| Summary: | A <i>multi-cut rearrangement</i> of a string <i>S</i> is a string <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>S</mi><mo>′</mo></msup></semantics></math></inline-formula> obtained from <i>S</i> by an operation called <i>k-cut rearrangement</i>, that consists of (1) cutting <i>S</i> at a given number <i>k</i> of places in <i>S</i>, making <i>S</i> the concatenated string <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>·</mo><msub><mi>X</mi><mn>2</mn></msub><mo>·</mo><msub><mi>X</mi><mn>3</mn></msub><mo>·</mo><mi>…</mi><mo>·</mo><msub><mi>X</mi><mi>k</mi></msub><mo>·</mo><msub><mi>X</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>X</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>X</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> are possibly empty, and (2) rearranging the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>X</mi><mi>i</mi></msub></semantics></math></inline-formula>s so as to obtain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>S</mi><mo>′</mo></msup><mo>=</mo><msub><mi>X</mi><mrow><mi>π</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msub><mo>·</mo><msub><mi>X</mi><mrow><mi>π</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msub><mo>·</mo><msub><mi>X</mi><mrow><mi>π</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msub><mo>·</mo><mi>…</mi><mo>·</mo><msub><mi>X</mi><mrow><mi>π</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula> being a permutation on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula> satisfying <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Given two strings <i>S</i> and <i>T</i> built on the same multiset of characters from an alphabet <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>Σ</mi></semantics></math></inline-formula>, the <span style="font-variant: small-caps;">Sorting by Multi-Cut Rearrangements</span> (<span style="font-variant: small-caps;">SMCR</span>) problem asks whether a given number <i>ℓ</i> of <i>k</i>-cut rearrangements suffices to transform <i>S</i> into <i>T</i>. The <span style="font-variant: small-caps;">SMCR</span> problem generalizes several classical genomic rearrangements problems, such as <span style="font-variant: small-caps;">Sorting by Transpositions</span> and <span style="font-variant: small-caps;">Sorting by Block Interchanges</span>. It may also model <i>chromoanagenesis</i>, a recently discovered phenomenon consisting of massive simultaneous rearrangements. In this paper, we study the <span style="font-variant: small-caps;">SMCR</span> problem from an algorithmic complexity viewpoint. More precisely, we investigate its classical and parameterized complexity, as well as its approximability, in the general case or when <i>S</i> and <i>T</i> are permutations.
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