Hardness of Vertex Deletion and Project Scheduling

  • Ola Svensson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7408)


Assuming the Unique Games Conjecture, we show strong inapproximability results for two natural vertex deletion problems on directed graphs: for any integer k ≥ 2 and arbitrary small ε > 0, the Feedback Vertex Set problem and the DAG Vertex Deletion problem are inapproximable within a factor k − ε even on graphs where the vertices can be almost partitioned into k solutions. This gives a more structured and therefore stronger UGC-based hardness result for the Feedback Vertex Set problem that is also simpler (albeit using the “It Ain’t Over Till It’s Over” theorem) 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 NP-hard to approximate within any constant assuming the Unique Games Conjecture. This explains the difficulty in obtaining good approximation algorithms for that problem and further motivates previous alternative approaches such as bicriteria approximations.


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  1. 1.
    Bansal, N., Khot, S.: Optimal long code test with one free bit. In: Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, pp. 453–462. IEEE Computer Society, Washington, DC (2009)CrossRefGoogle Scholar
  2. 2.
    Bansal, N., Khot, S.: Inapproximability of Hypergraph Vertex Cover and Applications to Scheduling Problems. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 250–261. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  3. 3.
    De, P., James Dunne, E., Ghosh, J.B., Wells, C.E.: The discrete time-cost tradeoff problem revisited. European Journal of Operational Research 81(2), 225–238 (1995)MATHCrossRefGoogle Scholar
  4. 4.
    Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Annals of Mathematics 162, 2005 (2004)MathSciNetGoogle Scholar
  5. 5.
    Even, G., (Seffi) Naor, J., Schieber, B., Sudan, M.: Approximating minimum feedback sets and multi-cuts in directed graphs. Algorithmica 20, 151–174 (1998)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Fulkerson, D.R.: A network flow computation for project cost curves. Management Science 7(2), 167–178 (1961)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Grigoriev, A., Woeginger, G.J.: Project scheduling with irregular costs: complexity, approximability, and algorithms. Acta Inf. 41(2-3), 83–97 (2004)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Guruswami, V., Håstad, J., Manokaran, R., Raghavendra, P., Charikar, M.: Beating the random ordering is hard: Every ordering CSP is approximation resistant. SIAM J. Comput. 40(3), 878–914 (2011)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Karp, R.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press (1972)Google Scholar
  10. 10.
    Kelley, J.E.: Critical-path planning and scheduling: Mathematical basis. Operations Research 9(3), 296–320 (1961)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Kelley Jr., J.E., Walker, M.R.: Critical-path planning and scheduling. Papers presented at the December 1-3, Eastern Joint IRE-AIEE-ACM Computer Conference, IRE-AIEE-ACM 1959 (Eastern), pp. 160–173. ACM, New York (1959)CrossRefGoogle Scholar
  12. 12.
    Khot, S.: On the power of unique 2-prover 1-round games. In: Reif, J.H. (ed.) STOC, pp. 767–775. ACM (2002)Google Scholar
  13. 13.
    Leighton, T., Rao, S.: An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms. In: Proceedings of the 29th Annual Symposium on Foundations of Computer Science, SFCS 1988, pp. 422–431. IEEE Computer Society, Washington, DC (1988)Google Scholar
  14. 14.
    Mossel, E., O’Donnell, R., Oleszkiewicz, K.: Noise stability of functions with low influences: Invariance and optimality. Annals of Mathematics 171(1) (2010)Google Scholar
  15. 15.
    Paik, D., Reddy, S., Sahni, S.: Deleting vertices to bound path length. IEEE Trans. Comput. 43(9), 1091–1096 (1994)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Seymour, P.D.: Packing directed circuits fractionally. Combinatorica 15(2), 281–288 (1995)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Skutella, M.: Approximation algorithms for the discrete time-cost tradeoff problem. Mathematics of Operations Research 23(4), 909–929 (1998)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ola Svensson
    • 1
  1. 1.EPFLSwitzerland

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