Rescheduling problems arise when unpredicted events occur, such as the arrival of new orders. These new jobs should be integrated in a proper way in the existing schedule of the so-called old jobs, with the aim of minimizing an objective function for the joint set of jobs. To avoid a major disruption of the original schedule, each old job is not allowed to deviate from its original completion time by more than a certain threshold. Filling a gap in the existing literature, we consider the minimization of the total weighted completion time. The resulting rescheduling problem is shown to be weakly NP-hard and several properties of the structure of an optimal schedule are derived. These can be used for the construction of an exact dynamic programming algorithm with pseudo-polynomial running time. A fully polynomial time approximation scheme is obtained from the dynamic program by three different scaling and reduction steps. Finally, for the minimization of the number of late jobs a strong NP-hardness result is derived. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: This work was partially supported by the Ministero dell’Istruzione, dell’Università e della Ricerca [Award TESUN-83486178370409 finanziamento dipartimenti di eccellenza CAP. 1694 TIT. 232 ART. 6]. U. Pferschy acknowledges support by the Field of Excellence COLIBRI at the University of Graz.

Abstract An instance of the non-preemptive tree packing problem consists of an undirected graph $$G=(V,E)$$ G = ( V , E ) together with a weight w(e) for every edge $$e\in E$$ e ∈ E . The goal is to activate every edge e for some time interval of length w(e), such that the activated edges keep G connected for the longest possible overall time. We derive a variety of results on this problem. The problem is strongly NP-hard even on graphs of treewidth 2, and it does not allow a polynomial time approximation scheme (unless P=NP). Furthermore, we discuss the performance of a simple greedy algorithm, and we construct and analyze a number of parameterized and exact algorithms.

This paper studies the allocation of indivisible items to agents, when each agent’s preferences are expressed by means of a directed acyclic graph. The vertices of each preference graph represent the subset of items approved of by the respective agent. An arc (a,b) in such a graph means that the respective agent prefers item a over item b. We introduce a new measure of dissatisfaction of an agent by counting the number of non-assigned items which are approved of by the agent and for which no more preferred item is allocated to the agent. Considering two problem variants, we seek an allocation of the items to the agents in a way that minimizes (i) the total dissatisfaction over all agents or (ii) the maximum dissatisfaction among the agents. For both optimization problems we study the status of computational complexity and obtain NP-hardness results as well as polynomial algorithms with respect to natural underlying graph structures, such as stars, trees, paths, and matchings. We also analyze the parameterized complexity of the two problems with respect to various parameters related to the number of agents, the dissatisfaction threshold, the vertex degrees of the preference graphs, and the treewidth.

We continue the study of $δ$-dispersion, a continuous facility location problem on a graph where all edges have unit length and where the facilities may also be positioned in the interior of the edges. The goal is to position as many facilities as possible subject to the condition that every two facilities have distance at least $δ$ from each other. Our main technical contribution is an efficient procedure to `round-up' distance $δ$. It transforms a $δ$-dispersed set $S$ into a $δ^\star$-dispersed set $S^\star$ of same size where distance $δ^\star$ is a slightly larger rational $\tfrac{a}{b}$ with a numerator $a$ upper bounded by the longest (not-induced) path in the input graph. Based on this rounding procedure and connections to the distance-$d$ independent set problem we derive a number of algorithmic results. When parameterized by treewidth, the problem is in XP. When parameterized by treedepth the problem is FPT and has a matching lower bound on its time complexity under ETH. Moreover, we can also settle the parameterized complexity with the solution size as parameter using our rounding technique: $δ$-\dispersion is FPT for every $δ\leq 2$ and W[1]-hard for every $δ> 2$. Further, we show that $δ$-dispersion is NP-complete for every fixed irrational distance $δ$, which was left open in a previous work.

Abstract We study a continuous facility location problem on a graph where all edges have unit length and where the facilities may also be positioned in the interior of the edges. The goal is to position as many facilities as possible subject to the condition that any two facilities have at least distance $$\delta$$ δ from each other. We investigate the complexity of this problem in terms of the rational parameter $$\delta$$ δ . The problem is polynomially solvable, if the numerator of $$\delta$$ δ is 1 or 2, while all other cases turn out to be NP-hard.

The feasible solutions in the robust recoverable selection problem are subsets of size p that are to be selected from a ground set of size n. The objective is to construct a feasible solution in two sequential stages with two separate (but interleaved) cost structures. The fastest algorithm for this problem in the literature up to now has quadratic running time. We improve on this by developing an algorithm with linear running time.

We study a continuous facility location problem on a graph where all edges have unit length and where the facilities may also be positioned in the interior of the edges. The goal is to position as many facilities as possible subject to the condition that any two facilities have at least distance delta from each other. We investigate the complexity of this problem in terms of the rational parameter delta. The problem is polynomially solvable, if the numerator of delta is 1 or 2, while all other cases turn out to be NP-hard.