Recent Advances in Graph Partitioning

We survey recent trends in practical algorithms for balanced graph partitioning together with applications and future research directions

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

Balanced Coarsening for Multilevel Hypergraph Partitioning via Wasserstein Discrepancy

We propose a balanced coarsening scheme for multilevel hypergraph partitioning. In addition, an initial partitioning algorithm is designed to improve the quality of k-way hypergraph partitioning. By assigning vertex weights through the LPT algorithm, we generate a prior hypergraph under a relaxed balance constraint. With the prior hypergraph, we have defined the Wasserstein discrepancy to coordinate the optimal transport of coarsening process. And the optimal transport matrix is solved by Sinkhorn algorithm. Our coarsening scheme fully takes into account the minimization of connectivity metric (objective function). For the initial partitioning stage, we define a normalized cut function induced by Fiedler vector, which is theoretically proved to be a concave function. Thereby, a three-point algorithm is designed to find the best cut under the balance constraint

유전 알고리즘에서의 적응적 짝짓기 제도

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2017. 2. 문병로.짝짓기 제도는 자식 해를 만들기 위하여 두 부모를 선택하는 방법을 말한다. 이는 유전 알고리즘의 동작 전반에 영향을 끼친다. 본 논문에서 는,헝가리안방법을사용한짝짓기제도에대해연구하였다.그제도들은 대응되는 거리의 합을 최소화하는 방법, 최대화하는 방법, 그리고 비교를 위해 랜덤하게 대응시키는 방법들을 가리킨다. 본 논문에서는 이 제도들 을잘알려진문제인순회판매원문제와그래프분할문제에적용하였다. 또한 세대별로 가장 좋은 해가 어떻게 변화하는지 분석하였다. 이러한 분 석에 기초하여, 본 논문에서는 간단히 결합된 짝짓기 제도를 제안하였다. 제안된 제도는 결합되지 않은 제도에 비해 더 좋은 결과를 보였다. 본 논문에서는 또한, 본 논문의 핵심 방법인 짝짓기 제도를 결합하는 방법을 제안한다. 본 논문의 적응적인 짝짓기 방법은 세 헝가리안 제도 중하나를선택한다.모든짝지어진쌍은다음세대를위한짝짓기방법을 결정할 투표권을 갖게 된다. 각각의 선호도는 부모해간 거리와 부모해와 자식해의 거리의 비율을 통해 결정된다. 제안된 적응적 방법은 모든 단일 헝가리안짝짓기제도,비적응적으로결합된방법,전통적인룰렛휠선택, 기존의다른거리기준방법들보다좋은결과를보였다.제안된적응적방 법은정기적인해집단의유입과지역최적화와결합된환경에서도적절한 제도를 선택했다. 본 논문에서는 헝가리안 방법을 최대 혹은 최소의 지역 최적점을찾는방법으로교체했다.이방식역시지역최적점을찾는단일 방법들보다 좋은 결과를 보였다I. Introduction 1 1.1 Motivation 1 1.2 Related Work 2 1.3 Contribution 4 1.4 Organization 6 II. Preliminary 7 2.1 Hungarian Method 7 2.2 Geometric Operators 10 2.2.1 Formal Definitions 10 2.3 Exploration Versus Exploitation Trade-off 11 2.4 Test Problems and Distance Metric 13 III. Hungarian Mating Scheme 15 3.1 Proposed Scheme 15 3.2 Tested GA 18 3.3 Observation 18 3.3.1 Traveling Salesman Problem 18 3.3.2 Graph Bisection Problem 21 IV. Hybrid and Adaptive Scheme 28 4.1 Simple Hybrid Scheme 28 4.2 Adaptive Scheme 30 4.2.1 Significance of Adaptive Scheme 30 4.2.2 Proposed Method 31 4.2.3 Theoretical Support 34 4.2.4 Experiments 36 4.2.5 Traveling Salesman Problem 36 4.2.6 Graph Bisection Problem 40 4.2.7 Comparison with Traditional Method 41 4.2.8 Comparison with Distance-based Methods 42 V. Tests in Various Environments 50 5.1 Hybrid GA 50 5.1.1 Experiment Settings 50 5.1.2 Results and Discussions 51 5.2 GA with New Individuals 52 5.2.1 Experiment Settings 52 5.2.2 Results and Discussions 53 VI. A Revised Version of Adaptive Method 62 6.1 Hungarian Mating Scheme 62 6.2 Experiment Settings 62 6.3 Results and Discussions 63 VII. Conclusion 67 7.1 Summary 67 7.2 Future Work 68Docto