$ is a parameter depending on the labels of the vertices. Finally, we exhibit polynomials that are equivalent to the Cayley permanent polynomial but are easy to compute over commutative domains.