The Target Set Selection problem takes as an input a graph G and a nonnegative integer threshold thr(v) for every vertex v. A vertex v can get active as soon as at least thr(v) of its neighbors have been activated. The objective is to select a smallest possible initial set of vertices, the target set, whose activation eventually leads to the activation of all vertices in the graph.We show that Target Set Selection is in FPT when parameterized with the combined parameters clique-width of the graph and the maximum threshold value. This generalizes all previous FPT-membership results for the parameterization by maximum threshold, and thereby solves an open question from the literature. We stress that the time complexity of our algorithm is surprisingly well-behaved and grows only single-exponentially in the parameters.

We investigate the so-called recoverable robust assignment problem on balanced bipartite graphs with 2n vertices, a mainstream problem in robust optimization: For two given linear cost functions c1 and c2 on the edges and a given integer k, the goal is to find two perfect matchings M1 and M2 that minimize the objective value c1(M1) + c2(M2), subject to the constraint that M1 and M2 have at least k edges in common.We derive a variety of results on this problem. First, we show that the problem is W[1]-hard with respect to the parameter k, and also with respect to the recoverability parameter k = n − k. This hardness result holds even in the highly restricted special case where both cost functions c1 and c2 only take the values 0 and 1. (On the other hand, containment of the problem in XP is straightforward to see.) Next, as a positive result we construct a polynomial time algorithm for the special case where one cost function is Monge, whereas the other one is Anti-Monge. Finally, we study the variant where matching M1 is frozen, and where the optimization goal is to compute the best corresponding matching M2, the second stage recoverable assignment problem. We show that this problem variant is contained in the randomized parallel complexity class RNC2, and that it is at least as hard as the infamous problem Exact Matching in Red-Blue Bipartite Graphs whose computational complexity is a long-standing open problem.

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We study a continuous facility location problem on undirected graphs where all edges have unit length and where the facilities may be positioned on the vertices as well as on interior points of the edges. The goal is to cover the entire graph with a minimum number of facilities with covering range $$\delta >0$$ δ > 0 . In other words, we want to position as few facilities as possible subject to the condition that every point on every edge is at distance at most $$\delta $$ δ from one of these facilities. We investigate this covering problem in terms of the rational parameter $$\delta $$ δ . We prove that the problem is polynomially solvable whenever $$\delta $$ δ is a unit fraction, and that the problem is NP-hard for all non unit fractions $$\delta $$ δ . We also analyze the parametrized complexity with the solution size as parameter: The resulting problem is fixed parameter tractable for $$\delta <3/2$$ δ < 3 / 2 , and it is W[2]-hard for $$\delta \ge 3/2$$ δ ≥ 3 / 2 .

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