An FPT algorithm for timeline cover

One of the most studied problem in theoretical computer science, VERTEX COVER, has been recently considered in the temporal graph framework. Here we study a VERTEX COVER variant, called k-TIMELINECOVER. Given a temporal graph k-TIMELINECOVER asks to define an interval for each vertex so that for every temporal edge existing in a timestamp t, at least one of the endpoints has an interval that includes t. The goal is to decide whether it is possible to cover every temporal edge while using vertex intervals of total span at most k. k-TIMELINECOVER has been shown to be NP-hard, but its parameterized complexity has not been fully understood when parameterizing by the span of the solution. We settle this open problem by giving an FPT algorithm that combines two techniques, a modified form of iterative compression and a reduction to DIGRAPH PAIR CUT.