Hitting and harvesting pumpkins

Abstract

The c-pumpkin is the graph with two vertices linked by c ≥ 1 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 hitting and packing c-pumpkin-models in a given graph in the realm of approximation algorithms and parameterized algorithms. We give a fixed-parameter tractable (FPT) algorithm running in time 2O(k)nO(1) deciding, for any fixed c ≥ 1, whether all c-pumpkin-models can be hit 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. Finally, we present an O(log n)-approximation algorithm for both the problems of hitting all c-pumpkin-models with a smallest number of vertices and packing a maximum number of vertex-disjoint c-pumpkin-models.