FPT algorithms over linear delta-matroids with applications

TL;DR

The paper addresses the parameterized complexity of problems defined over linear delta-matroids, contrasting the cardinality parameter $k$ with the rank parameter $r$ and showing a nuanced landscape that diverges from classical matroids. It extends determinantal sieving to delta-matroids, delivering FPT algorithms with runtimes like $O^*(2^r)$ (characteristic $2$) and $O^*(2^{(q-2)r})$ for $q$-Delta-matroid Intersection, as well as $O^*(2^k)$-type bounds for intersections and packings parameterized by $k$, together with randomized polynomial-space methods. It introduces Delta-matroid Triangle Cover, achieving an $O^*(k^{O(k)})$ algorithm, and leverages a new $ ext{ell}$-projection operation to connect to Colorful Delta-matroid Matching, enabling applications to Cluster Subgraph and Strong Triadic Closure, including FPT results on $K_4$-free graphs and a constant-factor FPT approximation for the latter. The work also clarifies the limits of purely algebraic approaches by highlighting $k$-parameterized hardness for certain problems, while providing a rich toolkit of representations, pivots, Pfaffian techniques, and kernelization-inspired operations that broaden the applicability of delta-matroid methods in parameterized algorithm design.

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 Wahlström (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.

Figures (1)

• Figure 1: A maximum strong set (thick lines) which does not induce a cluster graph.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4: Basis sieving EKW23
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10