A Complexity Dichotomy for Finding Disjoint Solutions of Vertex Deletion Problems

We investigate the computational complexity of a general “compression task” centrally occurring in the recently developed technique of iterative compression for exactly solving NP-hard minimization problems. The core issue (particularly but not only motivated by iterative compression) is to determine the computational complexity of, given an already inclusion-minimal solution for an underlying (typically NPhard) vertex deletion problem in graphs, to find a better disjoint solution. The complexity of this task is so far lacking a systematic study. We consider a large class of vertex deletion problems on undirected graphs and show that, except for few cases which are polynomial-time solvable, the others are NP-complete. This class includes problems such as Vertex Cover (here the corresponding compression task is decidable in polynomial time) or Undirected Feedback Vertex Set (here the corresponding compression task is NP-complete).

• Type: Conference
• Year: 2009
• Where: MFCS
• Authors: Michael R. Fellows, Jiong Guo, Hannes Moser, Rolf Niedermeier
• Keywords: Compression Task, Iterative Compression, MFCS 2009, Theoretical Computer Science, Vertex Deletion Problems