| Popis: |
Genome rearrangements are events in which large blocks of DNA exchange pieces during evolution. The analysis of such events is a tool for understanding evolutionary genomics, based on finding the minimum number of rearrangements to transform one genome into another. In a general scenario, more than two genomes are considered and we have new challenges. The {\sc Median} problem consists in finding, given three permutations and a distance metric, a permutation $s$ that minimizes the sum of the distances between $s$ and each input. We study the {\sc median} problem over \emph{swap} distances in permutations, for which the computational complexity has been open for almost 20 years (Eriksen, \emph{Theor. Compt. Sci.}, 2007). We consider this problem through some branches. We associate median solutions and interval convex sets, where the concept of graph convexity inspires the following investigation: Does a median permutation belong to every shortest path between one of the pairs of input permutations? We are able to partially answer this question, and as a by-product we solve a long open problem by proving that the {\sc Swap Median} problem is NP-hard. Furthermore, using a similar approach, we show that the {\sc Closest} problem, which seeks to minimize the maximum distance between the solution and the input permutations, is NP-hard even considering three input permutations. This gives a sharp dichotomy into the P vs. NP-hard approaches, since considering two input permutations the problem is easily solvable and considering any number of input permutations it is known to be NP-hard since 2007 (Popov, \emph{Theor. Compt. Sci.}, 2007). In addition, we show that {\sc Swap Median} and {\sc Swap Closest} are APX-hard problems. |
