Efficient Computation of Representative Sets with Applications in Parameterized and Exact Algorithms

TL;DR

This paper introduces two algorithms for computing representative families in matroids and demonstrates their application in designing efficient parameterized and exact algorithms for various graph problems.

Contribution

The paper presents novel algorithms for computing representative families and applies them to improve algorithms for multiple graph problems.

Findings

Algorithms for representative families in linear and uniform matroids

Single-exponential parameterized algorithms for graph problems

Applications to problems like longest directed cycle and k-subgraph isomorphism

Abstract

We give two algorithms computing representative families of linear and uniform matroids and demonstrate how to use representative families for designing single-exponential parameterized and exact exponential time algorithms. The applications of our approach include - LONGEST DIRECTED CYCLE - MINIMUM EQUIVALENT GRAPH (MEG) - Algorithms on graphs of bounded treewidth -k-PATH, k-TREE, and more generally, k-SUBGRAPH ISOMORPHISM, where the k-vertex pattern graph is of constant treewidth.