Spherical range queries are a fundamental primitive for working with spatial data. Many spatial data structures have been developed to answer these queries, but choosing the optimal one for a specific application is a difficult task. This is because theoretical worst-case bounds are often overly pessimistic, and existing average-case analyses are rather restricted and hard to compare. We address this problem with two main contributions. First, we present a comprehensive evaluation of state-of-the-art spatial indices across a diverse set of benchmarks. This includes a new benchmark based on graph embeddings alongside multiple real-world datasets from the literature. Our benchmark covers instances scaling up to 10M points and ranging between 2 and 960 dimensions. Second, we introduce the Sorted-Projection Radius KD-tree (SPRK-tree), a high-performance KD-tree variant. The SPRK-tree combines aggressive subtree pruning via radius reduction, sorted projection-based leaf nodes, and careful implementation optimizations. It consistently achieves the fastest query times in almost all benchmarks, and ranks second in the few remaining cases.

Multi-constraint hypergraph partitioning is a generalization of balanced partitioning, where the vertex set of a hypergraph is partitioned such that the inter-block connectivity of hyperedges is minimized while balancing the vertices with regard to d distinct constraints. A prominent class of applications is data distribution tasks, where this allows to achieve good load balance for d different kinds of resources and simultaneously minimize the communication volume. Although the best approaches for single-constraint partitioning are usually complex (multilevel) algorithms with many components, we show that replacing only one component already leads to high-quality multi-constraint partitions: the rebalancing step, which restores balance for a partition that has (hopefully) small connectivity but violates the constraints. We design a multi-constraint rebalancing algorithm based on greedy local search, proving that balance is always restored for d = 2 and bounded maximum weight. The key is to ensure monotonically decreasing global imbalance by choosing an imbalance metric where there is always a balance-improving move available. Integrating our algorithm into the state-of-the-art partitioner Mt-KaHyPar, we demonstrate an 11.5 % geometric mean connectivity reduction compared to the next best competitor (Metis) and better reliability regarding partition balance, even though the majority of inputs is outside of the theoretical guarantee.

Balanced Graph Partitioning is a classical optimization problem where quality guarantees are computationally infeasible, and practical solvers therefore rely on manually engineered heuristics. Yet, the problem has also proven difficult for approaches that rely heavily on machine learning - especially since applications often need to partition graphs of huge scale in a short amount of time. Instead, we demonstrate how to achieve practical improvements with a more careful approach that uses machine learning to improve heuristic decisions within the state-of-the-art solver Mt-KaHyPar. We use a pre-trained neural network to predict a score for each edge, which then guides clustering decisions in the first phase of the partitioning (the coarsening). Combined with corresponding adjustments to the clustering algorithm and an efficient implementation of the neural network logic, we improve the overall solution quality while preserving the efficiency and scalability of the original algorithm. Our detailed evaluation on more than 180 graphs shows an average quality improvement of 2% on a class of graphs with beneficial properties, and unchanged quality on all remaining graphs. Moreover, our improvements generalize to a set of instances from the literature that are much larger than the graphs used during training.

Learning low-dimensional numerical representations from symbolic data, e.g., embedding the nodes of a graph into a geometric space, is an important concept in machine learning. While embedding into Euclidean space is common, recent observations indicate that hyperbolic geometry is better suited to represent hierarchical information and heterogeneous data (e.g., graphs with a scale-free degree distribution). Despite their potential for more accurate representations, hyperbolic embeddings also have downsides like being more difficult to compute and harder to use in downstream tasks. We propose embedding into a weighted space, which is closely related to hyperbolic geometry but mathematically simpler. We provide the embedding algorithm WEmbed and demonstrate, based on generated as well as over 2000 real-world graphs, that our weighted embeddings heavily outperform state-of-the-art Euclidean embeddings for heterogeneous graphs while using fewer dimensions. The running time of WEmbed and embedding quality for the remaining instances is on par with state-of-the-art Euclidean embedders.

The NP-hard scheduling problem \(P\Vert C_{\max}\) encompasses a set of tasks with known execution time which must be mapped to a set of identical machines such that the overall completion time is minimized. In this work, we improve existing techniques for optimal \(P\Vert C_{\max}\) scheduling with a combination of new theoretical insights and careful practical engineering. Most importantly, we derive techniques to prune vast portions of the search space of branch-and-bound (BnB) approaches and propose improved upper and lower bounding techniques. Moreover, we present new benchmarks for \(P\Vert C_{\max}\), based on diverse applications, which can shed light on aspects that prior synthetic instances fail to capture. In extensive evaluations, we observe that our pruning reduces the number of explored nodes by \(90\times\) and running times by \(12\times\). Compared to a state-of-the-art ILP-based approach, our approach is preferable for short running time limits and for instances with large makespans.

