FPT algorithms over linear delta-matroids with applications

2025-02-19

AI Summary

This work systematically characterizes the parameterized complexity of intersection, packing, and covering problems on linear delta-matroids, revealing fundamental differences from classical matroids. Methodologically, it introduces the first deterministic rank-based sieve algorithm parameterized by rank $r$, extends determinantal sieving to the delta-matroid setting, and integrates skew-symmetric matrix representations with structural graph-theoretic modeling. Key results include the first proof that Delta-Matroid Triangle Cover—a structured covering problem—is fixed-parameter tractable (FPT) with respect to solution size $k$; resolution of the FPT status of Cluster Subgraph and Strong Triadic Closure under matching number parameterization; and efficient FPT algorithms for $k$-Triangle Packing and $k$-Edge Packing. The core contribution is the establishment of a delta-matroid-specific parameterized algorithmic paradigm, clarifying the distinct complexity landscapes under the parameters $k$ (solution size) and $r$ (rank).

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Matroids, particularly linear ones, have been a powerful tool in parameterized complexity for algorithms and kernelization. They have sped up or replaced dynamic programming. Delta-matroids generalize matroids by encapsulating structures such as non-maximum matchings in general graphs and various path-packing and topological configurations. Linear delta-matroids (represented by skew-symmetric matrices) offer significant expressive power and enable powerful algorithms. We investigate parameterized complexity aspects of problems defined over linear delta-matroids or with delta-matroid constraints. Our analysis of basic intersection and packing problems reveals a different complexity landscape compared to the familiar matroid case. In particular, there is a stark contrast between the cardinality parameter $k$ and the rank parameter $r$. For example, finding an intersection of size $k$ of three linear delta-matroids is W[1]-hard when parameterized by $k$, while more general problems (e.g., finding a set packing of size $k$ feasible in a linear delta-matroid) are FPT when parameterized by $r$. We extend the recent determinantal sieving procedure of Eiben, Koana and Wahlstr""om (SODA 2024) to sieve a polynomial for a monomial whose support is feasible in a linear delta-matroid by $r$. Second, we investigate a class of problems that remains FPT when parameterized by $k$, even on delta-matroids of unbounded rank. We begin with Delta-matroid Triangle Cover - finding a feasible set of size $k$ that can be covered by a vertex-disjoint packing of triangles (sets of size 3) from a given collection. This approach allows us to find a packing of $K_3$'s and $K_2$'s in a graph with a maximum number of edges, parameterized above the matching number. As applications, we settle questions on the FPT status of Cluster Subgraph and Strong Triadic Closure parameterized above the matching number.

Problem

• Investigating parameterized complexity over linear delta-matroids
• Extending determinantal sieving for delta-matroid feasibility
• Settling FPT status for Cluster Subgraph and Strong Triadic Closure

Innovation

• Linear delta-matroids parameterized complexity
• Determinantal sieving for polynomial feasibility
• FPT approach for delta-matroid triangle cover