Parameterized algorithms for Module Map problems

Frank Sommer, Christian Komusiewicz

Abstract: Motivated by applications in the analysis of genetic networks, we introduce and study the NP-hard Module Map problem which has as input a graph G=(V,E)<math><mrow is="true"><mi is="true">G</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mrow is="true"><mo is="true">(</mo><mi is="true">V</mi><mo is="true">,</mo><mi is="true">E</mi><mo is="true">)</mo></mrow></mrow></math> with red and blue edges and an integer k<math><mi is="true">k</mi></math> and asks to transform G<math><mi is="true">G</mi></math> by at most k<math><mi is="true">k</mi></math> edge modifications into a graph G′<math><msup is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mo is="true">′</mo></mrow></msup></math> which has the following properties: the vertex set of G′<math><msup is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mo is="true">′</mo></mrow></msup></math> can be partitioned into so-called clusters such that inside a cluster every pair of vertices is connected by a blue edge and for two distinct clusters A<math><mi is="true">A</mi></math> and B<math><mi is="true">B</mi></math> either all vertices u∈A<math><mrow is="true"><mi is="true">u</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">∈</mo><mi is="true">A</mi></mrow></math> and v∈B<math><mrow is="true"><mi is="true">v</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">∈</mo><mi is="true">B</mi></mrow></math> are connected by a red edge or there is no edge between A<math><mi is="true">A</mi></math> and B<math><mi is="true">B</mi></math>. We show that Module Map can be solved in O(3k⋅(|V|+|E|))<math><mrow is="true"><mi mathvariant="script" is="true">O</mi><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mn is="true">3</mn></mrow><mrow is="true"><mi is="true">k</mi></mrow></msup><mi is="true">⋅</mi><mrow is="true"><mo is="true">(</mo><mrow is="true"><mo is="true">|</mo><mi is="true">V</mi><mo is="true">|</mo></mrow><mo is="true">+</mo><mrow is="true"><mo is="true">|</mo><mi is="true">E</mi><mo is="true">|</mo></mrow><mo is="true">)</mo></mrow><mo is="true">)</mo></mrow></mrow></math> time and O(2k⋅|V|3)<math><mrow is="true"><mi mathvariant="script" is="true">O</mi><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">k</mi></mrow></msup><mi is="true">⋅</mi><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">|</mo><mi is="true">V</mi><mo is="true">|</mo></mrow></mrow><mrow is="true"><mn is="true">3</mn></mrow></msup><mo is="true">)</mo></mrow></mrow></math> time, respectively. Furthermore, we show that Module Map admits a kernel with O(k2)<math><mrow is="true"><mi mathvariant="script" is="true">O</mi><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">k</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo is="true">)</mo></mrow></mrow></math> vertices.