Hardness of vertex deletion and project scheduling.

*(English)*Zbl 1366.68083Summary: Assuming the Unique Games Conjecture, we show strong inapproximability results for two natural vertex deletion problems on directed graphs: for any integer \(k\geq 2\) and arbitrary small \(\epsilon > 0\), the Feedback Vertex Set problem and the DAG Vertex Deletion problem are inapproximable within a factor \(k-\epsilon\) even on graphs where the vertices can be almost partitioned into \(k\) solutions. This gives a more structured and yet simpler (albeit using the “It Ain’t Over Till It’s Over” theorem) UG-hardness result for the Feedback Vertex Set problem than the previous hardness result. {

} In comparison to the classical Feedback Vertex Set problem, the DAG Vertex Deletion problem has received little attention and, although we think it is a natural and interesting problem, the main motivation for our inapproximability result stems from its relationship with the classical Discrete Time-Cost Tradeoff Problem. More specifically, our results imply that the deadline version is UG-hard to approximate within any constant. This explains the difficulty in obtaining good approximation algorithms for that problem and further motivates previous alternative approaches such as bicriteria approximations.

} In comparison to the classical Feedback Vertex Set problem, the DAG Vertex Deletion problem has received little attention and, although we think it is a natural and interesting problem, the main motivation for our inapproximability result stems from its relationship with the classical Discrete Time-Cost Tradeoff Problem. More specifically, our results imply that the deadline version is UG-hard to approximate within any constant. This explains the difficulty in obtaining good approximation algorithms for that problem and further motivates previous alternative approaches such as bicriteria approximations.

##### MSC:

68Q17 | Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) |

68R10 | Graph theory (including graph drawing) in computer science |

68W25 | Approximation algorithms |

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\textit{O. Svensson}, Theory Comput. 9, Paper No. 24, 759--781 (2013; Zbl 1366.68083)

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[19] | [19] OLASVENSSON: Hardness of vertex deletion and project scheduling. In Proc. 15th Internat. Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX’12), pp. 301–312. Springer, 2012. [doi:10.1007/978-3-642-32512-0_26]759 AUTHOR Ola Svensson assistant professor École Polytechnique Fédérale de Lausanne ola.svenssonepfl ch http://theory.epfl.ch/osven THEORY OFCOMPUTING, Volume 9 (24), 2013, pp. 759–781780 HARDNESS OFVERTEXDELETION ANDPROJECTSCHEDULING ABOUT THE AUTHOR OLASVENSSON(not to be confused with the singerOla Svensson) graduated fromIDSIA in 2009; his advisor wasMonaldo Mastrolilli. The subject of his thesis was the approximability of graph and scheduling problems. After spending two years as a postdoc withJohan HåstadatKTH Royal Institute of Technology, Sweden, he is now back in Switzerland as an assistant professor in thetheory groupatEPFL. Apart from doing research, he enjoys the Alps that are in complete contrast to the flat but still beautiful landscape in South Sweden where he grew up. THEORY OFCOMPUTING, Volume 9 (24), 2013, pp. 759–781781 |

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