Efficient Linearly Ordered Colorings of Hypergraphs via Semidefinite Programming Rounding

Kernekoncepter

This paper presents an improved algorithm for linearly ordered (LO) coloring of 2-LO colorable 3-uniform hypergraphs, using semidefinite programming (SDP) rounding techniques to achieve a significantly better bound on the number of colors used compared to prior work.

Resumé

The paper considers the problem of linearly ordered (LO) coloring of hypergraphs. In an LO coloring, vertices are assigned ordered colors such that (i) no edge is monochromatic, and (ii) each edge has a unique maximum color.

The key contributions are:

1. The authors show how to use SDP-based rounding methods to produce an LO coloring with e^O(n^(1/5)) colors for 2-LO colorable 3-uniform hypergraphs. This improves upon the previous best bound of e^O(n^(1/3)) colors.

2. The authors first reduce the problem to cases with highly structured SDP solutions, called "balanced" hypergraphs. They then show how to apply classic SDP-rounding tools in this case.

3. The reduction to balanced hypergraphs is novel and could be of independent interest.

4. As a byproduct, the authors provide a simple proof of a result from prior work that given a 2-LO colorable 3-uniform hypergraph, it can be 2-colored in polynomial time.

The paper builds upon prior work on approximate graph and hypergraph coloring, leveraging SDP-based techniques to obtain improved bounds for the LO coloring problem.

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Vigtigste indsigter udtrukket fra

Improved linearly ordered colorings of hypergraphs via SDP rounding

by Anand Louis,... kl. arxiv.org 05-02-2024

https://arxiv.org/pdf/2405.00427.pdf

Dybere Forespørgsler

Can the techniques developed in this paper be extended to LO coloring of hypergraphs beyond the 3-uniform case

The techniques developed in the paper can potentially be extended to LO coloring of hypergraphs beyond the 3-uniform case. The key lies in adapting the SDP-based rounding methods and combinatorial tools to suit the specific structure and requirements of higher-order hypergraphs. By understanding the underlying principles of LO coloring and the SDP relaxation used in the paper, researchers can explore how to generalize these techniques to hypergraphs with different uniformity levels. This extension may involve modifying the algorithms to account for the increased complexity and connectivity present in higher-order hypergraphs. Additionally, considering the unique properties of hypergraphs beyond the 3-uniform case will be crucial in designing efficient and effective coloring algorithms for such scenarios.

Are there connections between the complexity of LO coloring and other promise constraint satisfaction problems that remain unclassified

There are potential connections between the complexity of LO coloring and other promise constraint satisfaction problems that remain unclassified. The classification of promise constraint satisfaction problems (PCSPs) on the binary alphabet has been a significant area of research, with recent advancements shedding light on the complexity of various symmetric PCSPs. Given the relationship between LO coloring and PCSPs, particularly in the context of 2-LO colorable 3-uniform hypergraphs, there may be unexplored connections with other PCSPs that involve different constraints and structures. By investigating these connections, researchers may uncover new insights into the computational complexity of LO coloring and its implications for a broader class of constraint satisfaction problems.

How might the insights from the balanced hypergraph reduction be applicable to other graph and hypergraph optimization problems

The insights from the balanced hypergraph reduction can be applicable to other graph and hypergraph optimization problems by providing a framework for efficiently handling structured instances. The reduction to balanced hypergraphs in the context of LO coloring demonstrates the importance of exploiting specific properties to simplify the problem and improve algorithmic efficiency. This approach can be extended to various optimization problems where structured instances offer opportunities for optimization and algorithm design. By identifying and leveraging balanced structures in different types of graphs and hypergraphs, researchers can develop specialized algorithms that capitalize on the inherent properties of the problem instances. This can lead to more effective solutions and potentially improved complexity bounds for a wide range of optimization problems.