Computational Complexity of Maximum Cut on Interval Graphs of Interval Count Two

Core Concepts

Maximum Cut is NP-complete on interval graphs of interval count two.

Abstract

The paper shows that the Maximum Cut problem, which is a fundamental and well-known NP-complete problem, remains NP-complete even on interval graphs of interval count two. The key insights are: The authors provide a polynomial-time reduction from Maximum Cut on cubic graphs to Maximum Cut on interval graphs of interval count two. They introduce several new gadgets, including vertex gadgets, edge gadgets, join gadgets, and switch gadgets, to construct the interval graph. These gadgets are carefully designed to ensure that the Maximum Cut partitions of the interval graph correspond to the Maximum Cut partitions of the original cubic graph. The authors prove that the coloring of each gadget under a Maximum Cut partition of the interval graph follows a specific pattern. This allows them to show that the reduction is correct and the Maximum Cut problem remains NP-complete on interval graphs of interval count two. The result brings the complexity of Maximum Cut on interval graphs closer to the open problem of its complexity on unit interval graphs (interval graphs of interval count one).

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Key Insights Distilled From

Maximum Cut on Interval Graphs of Interval Count Two is NP-complete

by Alexey Barsu... at arxiv.org 04-25-2024

https://arxiv.org/pdf/2203.06630.pdf

Deeper Inquiries

What other graph classes, beyond interval graphs, could the Maximum Cut problem be NP-complete on, and what techniques could be used to prove such results

The Maximum Cut problem has been proven to be NP-complete on various graph classes beyond interval graphs. Some of these graph classes include planar graphs, bipartite graphs, and graphs with bounded treewidth. To establish the NP-completeness of Maximum Cut on these graph classes, similar reduction techniques can be employed. For planar graphs, a common approach is to reduce from a known NP-complete problem on planar graphs, such as Planar 3-SAT or Planar 3-Colorability. The reduction would involve constructing gadgets that preserve the planarity of the graph while encoding the constraints of the original problem. By showing that a solution to the original problem can be translated into a solution to the Maximum Cut problem on planar graphs, the NP-completeness is established. Similarly, for bipartite graphs, a reduction from a known NP-complete problem on bipartite graphs, like Bipartite 3-SAT, could be used. The gadgets in the reduction would need to maintain the bipartite structure of the graph and ensure that the Maximum Cut problem captures the essence of the original problem. In the case of graphs with bounded treewidth, techniques involving dynamic programming and tree decomposition could be employed. By leveraging the properties of graphs with bounded treewidth, the reduction can be designed to showcase the complexity of Maximum Cut on such graph classes.

How could the techniques developed in this paper be extended to provide a complete characterization of the complexity of Maximum Cut on unit interval graphs

To provide a complete characterization of the complexity of Maximum Cut on unit interval graphs, the techniques developed in the paper could be extended by further refining the construction of gadgets and link chains. By carefully designing the intervals and gadgets in the reduction, it may be possible to establish the NP-completeness of Maximum Cut on unit interval graphs. One approach could involve creating specialized gadgets that mimic the properties of unit interval graphs more closely. By introducing gadgets that capture the unique characteristics of unit interval graphs, such as the restriction on interval lengths, the reduction could be tailored to specifically address the complexities of this graph class. Additionally, exploring the relationship between unit interval graphs and other graph classes for which the complexity of Maximum Cut is known could provide insights into the computational hardness of unit interval graphs. By drawing parallels and distinctions between unit interval graphs and other graph classes, a more comprehensive understanding of the complexity of Maximum Cut on unit interval graphs could be achieved.

What are the potential practical implications of establishing the computational complexity of Maximum Cut on various graph classes, and how could this knowledge be applied in real-world optimization problems

Establishing the computational complexity of Maximum Cut on various graph classes has significant practical implications in the field of optimization and decision-making. By determining that Maximum Cut is NP-complete on specific graph classes, it highlights the inherent difficulty of finding optimal solutions within those classes. In real-world applications, this knowledge can guide algorithm design and problem-solving strategies. For example, in network optimization problems where partitioning a network into two subnetworks with maximum cut capacity is crucial, understanding the complexity of Maximum Cut on different graph classes can help in selecting appropriate algorithms and heuristics to tackle the problem efficiently. Furthermore, the insights gained from studying the complexity of Maximum Cut on different graph classes can inform the development of approximation algorithms and heuristics for practical optimization problems. By leveraging the theoretical results on computational complexity, practitioners can tailor their approaches to address specific challenges in optimization tasks involving graph partitioning and network analysis.