Abstract
Partitioning genomes into syntenic blocks has many uses in comparative genomics, such as inferring phylogenies or ancestral gene order. These blocks are usually required to contain enough genes to be qualified as syntenic. This leads to the problem of finding a common partition of the genomes in which the size of the blocks are above a certain threshold (usually at least two). When this is not feasible, one can ask to remove a minimum number of “noisy” genes so that such a partition exists. This is known as the Strip Recovery problem and is similar to the well-known Minimum Common String Partition problem, but also quite different since the latter has no restriction on the block sizes.
The algorithmic aspects of Strip Recovery are not well-understood, especially in the presence of duplicated genes. In this work, we present several new complexity results. First, we solve an open problem mentioned by Bulteau and Weller in 2019 who asked whether, in polynomial time, one can decide if a common partition with block sizes at least two can be achieved without deleting any genes. We show that the problem is actually NP-hard for any fixed list of allowed block sizes, unless blocks sizes are all multiples of the minimum allowed size. The problem is also hard on fixed alphabets if this list is part of the input. However, the minimum number of required gene deletions can be found in polynomial time if both the allowed blocks sizes and alphabet are fixed.
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Notes
- 1.
Let us note that in practice, the same block can’t actually occur more than \(n/\min (F)\) times, where here n can be the minimum total length of \(\mathcal {W}_1\) or \(\mathcal {W}_2\). Hence, the smaller set \(\mathbb {C}(n/\min (F))\) should be considered in practice, but this does not help the complexity analysis.
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Lafond, M., Liyanage, A., Zhu, B., Zou, P. (2022). The Complexity of Finding Common Partitions of Genomes with Predefined Block Sizes. In: Jin, L., Durand, D. (eds) Comparative Genomics. RECOMB-CG 2022. Lecture Notes in Computer Science(), vol 13234. Springer, Cham. https://doi.org/10.1007/978-3-031-06220-9_7
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