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

Date and Time: Thursday, March 21, 2024, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Andrei-Costin Constantinescu

No Checkerboards Allowed! Permutations Satisfying (Non-)Betweenness Constraints in Polynomial Time

Given a binary matrix, can you permute its columns such that no 2x3 submatrix forms a checkerboard? Submatrices can skip rows/columns (e.g., rows 1, 3 and columns 2, 6, 10 induce a 2x3 submatrix). This problem occurs in voting theory when one wants to recover so-called single-crossing preferences from binary voting ballots. The existence of a polynomial-time algorithm has been asked as an outstanding problem in the works of Elkind and Lackner, and we hereby answer it in the affirmative. Our approach hinges on a new graph-theoretic formulation of the (non-)betweenness problem, allowing one to more easily study the combinatorial structure of instances resulting from reducing to it. While the (non-) betweenness problem is NP-hard in general, we show that in our case the resulting instances can be easily solved by identifying necessary conditions and proving that they are sufficient for the class of instances arising from the reduction. Our approach can also be used to solve other problems reducing to (non-)betweenness, and we believe it can find further use in solving ordering-with-constraints problems in general. Joint work with Roger Wattenhofer (https://arxiv.org/abs/2310.03736, WINE'23).

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