Hitting and Harvesting Pumpkins

TL;DR

This paper introduces fixed-parameter tractable algorithms and approximation methods for covering and packing c-pumpkin-models in graphs, generalizing classical problems like Vertex Cover and Feedback Vertex Set.

Contribution

It provides the first FPT algorithm for covering all c-pumpkin-models and a logarithmic approximation algorithm for both covering and packing c-pumpkin-models.

Findings

FPT algorithm for covering c-pumpkin-models with parameter k

O(log n)-approximation for covering and packing c-pumpkin-models

Generalization of Vertex Cover and Feedback Vertex Set algorithms

Abstract

The "c-pumpkin" is the graph with two vertices linked by c>0 parallel edges. A c-pumpkin-model in a graph G is a pair A,B of disjoint subsets of vertices of G, each inducing a connected subgraph of G, such that there are at least c edges in G between A and B. We focus on covering and packing c-pumpkin-models in a given graph: On the one hand, we provide an FPT algorithm running in time 2^O(k) n^O(1) deciding, for any fixed c>0, whether all c-pumpkin-models can be covered by at most k vertices. This generalizes known single-exponential FPT algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the cases c=1,2 respectively. On the other hand, we present a O(log n)-approximation algorithm for both the problems of covering all c-pumpkin-models with a smallest number of vertices, and packing a maximum number of vertex-disjoint c-pumpkin-models.