Dealing with NP-hard problems, kernelization is a fundamental notion for polynomial-time data reduction with performance guarantees: in polynomial time, a problem instance is reduced to an equivalent instance with size upper-bounded by a function of a parameter chosen in advance. Kernelization for weighted problems particularly requires to also shrink weights. Marx and V\'egh [ACM Trans. Algorithms 2015] and Etscheid et al. [J. Comput. Syst. Sci. 2017] used a technique of Frank and Tardos [Combinatorica 1987] to obtain polynomial-size kernels for weighted problems, mostly with additive goal functions. We characterize the function types that the technique is applicable to, which turns out to contain many non-additive functions. Using this insight, we systematically obtain kernelization results for natural problems in graph partitioning, network design, facility location, scheduling, vehicle routing, and computational social choice, thereby improving and generalizing results from the literature.

We study a problem of energy-efficiently connecting a symmetric wireless communication network: given an n-vertex graph with edge weights, find a connected spanning subgraph of minimum cost, where the cost is determined by each vertex paying the heaviest edge incident to it in the subgraph. The problem is known to be NP-hard. Strengthening this hardness result, we show that even o(log n)-approximating the difference d between the optimal solution cost and a natural lower bound is NP-hard. Moreover, we show that under the exponential time hypothesis, there are no exact algorithms running in 2o(n) time or in [Formula: see text] time for any computable function f. We also show that the special case of connecting c network components with minimum additional cost generally cannot be polynomial-time reduced to instances of size cO(1) unless the polynomial-time hierarchy collapses. On the positive side, we provide an algorithm that reconnects O(log n)-connected components with minimum additional cost in polynomial time. These algorithms are motivated by application scenarios of monitoring areas or where an existing sensor network may fall apart into several connected components because of sensor faults. In experiments, the algorithm outperforms CPLEX with known integer linear programming (ILP) formulations when n is sufficiently large compared with c. Summary of Contribution: Wireless sensor networks are used to monitor air pollution, water pollution, and machine health; in forest fire and landslide detection; and in natural disaster prevention. Sensors in wireless sensor networks are often battery-powered and disposable, so one may be interested in lowering the energy consumption of the sensors in order to achieve a long lifetime of the network. We study the min-power symmetric connectivity problem, which models the task of assigning transmission powers to sensors so as to achieve a connected communication network with minimum total power consumption. The problem is NP-hard. We provide perhaps the first parameterized complexity study of optimal and approximate solutions for the problem. Our algorithms work in polynomial time in the scenario where one has to reconnect a sensor network with n sensors and O(log n)-connected components by means of a minimum transmission power increase or if one can find transmission power lower bounds that already yield a network with O(log n)-connected components. In experiments, we show that, in this scenario, our algorithms outperform previously known exact algorithms based on ILP formulations.

Abstract Given a graph G = (V, E), two vertices s, t ∈ V, and two integers k, ℓ, the Short Secluded Path problem is to find a simple s‐t‐path with at most k vertices and ℓ neighbors. We study the parameterized complexity of the problem with respect to four structural graph parameters: the vertex cover number, treewidth, feedback vertex number, and feedback edge number. In particular, we completely settle the question of the existence of problem kernels with size polynomial in these parameters and their combinations with k and ℓ. We also obtain a 2O(tw) · ℓ2 · n‐time algorithm for n‐vertex graphs of treewidth tw, which yields subexponential‐time algorithms in several graph classes.

Abstract Given an undirected graph with edge weights and a subset R of its edges, the Rural Postman Problem (RPP) is to find a closed walk of minimum total weight containing all edges of R. We prove that RPP is WK[1]‐complete parameterized by the number and weight d of edges traversed additionally to the required ones. Thus RPP instances cannot be polynomial‐time compressed to instances of size polynomial in d unless the polynomial‐time hierarchy collapses. In contrast, denoting by b ≤ 2d the number of vertices incident to an odd number of edges of R and by c ≤ d the number of connected components formed by the edges in R, we show how to reduce any RPP instance I to an RPP instance I′ with 2b + O(c/ϵ) vertices in O(n3) time so that any α‐approximate solution for I′ gives an α(1 + ϵ)‐approximate solution for I, for any α ≥ 1 and ϵ > 0. That is, we provide a polynomial‐size approximate kernelization scheme (PSAKS). We experimentally evaluate it on wide‐spread benchmark data sets as well as on two real snow plowing instances from Berlin. We also make first steps toward a PSAKS for the parameter c.

