Random Contractions and Sampling for Hypergraph and Hedge Connectivity

We initiate the study of hedge connectivity of undirected graphs, motivated by dependent edge failures in real-world networks. In this model, edges are partitioned into groups called hedges that fail together. The hedge connectivity of a graph is the minimum number of hedges whose removal disconnects the graph. We give a polynomial-time approximation scheme and a quasi-polynomial exact algorithm for hedge connectivity. This provides strong evidence that the hedge connectivity problem is tractable, which contrasts with prior work that established the intractability of the corresponding s−t min-cut problem. Our techniques also yield new combinatorial and algorithmic results in hypergraph connectivity. Next, we study the behavior of hedge graphs under uniform random sampling of hedges. We show that unlike graphs, all cuts in the sample do not converge to their expected value in hedge graphs. Nevertheless, the min-cut of the sample does indeed concentrate around the expected value of the original min-cut. This leads to a sharp threshold on hedge survival probabilities for graph disconnection. To the best of our knowledge, this is the first network reliability analysis under dependent edge failures

Scalable High-Quality Graph and Hypergraph Partitioning

The balanced hypergraph partitioning problem (HGP) asks for a partition of the node set of a hypergraph into $k$ blocks of roughly equal size, such that an objective function defined on the hyperedges is minimized. In this work, we optimize the connectivity metric, which is the most prominent objective function for HGP. The hypergraph partitioning problem is NP-hard and there exists no constant factor approximation. Thus, heuristic algorithms are used in practice with the multilevel scheme as the most successful approach to solve the problem: First, the input hypergraph is coarsened to obtain a hierarchy of successively smaller and structurally similar approximations. The smallest hypergraph is then initially partitioned into $k$ blocks, and subsequently, the contractions are reverted level-by-level, and, on each level, local search algorithms are used to improve the partition (refinement phase). In recent years, several new techniques were developed for sequential multilevel partitioning that substantially improved solution quality at the cost of an increased running time. These developments divide the landscape of existing partitioning algorithms into systems that either aim for speed or high solution quality with the former often being more than an order of magnitude faster than the latter. Due to the high running times of the best sequential algorithms, it is currently not feasible to partition the largest real-world hypergraphs with the highest possible quality. Thus, it becomes increasingly important to parallelize the techniques used in these algorithms. However, existing state-of-the-art parallel partitioners currently do not achieve the same solution quality as their sequential counterparts because they use comparatively weak components that are easier to parallelize. Moreover, there has been a recent trend toward simpler methods for partitioning large hypergraphs that even omit the multilevel scheme. In contrast to this development, we present two shared-memory multilevel hypergraph partitioners with parallel implementations of techniques used by the highest-quality sequential systems. Our first multilevel algorithm uses a parallel clustering-based coarsening scheme, which uses substantially fewer locking mechanisms than previous approaches. The contraction decisions are guided by the community structure of the input hypergraph obtained via a parallel community detection algorithm. For initial partitioning, we implement parallel multilevel recursive bipartitioning with a novel work-stealing approach and a portfolio of initial bipartitioning techniques to compute an initial solution. In the refinement phase, we use three different parallel improvement algorithms: label propagation refinement, a highly-localized direct $k$-way FM algorithm, and a novel parallelization of flow-based refinement. These algorithms build on our highly-engineered partition data structure, for which we propose several novel techniques to compute accurate gain values of node moves in the parallel setting. Our second multilevel algorithm parallelizes the $n$-level partitioning scheme used in the highest-quality sequential partitioner KaHyPar. Here, only a single node is contracted on each level, leading to a hierarchy with approximately $n$ levels where $n$ is the number of nodes. Correspondingly, in each refinement step, only a single node is uncontracted, allowing a highly-localized search for improvements. We show that this approach, which seems inherently sequential, can be parallelized efficiently without compromises in solution quality. To this end, we design a forest-based representation of contractions from which we derive a feasible parallel schedule of the contraction operations that we apply on a novel dynamic hypergraph data structure on-the-fly. In the uncoarsening phase, we decompose the contraction forest into batches, each containing a fixed number of nodes. We then uncontract each batch in parallel and use highly-localized versions of our refinement algorithms to improve the partition around the uncontracted nodes. We further show that existing sequential partitioning algorithms considerably struggle to find balanced partitions for weighted real-world hypergraphs. To this end, we present a technique that enables partitioners based on recursive bipartitioning to reliably compute balanced solutions. The idea is to preassign a small portion of the heaviest nodes to one of the two blocks of each bipartition and optimize the objective function on the remaining nodes. We integrated the approach into the sequential hypergraph partitioner KaHyPar and show that our new approach can compute balanced solutions for all tested instances without negatively affecting the solution quality and running time of KaHyPar. In our experimental evaluation, we compare our new shared-memory (hyper)graph partitioner Mt-KaHyPar to $25$ different graph and hypergraph partitioners on over $800$ (hyper)graphs with up to two billion edges/pins. The results indicate that already our fastest configuration outperforms almost all existing hypergraph partitioners with regards to both solution quality and running time. Our highest-quality configuration ($n$-level with flow-based refinement) achieves the same solution quality as the currently best sequential partitioner KaHyPar, while being almost an order of magnitude faster with ten threads. In addition, we optimize our data structures for graph partitioning, which improves the running times of both multilevel partitioners by almost a factor of two for graphs. As a result, Mt-KaHyPar also outperforms most of the existing graph partitioning algorithms. While the shared-memory graph partitioner KaMinPar is still faster than Mt-KaHyPar, its produced solutions are worse by $10\%$ in the median. The best sequential graph partitioner KaFFPa-StrongS computes slightly better partitions than Mt-KaHyPar (median improvement is $1\%$), but is more than an order of magnitude slower on average

