Graph partition is a key component to achieve workload balance and reduce job completion time in parallel graph processing systems. Among the various partition strategies, edge partition has demonstrated more promising performance in power-law graphs than vertex partition and thereby has been more widely adopted as the default partition strategy by existing graph systems. The graph edge partition problem, which is to split the edge set into multiple balanced parts to minimize the total number of copied vertices, has been widely studied from the view of optimization and algorithms. In this paper, we study local search algorithms for this problem to further improve the partition results from existing methods. More specifically, we propose two novel concepts, namely adjustable edges and blocks. Based on these, we develop a greedy heuristic as well as an improved search algorithm utilizing the property of the max-flow model. To evaluate the performance of our algorithms, we first provide adequate theoretical analysis in terms of the approximation quality. We significantly improve the previously known approximation ratio for this problem. Then we conduct extensive experiments on a large number of benchmark datasets and state-of-the-art edge partition strategies. The results show that our proposed local search framework can further improve the quality of graph partition by a wide margin.
This paper initiates the study of the classic balanced graph partitioning problem from an online perspective: Given an arbitrary sequence of pairwise communication requests between $n$ nodes, with patterns that may change over time, the objective is to service these requests efficiently by partitioning the nodes into $\ell$ clusters, each of size $k$, such that frequently communicating nodes are located in the same cluster. The partitioning can be updated dynamically by migrating nodes between clusters. The goal is to devise online algorithms which jointly minimize the amount of inter-cluster …
Motivated by performance optimization of large-scale graph processing systems that distribute the graph across multiple machines, we consider the balanced graph partitioning problem. Compared to the previous work, we study the multi-dimensional variant when balance according to multiple weight functions is required. As we demonstrate by experimental evaluation, such multi-dimensional balance is important for achieving performance improvements for typical distributed graph processing workloads. We propose a new scalable technique for the multidimensional balanced graph partitioning problem. The…
We introduce the balanced crown decomposition that captures the structure imposed on graphs by their connected induced subgraphs of a given size. Such subgraphs are a popular modeling tool in various application areas, where the non-local nature of the connectivity condition usually results in very challenging algorithmic tasks. The balanced crown decomposition is a combination of a crown decomposition and a balanced partition which makes it applicable to graph editing as well as graph packing and partitioning problems. We illustrate this by deriving improved kernelization and approximation al…
Given a simple connected graph $G = (V, E)$, we seek to partition the vertex set $V$ into $k$ non-empty parts such that the subgraph induced by each part is connected, and the partition is maximally balanced in the way that the maximum cardinality of these $k$ parts is minimized. We refer this problem to as {\em min-max balanced connected graph partition} into $k$ parts and denote it as {\sc $k$-BGP}. The general vertex-weighted version of this problem on trees has been studied since about four decades ago, which admits a linear time exact algorithm; the vertex-weighted {\sc $2$-BGP} and {\sc …
The rapid growth of big spatial data urged the research community to develop several big spatial data systems. Regardless of their architecture, one of the fundamental requirements of all these systems is to spatially partition the data efficiently across machines. The core challenges of big spatial partitioning are building high spatial quality partitions while simultaneously taking advantages of distributed processing models by providing load balanced partitions. Previous works on big spatial partitioning are to reuse existing index search trees as-is, e.g., the R-tree family, STR, Kd-tree, …
Partitioning a connected graph into $k$~vertex-disjoint connected subgraphs of similar (or given) orders is a classical problem that has been intensively investigated since late seventies. Given a connected graph $G=(V,E)$ and a weight function $w : V \to \mathbb{Q}_\geq$, a connected $k$-partition of $G$ is a partition of $V$ such that each class induces a connected subgraph. The balanced connected $k$-partition problem consists in finding a connected $k$-partition in which every class has roughly the same weight. To model this concept of balance, one may seek connected $k$-partitions that ei…
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 definit…
A primary challenge in metagenomics is reconstructing individual microbial genomes from the mixture of short fragments created by sequencing. Recent work leverages the sparsity of the assembly graph to find $r$-dominating sets which enable rapid approximate queries through a dominator-centric graph partition. In this paper, we consider two problems related to reducing uncertainty and improving scalability in this setting. First, we observe that nodes with multiple closest dominators necessitate arbitrary tie-breaking in the existing pipeline. As such, we propose finding $\textit{sparse}$ domin…
Two kinds of approximation algorithms exist for the k-BALANCED PARTITIONING problem: those that are fast but compute unsatisfying approximation ratios, and those that guarantee high quality ratios but are slow. In this paper we prove that this tradeoff between runtime and solution quality is necessary. For the problem a minimum number of edges in a graph need to be found that, when cut, partition the vertices into k equal-sized sets. We develop a reduction framework which identifies some necessary conditions on the considered graph class in order to prove the hardness of the problem. We focus …
Georgia Institute of Technology（佐治亚理工学院）；Sandia National Laboratories（桑迪亚国家实验室）；Heidelberg University（海德堡大学）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；Humboldt-Universität zu Berlin（柏林洪堡大学）；Apple Inc.（苹果公司）
In recent years, significant advances have been made in the design and evaluation of balanced (hyper)graph partitioning algorithms. We survey trends of the last decade in practical algorithms for balanced (hyper)graph partitioning together with future research directions. Our work serves as an update to a previous survey on the topic. In particular, the survey extends the previous survey by also covering hypergraph partitioning and streaming algorithms, and has an additional focus on parallel algorithms.
