Combinatorial Persistency Criteria for Multicut and Max-Cut

Lange, Jan-Hendrik; Andres, Bjoern; Swoboda, Paul

doi:10.1109/cvpr.2019.00625

Cited by 11 publications

(18 citation statements)

References 43 publications

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“…To this end, we build on the works of Alush and Goldberger [3] and Lange et al [15,16] who establish partial optimality conditions for problems equivalent to correlation clustering with a linear objective function. Regarding their terminology, we remark that the correlation clustering problem, the clique partitioning problem, and the multicut problem are equivalent if the objective functions are linear.…”

Section: Related Workmentioning

confidence: 99%

Partial Optimality in Cubic Correlation Clustering

Stein¹,

Gregorio²,

Andres³

2023

Preprint

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The higher-order correlation clustering problem is an expressive model, and recently, local search heuristics have been proposed for several applications. Certifying optimality, however, is np-hard and practically hampered already by the complexity of the problem statement. Here, we focus on establishing partial optimality conditions for the special case of complete graphs and cubic objective functions. In addition, we define and implement algorithms for testing these conditions and examine their effect numerically, on two datasets.

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Section: Related Workmentioning

confidence: 99%

Partial Optimality in Cubic Correlation Clustering

Stein¹,

Gregorio²,

Andres³

2023

Preprint

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“…Preprocessing and Inprocessing: For fixing variables to their optimal values and shrinking the problem before or during optimization, persistency or partial optimality methods have been proposed in [3,37,38]. These methods apply a family of criteria that, when passed, prove that any solution can be improved if its values do not coincide with the persistently fixed variables.…”

Section: Related Workmentioning

confidence: 99%

RAMA: A Rapid Multicut Algorithm on GPU

Abbas¹,

Swoboda²

2021

Preprint

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We propose a highly parallel primal-dual algorithm for the multicut (a.k.a. correlation clustering) problem, a classical graph clustering problem widely used in machine learning and computer vision. Our algorithm consists of three steps executed recursively: (1) Finding conflicted cycles that correspond to violated inequalities of the underlying multicut relaxation, (2) Performing message passing between the edges and cycles to optimize the Lagrange relaxation coming from the found violated cycles producing reduced costs and (3) Contracting edges with high reduced costs through matrix-matrix multiplications.Our algorithm produces primal solutions and dual lower bounds that estimate the distance to optimum. We implement our algorithm on GPUs and show resulting one to two order-of-magnitudes improvements in execution speed without sacrificing solution quality compared to traditional serial algorithms that run on CPUs. We can solve very large scale benchmark problems with up to O(10 8 ) variables in a few seconds with small primal-dual gaps. We make our code available at https://github.com/ pawelswoboda/RAMA.

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“…However, designing practically efficient preprocessing rules for the general max-cut problem, which also provides theoretical guarantees on the kernel size, remains a challenge. Recent work in this direction was done by Lange et al [118], who designed reduction rules for general max-cut. They showed the efficacy of their rules on instances from computer vision, biomedical image analysis and statistical physics, and for those instances managed to obtain substantial size reductions.…”

Section: Maximum Cutsmentioning

confidence: 99%

Recent Advances in Practical Data Reduction

Abu-Khzam¹,

Lamm²,

Mnich³

et al. 2020

Preprint

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Over the last two decades, significant advances have been made in the design and analysis of fixedparameter algorithms for a wide variety of graph-theoretic problems. This has resulted in an algorithmic toolbox that is by now well-established. However, these theoretical algorithmic ideas have received very little attention from the practical perspective. We survey recent trends in data reduction engineering results for selected problems. Moreover, we describe concrete techniques that may be useful for future implementations in the area and give open problems and research questions.2 Recent Advances for NP-Hard Problems Maximum Independent Set and Minimum Vertex CoverGiven an undirected graph G = (V, E), the goal of the maximum independent set (MIS) problem is to compute a set of vertices I ⊆ V such that (1) no two vertices in I are adjacent to one another, (2) the set I has maximum cardinality among all such sets. The complement of an independent set I is called a vertex cover V \ I. The MIS problem and the complementary problem of finding a minimum vertex cover (MVC) are well-studied NP-hard optimization problems [75] that attract both researchers and practitioners alike. Furthermore, there is no polynomial time approximation algorithm for MIS that can provide an O(n 1−ε ) guarantee for any constant ε > 0, unless P=NP [174]. Finally, MIS is W [1]-hard [58] when parameterized by solution size k. This makes it unlikely that the problem is fixed-parameter tractable in k [58]. On the other hand, MVC is fixed-parameter tractable in solution size k [58]. Exact ApproachesIn recent years, the bridge between theoretically efficient algorithms and their practical applicability has been significantly reduced. In particular, the branch-and-reduce paradigm, i.e., branching algorithms that use a wide variety of reduction rules, have been (1) shown to achieve theoretical running times that are among the best for both MIS and MVC [71,170], and (2) are able to solve large real-world networks in practice [5]. However, most often the approaches used in practice only use a small subset of the reduction rules that have been proposed to achieve good theoretical running times.Abu-Khzam et al.[4] introduced and analyzed the crown reduction rule (and the usage of data reduction rules in this context in practice). Even though the crown rule is not as powerful as the linear programming (LP)-based rule [135] when considering the worst-case size of the resulting kernel, they experimentally verified that it often performs as well as the LP-based rule and is significantly faster in many cases. Furthermore, they show that the LP-based rule is most useful for fairly sparse graphs and should be avoided for dense graphs, as it yields little to no reduction in size.Later, Akiba and Iwata [5] were the first to show the practicality of the branch-and-reduce paradigm for MVC (and MIS) compared to other state-of-the-art approaches like branch-and-bound and branch-and-cut. Their algorithm uses a wide spectrum of reduction rules that form the foundation...

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Combinatorial Persistency Criteria for Multicut and Max-Cut

Cited by 11 publications

References 43 publications

Partial Optimality in Cubic Correlation Clustering

Partial Optimality in Cubic Correlation Clustering

RAMA: A Rapid Multicut Algorithm on GPU

Recent Advances in Practical Data Reduction

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