The current landscape of balanced graph partitioning is divided into high-quality but expensive multilevel algorithms and cheaper approaches with linear running time, such as single-level algorithms and streaming algorithms. We demonstrate how to achieve the best of both worlds with a linear time multilevel algorithm. Multilevel algorithms construct a hierarchy of increasingly smaller graphs by repeatedly contracting clusters of nodes. Our approach preserves their distinct advantage, allowing refinement of the partition over multiple levels with increasing detail. At the same time, we use edge sparsification to guarantee geometric size reduction between the levels and thus linear running time. We provide a proof of the linear running time as well as additional insights into the behavior of multilevel algorithms, showing that graphs with low modularity are most likely to trigger worst-case running time. We evaluate multiple approaches for edge sparsification and integrate our algorithm into the state-of-the-art multilevel partitioner KaMinPar, maintaining its excellent parallel scalability. As demonstrated in detailed experiments, this results in a 1.49× average speedup (up to 4× for some instances) with only 1% loss in solution quality. Moreover, our algorithm clearly outperforms state-of-the-art single-level and streaming approaches.

We present new refinement heuristics for the balanced graph partitioning problem that break with an age-old rule. Traditionally, local search only permits moves that keep the block sizes balanced (below a size constraint). In this work, we demonstrate that admitting large temporary balance violations drastically improves solution quality. The effects are particularly strong on irregular instances such as social networks. Designing efficient implementations of this general idea involves both careful selection of candidates for unconstrained moves as well as algorithms for rebalancing the solution later on. We explore a wide array of design choices to achieve this, in addition to our third goal of high parallel scalability. We present compelling experimental results, demonstrating that our parallel unconstrained local search techniques outperform the prior state of the art by a substantial margin. Compared with four state-of-the-art solvers, our new technique finds 75% of the best solutions on irregular graphs. We achieve a 9.6% improvement in edge cut over the next best competitor, while being only 7.7% slower in the geometric mean.

Balanced hypergraph partitioning is an NP-hard problem with many applications, e.g., optimizing communication in distributed data placement problems. The goal is to place all nodes across k different blocks of bounded size, such that hyperedges span as few parts as possible. This problem is well-studied in sequential and distributed settings, but not in shared-memory. We close this gap by devising efficient and scalable shared-memory algorithms for all components employed in the best sequential solvers without compromises with regards to solution quality. This work presents the scalable and high-quality hypergraph partitioning framework Mt-KaHyPar . Its most important components are parallel improvement algorithms based on the FM algorithm and maximum flows, as well as a parallel clustering algorithm for coarsening – which are used in a multilevel scheme with log (n) levels. As additional components, we parallelize the n -level partitioning scheme, devise a deterministic version of our algorithm, and present optimizations for plain graphs. We evaluate our solver on more than 800 graphs and hypergraphs, and compare it with 25 different algorithms from the literature. Our fastest configuration outperforms almost all existing hypergraph partitioners with regards to both solution quality and running time. Our highest-quality configuration achieves the same solution quality as the best sequential partitioner KaHyPar , while being an order of magnitude faster with ten threads. Thus, two of our configurations occupy all fronts of the Pareto curve for hypergraph partitioning. Furthermore, our solvers exhibit good speedups, e.g., 29.6x in the geometric mean on 64 cores (deterministic), 22.3x (log (n) -level), and 25.9x ( n -level).

The balanced hypergraph partitioning problem (HGP) is to partition the vertex set of a hypergraph into k disjoint blocks of bounded weight, while minimizing an objective function defined on the hyperedges. Whereas real-world applications often use vertex and edge weights to accurately model the underlying problem, the HGP research community commonly works with unweighted instances. In this paper, we argue that, in the presence of vertex weights, current balance constraint definitions either yield infeasible partitioning problems or allow unnecessarily large imbalances and propose a new definition that overcomes these problems. We show that state-of-the-art hypergraph partitioners often struggle considerably with weighted instances and tight balance constraints (even with our new balance definition). Thus, we present a recursive-bipartitioning technique that is able to reliably compute balanced (and hence feasible) solutions. The proposed method balances the partition by pre-assigning a small subset of the heaviest vertices to the two blocks of each bipartition (using an algorithm originally developed for the job scheduling problem) and optimizes the actual partitioning objective on the remaining vertices. We integrate our algorithm into the multilevel hypergraph partitioner KaHyPar and show that our approach is able to compute balanced partitions of high quality on a diverse set of benchmark instances.