High-Quality Hypergraph Partitioning

This dissertation focuses on computing high-quality solutions for the NP-hard balanced hypergraph partitioning problem: Given a hypergraph and an integer $k$, partition its vertex set into $k$ disjoint blocks of bounded size, while minimizing an objective function over the hyperedges. Here, we consider the two most commonly used objectives: the cut-net metric and the connectivity metric. Since the problem is computationally intractable, heuristics are used in practice - the most prominent being the three-phase multi-level paradigm: During coarsening, the hypergraph is successively contracted to obtain a hierarchy of smaller instances. After applying an initial partitioning algorithm to the smallest hypergraph, contraction is undone and, at each level, refinement algorithms try to improve the current solution. With this work, we give a brief overview of the field and present several algorithmic improvements to the multi-level paradigm. Instead of using a logarithmic number of levels like traditional algorithms, we present two coarsening algorithms that create a hierarchy of (nearly) $n$ levels, where $n$ is the number of vertices. This makes consecutive levels as similar as possible and provides many opportunities for refinement algorithms to improve the partition. This approach is made feasible in practice by tailoring all algorithms and data structures to the $n$-level paradigm, and developing lazy-evaluation techniques, caching mechanisms and early stopping criteria to speed up the partitioning process. Furthermore, we propose a sparsification algorithm based on locality-sensitive hashing that improves the running time for hypergraphs with large hyperedges, and show that incorporating global information about the community structure into the coarsening process improves quality. Moreover, we present a portfolio-based initial partitioning approach, and propose three refinement algorithms. Two are based on the Fiduccia-Mattheyses (FM) heuristic, but perform a highly localized search at each level. While one is designed for two-way partitioning, the other is the first FM-style algorithm that can be efficiently employed in the multi-level setting to directly improve $k$-way partitions. The third algorithm uses max-flow computations on pairs of blocks to refine $k$-way partitions. Finally, we present the first memetic multi-level hypergraph partitioning algorithm for an extensive exploration of the global solution space. All contributions are made available through our open-source framework KaHyPar. In a comprehensive experimental study, we compare KaHyPar with hMETIS, PaToH, Mondriaan, Zoltan-AlgD, and HYPE on a wide range of hypergraphs from several application areas. Our results indicate that KaHyPar, already without the memetic component, computes better solutions than all competing algorithms for both the cut-net and the connectivity metric, while being faster than Zoltan-AlgD and equally fast as hMETIS. Moreover, KaHyPar compares favorably with the current best graph partitioning system KaFFPa - both in terms of solution quality and running time

Isolating Cuts, (Bi-)Submodularity, and Faster Algorithms for Connectivity

Li and Panigrahi [Jason Li and Debmalya Panigrahi, 2020], in recent work, obtained the first deterministic algorithm for the global minimum cut of a weighted undirected graph that runs in time o(mn). They introduced an elegant and powerful technique to find isolating cuts for a terminal set in a graph via a small number of s-t minimum cut computations. In this paper we generalize their isolating cut approach to the abstract setting of symmetric bisubmodular functions (which also capture symmetric submodular functions). Our generalization to bisubmodularity is motivated by applications to element connectivity and vertex connectivity. Utilizing the general framework and other ideas we obtain significantly faster randomized algorithms for computing global (and subset) connectivity in a number of settings including hypergraphs, element connectivity and vertex connectivity in graphs, and for symmetric submodular functions

Cuts and connectivity in graphs and hypergraphs

In this thesis, we consider cut and connectivity problems on graphs, digraphs, hypergraphs and hedgegraphs. The main results are the following: - We introduce a faster algorithm for finding the reduced graph in element-connectivity computations. We also show its application to node separation. - We present several results on hypergraph cuts, including (a) a near linear time algorithm for finding a (2+epsilon)-approximate min-cut, (b) an algorithm to find a representation of all min-cuts in the same time as finding a single min-cut, (c) a sparse subgraph that preserves connectivity for hypergraphs and (d) a near linear-time hypergraph cut sparsifier. - We design the first randomized polynomial time algorithm for the hypergraph k-cut problem whose complexity has been open for over 20 years. The algorithm generalizes to hedgegraphs with constant span. - We address the complexity gap between global vs. fixed-terminal cuts problems in digraphs by presenting a 2-1/448 approximation algorithm for the global bicut problem

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions