Improved Combinatorial Approximations for the NP-Hard Cluster Deletion Problem

Conceptos Básicos

We present improved approximation algorithms for the NP-hard Cluster Deletion problem, which seeks to partition a graph into a disjoint set of cliques by deleting a minimum number of edges. Our algorithms are simpler, faster, and achieve better approximation guarantees than previous work.

Resumen

The Cluster Deletion problem is an NP-hard graph clustering objective with applications in computational biology and social network analysis. The goal is to delete a minimum number of edges to partition a graph into cliques.
The authors first provide a tighter analysis of two previous approximation algorithms, improving their approximation guarantees from 4 to 3. They show that both algorithms can be derandomized in a simple way by greedily selecting pivot nodes based on maximum degree.
The authors' final contribution is a new and purely combinatorial approach for solving the linear program (LP) relaxation of the Cluster Deletion problem. This leads to a faster, more scalable algorithm compared to using general-purpose LP solvers. The authors also prove that their simple degree-based pivoting strategy provides the same 3-approximation guarantee as more complicated pivoting schemes.
The authors accompany their theoretical results with practical implementations and numerical experiments. They include the first implemented deterministic algorithms for Cluster Deletion, which in practice produce solutions that are typically much less than 3 times the optimal. They also demonstrate that their combinatorial LP solver is significantly faster than using black-box LP software and scales to instances that are orders of magnitude larger.

Estadísticas

The graph G has n nodes and m edges.
The set of open wedges in G is denoted as W, and the set of open wedges centered at node k is Wk.

Citas

"We provide several improved theoretical results and practical implementations for combinatorial algorithms for this task."
"We significantly bridge the theory-practice gap by presenting algorithms that are simpler, faster, and have better approximation guarantees."

Ideas clave extraídas de

Combinatorial Approximations for Cluster Deletion: Simpler, Faster, and Better

by Vicente Balm... a las arxiv.org 04-26-2024

https://arxiv.org/pdf/2404.16131.pdf

Consultas más profundas

How can the ideas presented in this work be extended to obtain faster approximation algorithms for the more general Cluster Editing problem

The ideas presented in the work can be extended to obtain faster approximation algorithms for the more general Cluster Editing problem by adapting the combinatorial approach used for Cluster Deletion. One possible extension is to explore different pivot strategies that are tailored to the specific characteristics of Cluster Editing. By designing pivot selection strategies that take into account the unique constraints and objectives of Cluster Editing, it may be possible to improve the efficiency and accuracy of the approximation algorithms. Additionally, incorporating heuristics or optimization techniques that exploit the structure of Cluster Editing instances can lead to faster algorithms with better approximation guarantees. By leveraging the insights gained from the study of Cluster Deletion and applying them to the broader problem of Cluster Editing, it is possible to develop more efficient and effective approximation algorithms for graph clustering objectives.

Can the analysis for the STC LP rounding algorithm be further tightened to match the 3-approximation guarantee of the MatchFlipPivot algorithm

The analysis for the STC LP rounding algorithm can potentially be further tightened to match the 3-approximation guarantee of the MatchFlipPivot algorithm by refining the constraints and variables in the linear programming formulation. By carefully examining the LP relaxation and the rounding process, it may be possible to identify additional properties or relationships that can lead to a more precise analysis. This could involve exploring different LP formulations, introducing new constraints, or optimizing the rounding process to achieve a tighter approximation guarantee. Additionally, conducting a more detailed analysis of the LP relaxation and its connection to the Cluster Deletion problem may reveal insights that can help improve the approximation ratio of the rounding algorithm. By delving deeper into the mathematical properties of the LP relaxation and the rounding technique, it is possible to enhance the analysis and potentially match the 3-approximation guarantee of the MatchFlipPivot algorithm.

Are there other graph clustering objectives beyond Cluster Deletion where the techniques of combining STC labelings with pivot-based rounding can be applied effectively

The techniques of combining STC labelings with pivot-based rounding can be applied effectively to other graph clustering objectives beyond Cluster Deletion. One such objective is correlation clustering, where the goal is to group similar items based on pairwise similarity or dissimilarity measures. By adapting the STC labelings and pivot-based rounding strategies to the context of correlation clustering, it is possible to develop approximation algorithms that efficiently partition the graph into clusters that maximize the correlation within clusters and minimize the correlation between clusters. Additionally, these techniques can be extended to other clustering problems in various domains such as social network analysis, image segmentation, and community detection. By leveraging the principles of STC labelings and pivot-based rounding, it is possible to design effective and scalable algorithms for a wide range of graph clustering objectives.