AI 中文总结
研究树上的斯塔克尔伯格顶点覆盖问题，提出三种新算法，包括一般树的伪多项式算法、特定树的强多项式算法和树的FPT算法，基于引理拆分实例，引入承诺概念，还证明带承诺问题是弱NP完全的。
AI 中文摘要
斯塔克尔伯格顶点覆盖问题是图\(G = (F \cup P, E)\)上两个参与者的双层优化问题，\(F\)中每个顶点有一个权重，第一个参与者为\(P\)中每个顶点选择一个价格。之后，第二个参与者找到最小顶点覆盖\(X\)，第一个参与者从\(X \cap P\)中的顶点获得设定价格，目标是最大化第一个参与者的收益。该问题最近被证明对于二分图是NP完全的，而在路径上可在线性时间内解决。我们提出三种新算法来解决特定类型树上的斯塔克尔伯格顶点覆盖问题：一是所有权重为整数时在一般树上的伪多项式算法，以最大权重为参数是FPT；二是对于\(P\)中任意两个顶点的最近公共祖先仍在\(P\)中的树的强多项式算法（包括路径情况）；三是对于树的FPT算法，参数是\(F\)顶点\(u\)在不使用其他\(P\)顶点时能到达的最大\(P\)顶点数\(v_i\)。这些算法基于一个引理，该引理允许我们在顶点\(u\)处将实例拆分为多个子实例，这源于二分图上顶点覆盖LP对偶性和整数性。引理要求子实例的最小顶点覆盖在\(u\)上一致（要么都包含\(u\)，要么都不包含）。为此我们引入了承诺的概念。最后，我们表明带有承诺的斯塔克尔伯格顶点覆盖问题是弱NP完全的。
英文摘要
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 weight 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 four new algorithms for solving Stackelberg Vertex Cover on certain kinds of graphs: (1) a pseudo-polynomial algorithm working on general trees when all weights are integer with a runtime linear in the number of vertices and cubic in the maximum weight (2) a generalization of (1) for bipartite graphs with integer weights and a tree decomposition that is FPT in the maximum weight and the treewidth, (3) a strongly polynomial algorithm for rooted trees having the property that the least common ancestor of any two vertices from P is again in P (this case includes paths); and (4) 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. We show that the Stackelberg Vertex Cover problem with commitments is weakly NP-complete. An open question is the non-bipartite case as there is an explicit counterexample showing that the split-and-join technique does not work.
