Complexity and algorithms for Swap median and relation to other consensus problems (2409.09734v2)

Abstract: 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, which can be modeled as permutations of integers. In a general scenario, more than two genomes are considered, and new challenges arise. Given three input permutations, the Median problem consists of finding a permutation s that minimizes the sum of the distances between s and each of the three input permutations, according to a specified distance measure. We prove that Median problem over swap distances is NP-complete, a problem whose computational complexity has remained unsolved for nearly 20 years (Eriksen, Theor. Comput. Sci., 2007). To tackle this problem, we introduce a graph-based perspective by the class called 2-circles-intersection graphs. We show that for each 2-circles-intersection graph G, we can associate three permutations such that G has a large independent set iff the median of the three associated permutations reaches a specific lower bound. We then prove that maximum independent set is NP-complete in this graph class. By this approach, we also establish that the Closest problem which aims to minimize the maximum distance between the solution and the input permutations is NP-complete even with three input permutations. This last result closes the complexity gap in the dichotomy between P and NP-complete cases: with two input permutations, the problem is easily solvable, while for an arbitrary number of input permutations, the Closest problem was known to be NP-hard since 2007 (Popov, Theor. Comput. Sci., 2007). Additionally, we show that both the Swap Median and Swap Closest problems are APX-hard, further emphasizing the computational complexity of these genome-related problems through graph theory.