Polynomial kernels for tracking shortest paths

Václav Blazej, Pratibha Choudhary, Dusan Knop, Jan Matyás Kristan, Ondrej Suchý, Tomás Valla

Abstract: Highlights•A kernel with O(k2)<math><mi mathvariant="script" is="true">O</mi><mo stretchy="false" 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 stretchy="false" is="true">)</mo></math> edges for Tracking Paths in DAG, where k is the size of the solution.•A kernel with O(k)<math><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">k</mi><mo stretchy="false" is="true">)</mo></math> edges for Tracking Paths in DAG, when the input graph is planar.•For Tracking Shortest Paths, this implies a kernel with O(k4)<math><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">k</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math> edges in general graphs and O(k2)<math><mi mathvariant="script" is="true">O</mi><mo stretchy="false" 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 stretchy="false" is="true">)</mo></math> edges in planar graphs.•Single exponential algorithm for Tracking Shortest Paths in planar graphs.