### Approximability and Fixed-Parameter Tractability for the Exemplar Genomic Distance Problems [chapter]

Binhai Zhu
<span title="">2009</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
In this paper, we present a survey of the approximability and fixed-parameter tractability results for some Exemplar Genomic Distance problems. We mainly focus on three problems: the exemplar breakpoint distance problem and its complement (i.e., the exemplar non-breaking similarity or the exemplar adjacency number problem), and the maximal strip recovery (MSR) problem. The following results hold for the simplest case between only two genomes (genomic maps) G and H, each containing only one
more &raquo; ... nce of genes (gene markers), possibly with repetitions. 1. For the general Exemplar Breakpoint Distance problem, it was shown that deciding if the optimal solution value of some given instance is zero is NP-hard. This implies that the problem does not admit any approximation, neither any FPT algorithm, unless P=NP. In fact, this result holds even when a gene appears in G (H) at most two times. 2. For the Exemplar Non-breaking Similarity problem, it was shown that the problem is linearly reducible from Independent Set. Hence, it does not admit any factor-O(n ) approximation unless P=NP and it is W[1]-complete (loosely speaking, there is no way to obtain an O(n o(k) ) time exact algorithm unless FPT=W[1], here k is the optimal solution value of the problem). 3. For the MSR problem, after quite a lot of struggle, we recently showed that the problem is NP-complete. On the other hand, the problem was previously known to have a factor-4 approximation and we showed recently that it admits a simple FPT algorithm which runs in O(2 3.61k n + n 2 ) time, where k is the optimal solution value of the problem.
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