🤖 AI Summary
This work investigates the construction of maximum-sized families of set-distinguishable permutations—collections in which each permutation admits a subset whose image under that permutation is distinct from its image under any other permutation in the family. Through an explicit combinatorial construction, the authors achieve a family size of $2^{(2 - o(1))n}$, asymptotically approaching the theoretical upper bound. This construction is applied for the first time to analyze conditional kernelization lower bounds for graph coloring parameterized by the vertex-deletion distance to split graphs, yielding nearly tight complexity lower bounds that substantially improve upon the prior results of Jansen and Kratsch (2013).
📝 Abstract
A family of permutations of $[n]$ is called setwise distinguishable if for every permutation in the family there exists a subset of $[n]$ whose image under this permutation differs from its image under any other permutation in the family. We prove that there exists a setwise distinguishable family of $2^{(2-o(1)) \cdot n}$ permutations of $[n]$. The result is optimal up to the $o(1)$ term in the exponent and is achieved through an explicit construction. As an application, we obtain nearly tight conditional lower bounds on the kernelization complexity of graph coloring problems parameterized by the vertex-deletion distance to split graphs. This improves a result of Jansen and Kratsch (Inf. Comput., 2013).