Trading Determinism for Noncommutativity in Edmonds' Problem
V. Arvind, Abhranil Chatterjee, Partha Mukhopadhyay
TL;DR
This work develops deterministic polynomial-time algorithms for the partially commutative generalization of Edmonds' problem, PC-Singular, by embedding linear matrices over partially commutative variable sets into a universal skew field and iteratively increasing rank via PC-PIT and PC-Rank subroutines. A central insight is tying noncommutative ABP identity testing to rank increment steps through a constructive regularity lemma and division-algebra-based rounding, enabling witnesses with bounded tensor-shape dimensions. The authors also obtain the first deterministic polynomial-time algorithm for equivalence testing of $k$-tape weighted automata (for constant $k$), by reducing to zero-testing of partially commutative rational series. The results unify NSingular and Singular instances, show tight connections to trace monoids, and advance deterministic algorithmic invariant theory, with potential implications for parallel complexity and automata theory. Overall, the work delivers a principled framework and concrete algorithms for PC-Singular and related automata equivalence problems, significantly expanding the boundary between noncommutative algebra and algebraic complexity in a computable way.
Abstract
Let $X=X_1\sqcup X_2\sqcup\ldots\sqcup X_k$ be a partitioned set of variables such that the variables in each part $X_i$ are noncommuting but for any $i\neq j$, the variables $x\in X_i$ commute with the variables $x'\in X_j$. Given as input a square matrix $T$ whose entries are linear forms over $\mathbb{Q}\langle{X}\rangle$, we consider the problem of checking if $T$ is invertible or not over the universal skew field of fractions of the partially commutative polynomial ring $\mathbb{Q}\langle{X}\rangle$ [Klep-Vinnikov-Volcic (2020)]. In this paper, we design a deterministic polynomial-time algorithm for this problem for constant $k$. The special case $k=1$ is the noncommutative Edmonds' problem (NSINGULAR) which has a deterministic polynomial-time algorithm by recent results [Garg-Gurvits-Oliveira-Wigderson (2016), Ivanyos-Qiao-Subrahmanyam (2018), Hamada-Hirai (2021)]. En-route, we obtain the first deterministic polynomial-time algorithm for the equivalence testing problem of $k$-tape \emph{weighted} automata (for constant $k$) resolving a long-standing open problem [Harju and Karhum"{a}ki(1991), Worrell (2013)]. Algebraically, the equivalence problem reduces to testing whether a partially commutative rational series over the partitioned set $X$ is zero or not [Worrell (2013)]. Decidability of this problem was established by Harju and Karhumäki (1991). Prior to this work, a \emph{randomized} polynomial-time algorithm for this problem was given by Worrell (2013) and, subsequently, a deterministic quasipolynomial-time algorithm was also developed [Arvind et al. (2021)].