Segment Routing is a recent network technology that helps optimizing network throughput by providing finer control over the routing paths. Instead of routing directly from a source to a target, packets are routed via intermediate waypoints. Between consecutive waypoints, the packets are routed according to traditional shortest path routing protocols. Bottlenecks in the network can be avoided by such rerouting, preventing overloading parts of the network. The associated NP-hard computational problem is Segment Routing: Given a network and a set of traffic demands (vertex pairs), the task is to find for each demand pair the placement of a given number of waypoints such that with shortest path routing along these waypoints, all demands are fulfilled without exceeding the capacities of the network. We investigate if special structures of real-world communication networks could be exploited algorithmically. Our results comprise NP-hardness on graphs with constant treewidth even if only one waypoint per demand is allowed. We further exclude (under standard complexity assumptions) the existence of efficient exact algorithms even if we assume a fixed number of waypoints per demand and a “small” amount of traffic demands. We complement these lower bounds with polynomial-time solvable special cases.

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We consider the problem of partitioning a graph into a non-fixed number of non-overlapping subgraphs of maximum density. The density of a partition is the sum of the densities of the subgraphs, where the density of a subgraph is half its average degree, that is, the ratio of its number of edges and its number of vertices. This problem, called Dense Graph Partition, is known to be NP-hard on general graphs and polynomial-time solvable on trees, and polynomial-time 2-approximable. In this paper we study the restriction of Dense Graph Partition to particular sparse and dense graph classes. In particular, we prove that it is NP-hard on dense bipartite graphs as well as on cubic graphs. On dense graphs on n vertices, it is polynomial-time solvable on graphs with minimum degree n − 3 and NP-hard on ( n − 4 ) -regular graphs. Some polynomial-time approximation results are also established.

We analyze the computational complexity of the following computational problems called Bounded-Density Edge Deletion and Bounded-Density Vertex Deletion : Given a graph G , a budget k and a target density τ ρ , are there k edges ( k vertices) whose removal from G results in a graph where the densest subgraph has density at most τ ρ ? Here, the density of a graph is the number of its edges divided by the number of its vertices. We prove that both problems are polynomial-time solvable on trees and cliques but are NP-complete on planar bipartite graphs and split graphs. From a parameterized point of view, we show that both problems are fixed-parameter tractable with respect to the vertex cover number but W[1]-hard with respect to the solution size. Furthermore, we prove that Bounded-Density Edge Deletion is W[1]-hard with respect to the feedback edge number, demonstrating that the problem remains hard on very sparse graphs.

Abstract Given a directed temporal graph, a start node , and objectives, the task in the single‐source multiobjective temporal shortest path problem (SSMTSPP) consists of computing the set of nondominated images of temporal ‐‐paths for each node as well as one corresponding efficient path for each of these images. This problem generalizes both the multiobjective shortest path problem in static graphs and the single‐objective temporal shortest path problem. In this article, we provide a general label setting algorithm for the SSMTSPP that can handle a large variety of different objectives. The only condition imposed on the objectives is a monotonicity property that generalizes the nonnegativity of the arc costs required for the well‐known label setting algorithm for solving the static single‐source shortest path problem in both the single objective and the multiobjective case. Our analysis of the presented algorithm shows that its worst‐case running time is polynomial in the sum of the input size of the problem instance and the number of nondominated images, which implies that it runs in polynomial time as long as the number of nondominated images is polynomial in the instance size (i.e., for all tractable versions of the problem). To complement this result, we provide a complete classification into tractable and intractable problems for all SSMTSPPs involving a large variety of objectives. In particular, using our general analysis, this provides a large range of specific SSMTSPPs for which our general label setting algorithm runs in polynomial time.

Nested Rollout Policy Adaptation (NRPA) is an approach using online learning policies in a nested structure. It has achieved a great result in a variety of difficult combinatorial optimization problems. In this paper, we propose Meta-NRPA, which combines optimal stopping theory with NRPA for warm-starting and significantly improves the performance of NRPA. We also present several exploratory techniques for NRPA which enable it to perform better exploration. We establish this for three notoriously difficult problems ranging from telecommunication, transportation and coding theory namely Minimum Congestion Shortest Path Routing, Traveling Salesman Problem with Time Windows and Snake-in-the-Box. We also improve the lower bounds of the Snake-in-the-Box problem for multiple dimensions.

