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

Date and Time: Tuesday, September 19, 2023, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Josiah Rohrer

Extended L-Graphs of Unique Sink Orientations (Master Thesis)

Unique Sink Orientations or short USOs are a combinatorial abstraction of important optimization problems including Linear Programming, Quadratic Programming and the P-Matrix Linear Complementary Problem (PLCP). Currently, no strongly polynomial time algorithm is known for Linear Programming and PLCP is not even known to have a polynomial time algorithm. Polynomial time sink finding in USOs would provide us with both of these algorithms and therefore have major implications in mathematical optimization. The number of realizable USO, those that are practically relevant and arise from these optimization problem, are a tiny fraction with respect to the number of all USOs. Therefore, we hope to find additional structure in realizable USOs that could be algorithmically exploited. One of these structures are L-graphs, which where recently shown to be acyclic on D-cubes, a key subclass of realizable USOs. Unfortunately, the property of having acyclic L-graph is not preserved under isomorphism. In this thesis we extend the definition of L-graphs and prove a weaker property that is preserved under isomorphism. Through this we introduce B-cubes, a new algebraic class that we prove is equivalent to isomorphic copies of D-cubes. In the second part we extend the Matousek-type USOs, the class of USOs with exactly one L-graph, and prove that USOs with up to six L-graphs either do not exist or are recursively combed. The thesis was supervised by E. Welzl, B. Gärtner, S. Weber.

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