A note on algebraic techniques for subgraph detection

Abstract

The k-path problem asks whether a given graph contains a simple path of length k. Along with other prominent parameterized problems, it reduces to the problem of detecting multilinear terms of degree k (k-MLD), making the latter a fundamental problem in parameterized algorithms. This has generated significant efforts directed at devising fast deterministic algorithms for k-MLD, and there are now at least two independent approaches that yield the same record bound on the running time. Namely the combinatorial representative-set based approach of Fomin et al. (JACM'16), and the algebraic techniques of Pratt (FOCS'19) and Brand and Pratt (ICALP'21). In this note, we explore the relationship between the latter results, based on partial differentials, and a previous algebraic approach based on the exterior algebra (Brand, ESA'19; Brand, Dell and Husfeldt, STOC'18). We do so by studying the relevant algebraic objects. These are on the one hand (1) the subalgebras of the tensor square of the exterior algebra generated in degree one. On the other hand, we consider (2) the space of partial derivatives of generic determinants, and closely related, (3) the space of minors of generic matrices. We prove that (2) arises as a quotient of (1), and that there is an isomorphism between the objects (1) and (3). Hence, the techniques are essentially equivalent, and the quotient relation between (2) and both of (1) and (3) hints at a possible refinement of the techniques.