Abstract We determine the power of the weighted sum scalarization with respect to the computation of approximations for general multiobjective minimization and maximization problems. Additionally, we introduce a new multi-factor notion of approximation that is specifically tailored to the multiobjective case and its inherent trade-offs between different objectives. For minimization problems, we provide an efficient algorithm that computes an approximation of a multiobjective problem by using an exact or approximate algorithm for its weighted sum scalarization. In case that an exact algorithm for the weighted sum scalarization is used, this algorithm comes arbitrarily close to the best approximation quality that is obtainable by supported solutions – both with respect to the common notion of approximation and with respect to the new multi-factor notion. Moreover, the algorithm yields the currently best approximation results for several well-known multiobjective minimization problems. For maximization problems, however, we show that a polynomial approximation guarantee can, in general, not be obtained in more than one of the objective functions simultaneously by supported solutions.

Abstract Papadimitriou and Yannakakis (Proceedings of the 41st annual IEEE symposium on the Foundations of Computer Science (FOCS), pp 86–92, 2000) show that the polynomial-time solvability of a certain auxiliary problem determines the class of multiobjective optimization problems that admit a polynomial-time computable $$(1+\varepsilon , \dots , 1+\varepsilon )$$ ( 1 + ε , ⋯ , 1 + ε ) -approximate Pareto set (also called an $$\varepsilon $$ ε -Pareto set). Similarly, in this article, we characterize the class of multiobjective optimization problems having a polynomial-time computable approximate $$\varepsilon $$ ε -Pareto set that is exact in one objective by the efficient solvability of an appropriate auxiliary problem. This class includes important problems such as multiobjective shortest path and spanning tree, and the approximation guarantee we provide is, in general, best possible. Furthermore, for biobjective optimization problems from this class, we provide an algorithm that computes a one-exact $$\varepsilon $$ ε -Pareto set of cardinality at most twice the cardinality of a smallest such set and show that this factor of 2 is best possible. For three or more objective functions, however, we prove that no constant-factor approximation on the cardinality of the set can be obtained efficiently.

Abstract We study the NP‐hard shortest path most vital edges problem arising in the context of analyzing network robustness. For an undirected graph with positive integer edge lengths and two designated vertices s and t, the goal is to delete as few edges as possible in order to increase the length of the (new) shortest st‐path as much as possible. This scenario has been studied from the viewpoint of parameterized complexity and approximation algorithms. We contribute to this line of research by providing refined computational tractability as well as hardness results. We achieve this by a systematic investigation of various problem‐specific parameters and their influence on the computational complexity. Charting the border between tractability and intractability, we also identify numerous challenges for future research.

We investigate the structural and algorithmic properties of 2-community structures in graphs introduced recently by Olsen (Math Soc Sci 66(3):331–336, 2013). A 2-community structure is a partition of a vertex set into two parts such that for each vertex the numbers of neighbours in/outside its own part and the sizes of the parts are correlated. We show that some well studied graph classes as graphs of maximum degree 3, minimum degree at least $$|V|-3$$ , trees and also others, have always a 2-community structure. Furthermore, a 2-community structure can be found in polynomial time in all these classes, even with additional request of connectivity in both parts. We introduce a concept of a weak 2-community and prove that in general graphs it is NP-complete to find a balanced weak 2-community structure with or without request for connectivity in both parts. On the other hand, we present a polynomial-time algorithm to solve the problem (without the condition for connectivity of parts) in graphs of degree at most 3.

Classical clustering problems search for a partition of objects into a fixed number of clusters. In many scenarios however the number of clusters is not known or necessarily fixed. Further, clusters are sometimes only considered to be of significance if they have a certain size. We discuss clustering into sets of minimum cardinality k without a fixed number of sets and present a general model for these types of problems. This general framework allows the comparison of different measures to assess the quality of a clustering. We specifically consider nine quality-measures and classify the complexity of the resulting problems with respect to k. Further, we derive some polynomial-time solvable cases for k = 2 with connections to matching-type problems which, among other graph problems, then are used to compute approximations for larger values of k.