A Polynomial-Time Randomized Reduction from Tournament Isomorphism to Tournament Asymmetry

The paper develops a new technique to extract a characteristic subset from a random source that repeatedly samples from a set of elements. Here a characteristic subset is a set that when containing an element contains all elements that have the same probability. With this technique at hand the paper looks at the special case of the tournament isomorphism problem that stands in the way towards a polynomial-time algorithm for the graph isomorphism problem. Noting that there is a reduction from the automorphism (asymmetry) problem to the isomorphism problem, a reduction in the other direction is nevertheless not known and remains a thorny open problem. Applying the new technique, we develop a randomized polynomial-time Turing-reduction from the tournament isomorphism problem to the tournament automorphism problem. This is the first such reduction for any kind of combinatorial object not known to have a polynomial-time solvable isomorphism problem

Stable Marriage with Multi-Modal Preferences

We introduce a generalized version of the famous Stable Marriage problem, now based on multi-modal preference lists. The central twist herein is to allow each agent to rank its potentially matching counterparts based on more than one "evaluation mode" (e.g., more than one criterion); thus, each agent is equipped with multiple preference lists, each ranking the counterparts in a possibly different way. We introduce and study three natural concepts of stability, investigate their mutual relations and focus on computational complexity aspects with respect to computing stable matchings in these new scenarios. Mostly encountering computational hardness (NP-hardness), we can also spot few islands of tractability and make a surprising connection to the \textsc{Graph Isomorphism} problem

Kernelization of Whitney Switches

A fundamental theorem of Whitney from 1933 asserts that 2-connected graphs G and H are 2-isomorphic, or equivalently, their cycle matroids are isomorphic, if and only if G can be transformed into H by a series of operations called Whitney switches. In this paper we consider the quantitative question arising from Whitney's theorem: Given two 2-isomorphic graphs, can we transform one into another by applying at most k Whitney switches? This problem is already NP-complete for cycles, and we investigate its parameterized complexity. We show that the problem admits a kernel of size O(k), and thus, is fixed-parameter tractable when parameterized by k.Comment: To appear at ESA 202

Kernelization of Whitney Switches

A fundamental theorem of Whitney from 1933 asserts that 2-connected graphs $G$ and $H$ are 2-isomorphic, or equivalently, their cycle matroids are isomorphic if and only if $G$ can be transformed into $H$ by a series of operations called Whitney switches. In this paper we consider the quantitative question arising from Whitney's theorem: Given two 2-isomorphic graphs, can we transform one into another by applying at most $k$ Whitney switches? This problem is already \sf NP-complete for cycles, and we investigate its parameterized complexity. We show that the problem admits a kernel of size $\mathcal{O}(k)$ and thus is fixed-parameter tractable when parameterized by $k$.publishedVersio

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Glosarium Matematika

273 p.; 24 cm