Solving Stackelberg Vertex Cover on trees using split and join

The Stackelberg Vertex Cover problem is a bilevel optimization problem with two players on a graph $G = (F \cup P, E)$ where each vertex from $F$ has a weight and the first player selects a price for each vertex in $P$. Afterwards, the second player finds a minimum vertex cover $X$ and the first player receives the set price for each vertex from $X \cap P$. The goal is to maximize the revenue of the first player. This problem was recently shown to be NP-complete for bipartite graphs while being solvable in linear time on paths. We present three new algorithms for solving Stackelberg Vertex Cover on certain kinds of trees: (1) a pseudo-polynomial algorithm working on general trees when all weights are integer, i.e., it is FPT with the maximum weight as a parameter; (2) a strongly polynomial algorithm for trees having the property that the least common ancestor of any two vertices from $P$ is again in $P$ (this case includes paths); and (3) an FPT-algorithm for trees, where the parameter is the maximum number $P$-vertices $v_i$ that an $F$-vertex $u$ can reach while using no other $P$-vertices. These algorithms are based on a lemma that allows us to split instances at a vertex $u$ into multiple sub-instances, which follows from LP duality and integrality of the vertex cover LP on bipartite graphs. The lemma requires that the minimum vertex covers of the sub-instances agree on $u$ (either all include $u$ or all don't). For this we introduce the concept of commitments. Finally, we show that the Stackelberg Vertex Cover problem with commitments is weakly NP-complete.