## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Thursday, September 29, 2022, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Joel Widmer

## Solving systems of linear equations of Matousek matrices in the matrix-vector query model (Master Thesis)

In the matrix-vector query model, we are given an oracle which for a given vector v returns Mv, where M is a fixed square matrix of size n unknown to us. This is called a (matrix-vector) query. We are interested in solving a system of linear equations Mx = y, where M is given by such an oracle. In other words, for a given vector y, we want to find a vector x such that the oracle returns y when given x. An algorithm solving this problem can query vectors one by one and may perform any number of other computations in between. The performance is measured only by the number of queries made. We consider the case, where M is the adjacency matrix of a directed acyclic graph with an additional loop at every vertex. These matrices are called Matousek matrices. We show that for a general Matousek matrix, any deterministic algorithm requires at least n - 1 queries to find x in the worst case. We also consider a subclass of Matousek matrices, called realizable Matousek matrices. A realizable Matousek matrix is the adjacency matrix of a graph which is the transitive closure of a branching with an additional loop at every vertex. For this case, we present an algorithm that finds x in O(log^2(n)) queries. In particular our results imply that solving this problem on realizable Matousek matrices is strictly easier than on non-realizable Matousek matrices. We translate this result to the setting of unique sink orientations (USOs) and show that solving realizable Matousek-type USOs is strictly easier than solving non-realizable Matousek-type USOs.
The thesis was supervised by E. Welzl, B. Gärtner, S. Weber.

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