We study a problem that models safely routing a convoy through a transportation network, where any vertex adjacent to the travel path of the convoy requires additional precaution: Given a graph G=(V,E), two vertices s,t in V, and two integers k,l, we search for a simple s-t-path with at most k vertices and at most l neighbors. We study the problem in two types of transportation networks: graphs with small crossing number, as formed by road networks, and tree-like graphs, as formed by waterways. For graphs with constant crossing number, we provide a subexponential 2^O(sqrt n)-time algorithm and prove a matching lower bound. We also show a polynomial-time data reduction algorithm that reduces any problem instance to an equivalent instance (a so-called problem kernel) of size polynomial in the vertex cover number of the input graph. In contrast, we show that the problem in general graphs is hard to preprocess. Regarding tree-like graphs, we obtain a 2^O(tw) * l^2 * n-time algorithm for graphs of treewidth tw, show that there is no problem kernel with size polynomial in tw, yet show a problem kernel with size polynomial in the feedback edge number of the input graph.

We prove that any polynomial‐time ‐approximation algorithm for then‐vertex metric asymmetric Traveling Salesperson Problem yields a polynomial‐time ‐approximation algorithm for the mixed and windy Capacitated Arc Routing Problem, where is the number of weakly connected components in the subgraph induced by the positive‐demand arcs—a small number in many applications. In conjunction with known results, we obtain constant‐factor approximations for and ‐approximations in general. Experiments show that our algorithm, together with several heuristic enhancements, outperforms many previous polynomial‐time heuristics. Finally, since the solution quality achievable in polynomial time appears to mainly depend onCand sinceC = 1 in almost all benchmark instances, we propose the Ob benchmark set, simulating cities that are divided into several components by a river. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(3), 262–278 2017

The h-index is an important bibliographic measure used to assess the performance of researchers. Van Bevern et al. [Artif. Intel., to appear] showed that, despite computational worst-case hardness results, substantial manipulation of the h-index of Google Scholar author profiles is possible by merging articles. Complementing this work, we study the opposite operation, the splitting of articles, which is arguably the more natural operation for manipulation and which is also allowed within Google Scholar. We present numerous results on computational complexity (from linear-time algorithms to parameterized computational hardness results) and empirically indicate that at least small improvements of the h-index by splitting merged articles are easily achievable.

Finding a vertex subset in a graph that satisfies a certain property is one of the most-studied topics in algorithmic graph theory. The focus herein is often on minimizing or maximizing the size of the solution, that is, the size of the desired vertex set. In several applications, however, we also want to limit the "exposure" of the solution to the rest of the graph. This is the case, for example, when the solution represents persons that ought to deal with sensitive information or a segregated community. In this work, we thus explore the (parameterized) complexity of finding such secluded vertex subsets for a wide variety of properties that they shall fulfill. More precisely, we study the constraint that the (open or closed) neighborhood of the solution shall be bounded by a parameter and the influence of this constraint on the complexity of minimizing separators, feedback vertex sets, F-free vertex deletion sets, dominating sets, and the maximization of independent sets.

We show that any alpha(n)-approximation algorithm for the n-vertex metric asymmetric Traveling Salesperson problem yields O(alpha(C))-approximation algorithms for various mixed, windy, and capacitated arc routing problems. Herein, C is the number of weakly-connected components in the subgraph induced by the positive-demand arcs, a number that can be expected to be small in applications. In conjunction with known results, we derive constant-factor approximations if C is in O(log n) and O(log(C)/log(log(C)))-approximations in general.

We introduce a graph-theoretic vertex dissolution model that applies to a number of redistribution scenarios, such as gerrymandering in political districting or work balancing in an online situation. The central aspect of our model is the deletion of certain vertices and the redistribution of their load to neighboring vertices in a completely balanced way. We investigate how the underlying graph structure, the knowledge of which vertices should be deleted, and the relation between old and new vertex loads influence the computational complexity of the underlying graph problems. Our results establish a clear borderline between tractable and intractable cases.

We study the NP-hard LIST-COLORED GRAPH MOTIF problem which, given an undirected list-colored graph G = (V, E) and a multiset M of colors, asks for maximum-cardinality sets S ⊆ V and M' ⊆ M such that G[S] is connected and contains exactly (with respect to multiplicity) the colors in M'. LIST-COLORED GRAPH MOTIF has applications in the analysis of biological networks. We study LIST-COLORED GRAPH MOTIF with respect to three different parameterizations. For the parameters motif size |M| and solution size |S|, we present fixed-parameter algorithms, whereas for the parameter |V| - |M|, we show W[1]-hardness for general instances and achieve fixed-parameter tractability for a special case of LIST-COLORED GRAPH MOTIF. We implemented the fixed-parameter algorithms for parameters |M| and |S|, developed further speed-up heuristics for these algorithms, and applied them in the context of querying protein-interaction networks, demonstrating their usefulness for realistic instances. Furthermore, we show that extending the request for motif connectedness to stronger demands, such as biconnectedness or bridge-connectedness leads to W[1]-hard problems when the parameter is the motif size |M|.