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ExCYT: A Graphical User Interface for Streamlining Analysis of High-Dimensional Cytometry Data

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Diverse collections in matroids and graphs.

Fedor V Fomin¹, Petr A Golovach¹, Fahad Panolan²

¹University of Bergen, Bergen, Norway.

Mathematical Programming

|February 19, 2024

Summary

This study explores the parameterized complexity of finding diverse solutions for combinatorial problems like Weighted Diverse Bases and Diverse Perfect Matchings. We developed fixed-parameter tractable algorithms, offering efficient solutions when solution diversity is the key parameter.

Keywords:

Diversity of solutions Graphs Matroids Parameterized complexity

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Area of Science:

Discrete Mathematics
Theoretical Computer Science
Combinatorial Optimization

Background:

Combinatorial problems often involve finding specific structures within mathematical objects.
The complexity of these problems can be analyzed using parameterized complexity, focusing on specific input parameters.
Diverse solutions are crucial in applications requiring multiple distinct options.

Purpose of the Study:

To investigate the parameterized complexity of three fundamental combinatorial problems: Weighted Diverse Bases, Weighted Diverse Common Independent Sets, and Diverse Perfect Matchings.
To determine if these problems are computationally tractable when parameterized by the number of desired diverse solutions.
To develop efficient algorithms and data structures for finding diverse sets of solutions.

Main Methods:

Parameterized complexity analysis was employed, focusing on the parameter k (number of solutions).
Fixed-parameter tractable (FPT) algorithms were designed for each of the three problems.
A kernelization technique was used to derive a polynomial-sized kernel for the Weighted Diverse Bases problem.

Main Results:

Demonstrated that none of the studied problems are solvable in polynomial time unless P=NP.
Developed fixed-parameter tractable algorithms for all three problems, with complexity depending polynomially on k.
Established a kernel of size related to k for the Weighted Diverse Bases problem, indicating structural properties.

Conclusions:

The problems of finding diverse bases, common independent sets, and perfect matchings are computationally hard in general.
Parameterized complexity offers a viable approach to solving these problems efficiently when the number of solutions (k) is small.
The derived FPT algorithms and kernel provide theoretical guarantees and practical implications for diverse solution finding.