A Complexity Dichotomy for Finding Disjoint Solutions of Vertex Deletion Problems

Authors:
Michael R. Fellows;Jiong Guo;Hannes Moser;Rolf Niedermeier
Affiliations:
Charles Darwin University;Universität des Saarlandes;Friedrich-Schiller-Universität Jena;TU Berlin
Venue:
ACM Transactions on Computation Theory (TOCT)
Year:
2011

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Abstract

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 the following task: given an already inclusion-minimal solution for an underlying (typically NP-hard) vertex deletion problem in graphs, find a smaller 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 a few cases are polynomial-time solvable, and the others are NP-hard. The considered class of vertex deletion problems includes Vertex Cover (where the compression task is polynomial time) and Undirected Feedback Vertex Set (where the compression task is NP-complete).