Dealing with several parameterized problems by random methods

Qilong Feng, Neng Huang, Xiong Jiang, Jianxin Wang

Abstract: In this paper, we apply random methods to deal with several parameterized problems. For the Parameterized Weighted P3<math><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mn is="true">3</mn></mrow></msub></math>-Packing problem, by randomly partitioning the vertices in given graph, a tripartite graph can be obtained. We prove that the Parameterized Weighted P3<math><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mn is="true">3</mn></mrow></msub></math>-Packing problem can be solved in polynomial time on tripartite graphs. Based on the algorithm on tripartite graphs, a randomized parameterized algorithm of running time ⁎O⁎(32k)<math><msup is="true"><mrow is="true"><mi is="true">O</mi></mrow><mrow is="true"><mo is="true">⁎</mo></mrow></msup><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mn is="true">32</mn></mrow><mrow is="true"><mi is="true">k</mi></mrow></msup><mo stretchy="false" is="true">)</mo></math> is given for the Parameterized Weighted P3<math><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mn is="true">3</mn></mrow></msub></math>-Packing problem. For the Parameterized Weighted Load Coloring problem, by randomly partitioning the vertices in given graph into two parts and studying the structure properties of the connected components in two parts, a randomized parameterized algorithm of running time ⁎O⁎(11.32k)<math><msup is="true"><mrow is="true"><mi is="true">O</mi></mrow><mrow is="true"><mo is="true">⁎</mo></mrow></msup><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mn is="true">11.32</mn></mrow><mrow is="true"><mi is="true">k</mi></mrow></msup><mo stretchy="false" is="true">)</mo></math> is presented. For the Parameterized Claw-free Edge Deletion problem on Diamond-free Graphs, by combining random with branching methods, a parameterized algorithm of running time ⁎O⁎(2.42k)<math><msup is="true"><mrow is="true"><mi is="true">O</mi></mrow><mrow is="true"><mo is="true">⁎</mo></mrow></msup><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mn is="true">2.42</mn></mrow><mrow is="true"><mi is="true">k</mi></mrow></msup><mo stretchy="false" is="true">)</mo></math> is given.