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Michael Skotnica

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TCS Journal 2024 Journal Article

Counting vanishing matrix-vector products

Cornelius Brand
Viktoriia Korchemna
Kirill Simonov
Michael Skotnica

Consider the following parameterized counting variation of the classic subset sum problem, which arises notably in the context of higher homotopy groups of topological spaces. Let v ∈ Q d be a rational vector, ( T 1, T 2 …, T m ) a list of d × d rational matrices, S ∈ Q h × d a rational matrix not necessarily square and k a parameter. The goal is to compute the number of ways one can choose k matrices T i 1, T i 2, …, T i k from the list such that S T i k ⋯ T i 1 v = 0 ∈ Q h. In this paper, we show that this problem is # W [ 2 ] -hard for parameter k. As a consequence, computing the k-th homotopy group of a d-dimensional 1-connected topological space for d > 3 is # W [ 2 ] -hard for parameter k. We also discuss a decision version of the problem and its several modifications for which we show W [ 1 ] / W [ 2 ] -hardness. This is in contrast to the parameterized k-sum problem, which is only W [ 1 ] -hard (Abboud-Lewi-Williams, ESA'14). In addition, we show that the decision version of the problem without parameter is an undecidable problem, and we give a fixed-parameter tractable algorithm for matrices of bounded size over finite fields, parameterized by the matrix dimensions and the order of the field.

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MFCS Conference 2023 Conference Paper

Deterministic Constrained Multilinear Detection

Cornelius Brand
Viktoriia Korchemna
Michael Skotnica

We extend the algebraic techniques of Brand and Pratt (ICALP'21) for deterministic detection of k-multilinear monomials in a given polynomial with non-negative coefficients to the more general situation of detecting colored k-multilinear monomials that satisfy additional constraints on the multiplicities of the colors appearing in them. Our techniques can be viewed as a characteristic-zero generalization of the algebraic tools developed by Guillemot and Sikora (MFCS'10) and Björklund, Kaski and Kowalik (STACS'13) As applications, we recover the state-of-the-art deterministic algorithms for the Graph Motif problem due to Pinter, Schachnai and Zehavi (MFCS'14), and give new deterministic algorithms for generalizations of certain questions on colored directed spanning trees or bipartite planar matchings running in deterministic time O^∗(4^k), studied originally by Gutin, Reidl, Wahlström and Zehavi (J. Comp. Sys. Sci. 95, '18). Finally, we give improved randomized algorithms for intersecting three and four matroids of rank k in characteristic zero, improving the record bounds of Brand and Pratt (ICALP'21) from O^∗(64^k) and O^∗(256^k), respectively, to O^∗(4^k).

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