Given a graph $G = (V,E)$ with vertex weights $w(v)$ and a desired number of parts $k$, the goal in graph partitioning problems is to partition the vertex set V into parts $V_1,\ldots,V_k$. Metrics for compactness, contiguity, and balance of the parts $V_i$ are frequent objectives, with much existing literature focusing on compactness and balance. Revisiting an old method known as striping, we give the first polynomial-time algorithms with guaranteed contiguity and provable bicriteria approximations for compactness and balance for planar grid graphs. We consider several types of graph partitio…
Graph partitioning problems are a central topic of study in algorithms and complexity theory. Edge expansion and vertex expansion, two popular graph partitioning objectives, seek a $2$-partition of the vertex set of the graph that minimizes the considered objective. However, for many natural applications, one might require a graph to be partitioned into $k$ parts, for some $k \geq 2$. For a $k$-partition $S_1, \ldots, S_k$ of the vertex set of a graph $G = (V,E)$, the $k$-way edge expansion (resp. vertex expansion) of $\{S_1, \ldots, S_k\}$ is defined as $\max_{i \in [k]} Φ(S_i)$, and the bala…
Frigo et al. proposed an ideal cache model and a recursive technique to design sequential cache-efficient algorithms in a cache-oblivious fashion. Ballard et al. pointed out that it is a fundamental open problem to extend the technique to an arbitrary architecture. Ballard et al. raised another open question on how to parallelize Strassen's algorithm exactly and efficiently on an arbitrary number of processors. We propose a novel way of partitioning a cache-oblivious algorithm to achieve perfect strong scaling on an arbitrary number, even a prime number, of processors within a certain range in…
Given an undirected graph $G$ and $q$ integers $n_1,n_2,n_3, \cdots, n_q$, balanced connected $q$-partition problem ($BCP_q$) asks whether there exists a partition of the vertex set $V$ of $G$ into $q$ parts $V_1,V_2,V_3,\cdots, V_q$ such that for all $i\in[1,q]$, $|V_i|=n_i$ and the graph induced on $V_i$ is connected. A related problem denoted as the balanced connected $q$-edge partition problem ($BCEP_q$) is defined as follows. Given an undirected graph $G$ and $q$ integers $n_1,n_2,n_3, \cdots, n_q$, $BCEP_q$ asks whether there exists a partition of the edge set of $G$ into $q$ parts $E_1,…
Balanced hypergraph partitioning is a classical NP-hard optimization problem with applications in various domains such as VLSI design, simulating quantum circuits, optimizing data placement in distributed databases or minimizing communication volume in high-performance computing. Engineering parallel heuristics for this problem is a topic of recent research. Most of them are non-deterministic though. In this work, we design and implement a highly scalable deterministic algorithm in the state-of-the-art parallel partitioning framework Mt-KaHyPar. On our extensive set of benchmark instances, it …
Space partitions of $\mathbb{R}^d$ underlie a vast and important class of fast nearest neighbor search (NNS) algorithms. Inspired by recent theoretical work on NNS for general metric spaces [Andoni, Naor, Nikolov, Razenshteyn, Waingarten STOC 2018, FOCS 2018], we develop a new framework for building space partitions reducing the problem to balanced graph partitioning followed by supervised classification. We instantiate this general approach with the KaHIP graph partitioner [Sanders, Schulz SEA 2013] and neural networks, respectively, to obtain a new partitioning procedure called Neural Locali…
A connected partition is a partition of the vertices of a graph into sets that induce connected subgraphs. Such partitions naturally occur in many application areas such as road networks, and image processing. We consider Balanced Connected Partitions (BCP), where the two classical objectives for BCP are to maximize the weight of the smallest, or minimize the weight of the largest component. We study BCP on c-claw-free graphs, the class of graphs that do not have $K_{1,c}$ as an induced subgraph, and present efficient (c-1)-approximation algorithms for both objectives. In particular, due to th…
Partitioning a graph into balanced blocks such that few edges run between blocks is a key problem for large-scale distributed processing. A current trend for partitioning huge graphs are streaming algorithms, which use low computational resources. In this work, we present a shared-memory streaming multi-recursive partitioning scheme that performs recursive multi-sections on the fly without knowing the overall input graph. Our approach has a considerably lower running time complexity in comparison with state-of-the-art non-buffered one-pass partitioning algorithms for the standard graph partiti…
Motivated by applications in gerrymandering detection, we study a reconfiguration problem on connected partitions of a connected graph $G$. A partition of $V(G)$ is \emph{connected} if every part induces a connected subgraph. In many applications, it is desirable to obtain parts of roughly the same size, possibly with some slack $s$. A \emph{Balanced Connected $k$-Partition with slack $s$}, denoted \emph{$(k,s)$-BCP}, is a partition of $V(G)$ into $k$ nonempty subsets, of sizes $n_1,\ldots , n_k$ with $|n_i-n/k|\leq s$, each of which induces a connected subgraph (when $s=0$, the $k$ parts are …
