Abstract The generalized maximum-entropy sampling problem (GMESP) is to select an order- s principal submatrix from an order- n covariance matrix, to maximize the product of its t greatest eigenvalues, $$0<t\le s <n$$ 0 < t ≤ s < n . Introduced more than 25 years ago, GMESP is a natural generalization of two fundamental problems in statistical design theory: (i) maximum-entropy sampling problem (MESP); (ii) binary D-optimality (D-Opt). In the general case, it can be motivated by a selection problem in the context of principal component analysis (PCA). We introduce the first convex-optimization based relaxation for GMESP, study its behavior, compare it to an earlier spectral bound, and demonstrate its use in a branch-and-bound scheme. We find that such an approach is practical when $$s-t$$ s - t is very small.

While some common fitness landscape characteristics are critical when determining the runtime of evolutionary algorithms (EAs), the relationship between fitness landscape structure and the runtime of EAs is poorly understood. Recently, Dang, Eremeev, and Lehre introduced a classification of pseudo-Boolean problems showing that "sparsity" of local optima and the "density" of fitness valleys can be crucial characteristics when determining the runtime of EAs Dang et al. (in Proceedings of the Genetic and Evolutionary Computation Conference. Association for Computing Machinery, New York, NY, USA, GECCO'21, pp 1133-1141, 10.1145/3449639.3459398, 2021c). However, their approach could only classify some classes of pseudo-Boolean functions and thus defined an incomplete hierarchy. We generalise the previous work to a complete hierarchy for all pseudo-Boolean functions, denoted Slo[Formula: see text]. The hierarchy is consistent with existing results for the runtime of EAs. The easiest problems are in Slo[Formula: see text] for [Formula: see text] and [Formula: see text]. As we increase [Formula: see text] and decrease [Formula: see text], the function class contains more interesting functions, including instances of hard combinatorial optimisation problems and problems perturbed by static noise. For [Formula: see text] and [Formula: see text] the problem class contains every problem, including problems closed under permutation (No Free Lunch). Problem classes where local optima sparsity exceed fitness valley density are shown to have exponential black-box complexity. We also study how random perturbations of a function can change its classification. E.g., randomly perturbing search points in OneMax with constant probability leads to a problem class that can still be optimised efficiently with appropriately tuned non-elitist EAs.

Abstract Hierarchical Clustering is a popular tool for understanding the hereditary properties of a data set. Such a clustering is actually a sequence of clusterings that starts with the trivial clustering in which every data point forms its own cluster and then successively merges two existing clusters until all points are in the same cluster. A hierarchical clustering achieves an approximation factor of $$\alpha $$ if the costs of each k-clustering in the hierarchy are at most $$\alpha $$ times the costs of an optimal k-clustering. We study as cost functions the maximum (discrete) radius of any cluster (k-center problem) and the maximum diameter of any cluster (k-diameter problem). In general, the optimal clusterings do not form a hierarchy and hence an approximation factor of 1 cannot be achieved. We call the smallest approximation factor that can be achieved for any instance the price of hierarchy. For the k-diameter problem we improve the upper bound on the price of hierarchy to $$3+2\sqrt{2}\approx 5.83$$ . Moreover we significantly improve the lower bounds for k-center and k-diameter, proving a price of hierarchy of exactly 4 and $$3+2\sqrt{2}$$ , respectively.

Abstract Spanner construction is a well-studied problem and Delaunay triangulations are among the most popular spanners. Tight bounds are known if the Delaunay triangulation is constructed using an equilateral triangle, a square, or a regular hexagon. However, all other shapes have remained elusive. In this paper, we extend the restricted class of spanners for which tight bounds are known. We prove that Delaunay triangulations constructed using rectangles with aspect ratio $$A$$ A have spanning ratio at most $$\sqrt{2} \sqrt{1+A^2 + A\sqrt{A^2 + 1}}$$ 2 1 + A 2 + A A 2 + 1 , which matches the known lower bound.

The amendment procedure and the successive procedure have been widely employed in parliamentary and legislative decision making and have undergone extensive study in the literature from various perspectives. However, investigating them through the lens of computational complexity theory has not been as thoroughly conducted as for many other prevalent voting procedures heretofore. To the best of our knowledge, there is only one paper which explores the complexity of several strategic voting problems under these two procedures, prior to our current work. To provide a better understanding of to what extent the two procedures resist strategic behavior, we study the parameterized complexity of constructive/destructive control by adding/deleting voters/candidates for both procedures. To enhance the generalizability of our results, we also examine a more generalized form of the amendment procedure. Our exploration yields a comprehensive (parameterized) complexity landscape of these problems with respect to numerous parameters.

Understanding how evolutionary algorithms perform on constrained problems has gained increasing attention in recent years. In this paper, we study how evolutionary algorithms optimize constrained versions of the classical LeadingOnes problem. We first provide a run time analysis for the classical (1+1) EA on the LeadingOnes problem with a deterministic cardinality constraint, giving Θ(n(n-B)log(B)+nB) as the tight bound. Our results show that the behaviour of the algorithm is highly dependent on the constraint bound of the uniform constraint. Afterwards, we consider the problem in the context of stochastic constraints and provide insights using theoretical and experimental studies on how the (μ+1) EA is able to deal with these constraints in a sampling-based setting.

In this paper, we study the Eulerian Strong Component Arc Deletion problem, where the input is a directed multigraph and the goal is to delete the minimum number of arcs to ensure every strongly connected component of the resulting digraph is Eulerian. This problem is a natural extension of the Directed Feedback Arc Set problem and is also known to be motivated by certain scenarios arising in the study of housing markets. The complexity of the problem, when parameterized by solution size (i.e., size of the deletion set), has remained unresolved and has been highlighted in several papers. In this work, we answer this question by ruling out (subject to the usual complexity assumptions) a fixed-parameter algorithm (FPT algorithm) for this parameter and conduct a broad analysis of the problem with respect to other natural parameterizations. We prove both positive and negative results. Among these, we demonstrate that the problem is also hard (W[1]-hard or even para-NP-hard) when parameterized by either treewidth or maximum degree alone. Complementing our lower bounds, we establish that the problem is in XP when parameterized by treewidth and FPT when parameterized either by both treewidth and maximum degree or by both treewidth and solution size. We show that on simple digraphs, these algorithms have near-optimal asymptotic dependence on the treewidth assuming the Exponential Time Hypothesis.