Reconfiguration and Enumeration of Optimal Cyclic Ladder Lotteries

Abstract

A ladder lottery of a permutation $\pi$ of $\{1,2,\ldots ,n\}$ is a network with n vertical lines and zero or more horizontal lines each of which connects two consecutive vertical lines and corresponds to an adjacent transposition. The composition of all the adjacent transpositions coincides with $\pi$. A cyclic ladder lottery of $\pi$ is a ladder lottery of $\pi$ that is allowed to have horizontal lines between the first and last vertical lines. A cyclic ladder lottery of $\pi$ is optimal if it has the minimum number of horizontal lines. In this paper, for optimal cyclic ladder lotteries, we consider the reconfiguration and enumeration problems. First, we investigate the two problems when a permutation $\pi$ and its optimal displacement vector $\boldsymbol{x}$ are given. Then, we show that any two optimal cyclic ladder lotteries of $\pi$ and $\boldsymbol{x}$ are always reachable under braid relations and one can enumerate all the optimal cyclic ladder lotteries in polynomial delay. Next, we consider the two problems for optimal displacement vectors when a permutation $\pi$ is given. Then, we present a characterization of the length of a shortest reconfiguration sequence of two optimal displacement vectors and show that there exists a constant-delay algorithm that enumerates all the optimal displacement vectors of $\pi$.

This work was supported by JSPS KAKENHI Grant Numbers JP18H04091, JP20H05793, JP20K14317, and JP22K17849.