Improved deterministic algorithms for weighted matching and packing problems

Jianer Chen, Qilong Feng, Yang Liu, Songjian Lu, Jianxin Wang

Abstract: Based on the method of (n,k)<math><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">,</mo><mi is="true">k</mi><mo is="true">)</mo></mrow></math>-universal sets, we present a deterministic parameterized algorithm for the weighted rD-MATCHING<math><mi is="true">r</mi><mtext is="true"><mi mathsize="small" is="true">D-MATCHING</mi></mtext></math> problem with time complexity O∗(4(r−1)k+o(k))<math><msup is="true"><mrow is="true"><mi is="true">O</mi></mrow><mrow is="true"><mo is="true">∗</mo></mrow></msup><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mn is="true">4</mn></mrow><mrow is="true"><mrow is="true"><mo is="true">(</mo><mi is="true">r</mi><mo is="true">−</mo><mn is="true">1</mn><mo is="true">)</mo></mrow><mi is="true">k</mi><mo is="true">+</mo><mi is="true">o</mi><mrow is="true"><mo is="true">(</mo><mi is="true">k</mi><mo is="true">)</mo></mrow></mrow></msup><mo is="true">)</mo></mrow></math>, improving the previous best upper bound O∗(4rk+o(k))<math><msup is="true"><mrow is="true"><mi is="true">O</mi></mrow><mrow is="true"><mo is="true">∗</mo></mrow></msup><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mn is="true">4</mn></mrow><mrow is="true"><mi is="true">r</mi><mi is="true">k</mi><mo is="true">+</mo><mi is="true">o</mi><mrow is="true"><mo is="true">(</mo><mi is="true">k</mi><mo is="true">)</mo></mrow></mrow></msup><mo is="true">)</mo></mrow></math>. In particular, the algorithm applied to the unweighted 3d-matching problem results in a deterministic algorithm with time O∗(16k+o(k))<math><msup is="true"><mrow is="true"><mi is="true">O</mi></mrow><mrow is="true"><mo is="true">∗</mo></mrow></msup><mrow is="true"><mo is="true">(</mo><mn is="true">1</mn><msup is="true"><mrow is="true"><mn is="true">6</mn></mrow><mrow is="true"><mi is="true">k</mi><mo is="true">+</mo><mi is="true">o</mi><mrow is="true"><mo is="true">(</mo><mi is="true">k</mi><mo is="true">)</mo></mrow></mrow></msup><mo is="true">)</mo></mrow></math>, improving the previous best result O∗(21.26k)<math><msup is="true"><mrow is="true"><mi is="true">O</mi></mrow><mrow is="true"><mo is="true">∗</mo></mrow></msup><mrow is="true"><mo is="true">(</mo><mn is="true">21.2</mn><msup is="true"><mrow is="true"><mn is="true">6</mn></mrow><mrow is="true"><mi is="true">k</mi></mrow></msup><mo is="true">)</mo></mrow></math>. For the weighted r<math><mi is="true">r</mi></math>-set packing problem, we present a deterministic parameterized algorithm with time complexity O∗(2(2r−1)k+o(k))<math><msup is="true"><mrow is="true"><mi is="true">O</mi></mrow><mrow is="true"><mo is="true">∗</mo></mrow></msup><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mrow is="true"><mo is="true">(</mo><mn is="true">2</mn><mi is="true">r</mi><mo is="true">−</mo><mn is="true">1</mn><mo is="true">)</mo></mrow><mi is="true">k</mi><mo is="true">+</mo><mi is="true">o</mi><mrow is="true"><mo is="true">(</mo><mi is="true">k</mi><mo is="true">)</mo></mrow></mrow></msup><mo is="true">)</mo></mrow></math>, improving the previous best result O∗(22rk+o(k))<math><msup is="true"><mrow is="true"><mi is="true">O</mi></mrow><mrow is="true"><mo is="true">∗</mo></mrow></msup><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mn is="true">2</mn><mi is="true">r</mi><mi is="true">k</mi><mo is="true">+</mo><mi is="true">o</mi><mrow is="true"><mo is="true">(</mo><mi is="true">k</mi><mo is="true">)</mo></mrow></mrow></msup><mo is="true">)</mo></mrow></math>. The algorithm, when applied to the unweighted 3-set packing problem, has running time O∗(32k+o(k))<math><msup is="true"><mrow is="true"><mi is="true">O</mi></mrow><mrow is="true"><mo is="true">∗</mo></mrow></msup><mrow is="true"><mo is="true">(</mo><mn is="true">3</mn><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">k</mi><mo is="true">+</mo><mi is="true">o</mi><mrow is="true"><mo is="true">(</mo><mi is="true">k</mi><mo is="true">)</mo></mrow></mrow></msup><mo is="true">)</mo></mrow></math>, improving the previous best result O∗(43.62k+o(k))<math><msup is="true"><mrow is="true"><mi is="true">O</mi></mrow><mrow is="true"><mo is="true">∗</mo></mrow></msup><mrow is="true"><mo is="true">(</mo><mn is="true">43.6</mn><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">k</mi><mo is="true">+</mo><mi is="true">o</mi><mrow is="true"><mo is="true">(</mo><mi is="true">k</mi><mo is="true">)</mo></mrow></mrow></msup><mo is="true">)</mo></mrow></math>. Moreover, for the weighted r<math><mi is="true">r</mi></math>-set packing and weighted rD-MATCHING<math><mi is="true">r</mi><mtext is="true"><mi mathsize="small" is="true">D-MATCHING</mi></mtext></math> problems, we give a kernel of size O(kr)<math><mi 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"><mi is="true">r</mi></mrow></msup><mo is="true">)</mo></mrow></math>, which is the first kernelization algorithm for the problems on weighted versions.