, Volume 80, Issue 7, pp 2134–2159 | Cite as

Optimal Composition Ordering Problems for Piecewise Linear Functions

  • Yasushi Kawase
  • Kazuhisa Makino
  • Kento Seimi
Part of the following topical collections:
  1. Special Issue on Algorithms and Computation


In this paper, we introduce maximum composition ordering problems. The input is n real functions \(f_1,\dots ,f_n:\mathbb {R}\rightarrow \mathbb {R}\) and a constant \(c\in \mathbb {R}\). We consider two settings: total and partial compositions. The maximum total composition ordering problem is to compute a permutation \(\sigma :[n]\rightarrow [n]\) which maximizes \(f_{\sigma (n)}\circ f_{\sigma (n-1)}\circ \dots \circ f_{\sigma (1)}(c)\), where \([n]=\{1,\dots ,n\}\). The maximum partial composition ordering problem is to compute a permutation \(\sigma :[n]\rightarrow [n]\) and a nonnegative integer \(k~(0\le k\le n)\) which maximize \(f_{\sigma (k)}\circ f_{\sigma (k-1)}\circ \dots \circ f_{\sigma (1)}(c)\). We propose \(\mathrm {O}(n\log n)\) time algorithms for the maximum total and partial composition ordering problems for monotone linear functions \(f_i\), which generalize linear deterioration and shortening models for the time-dependent scheduling problem. We also show that the maximum total composition ordering problem can be solved in polynomial time if \(f_i\) is of the form \(\max \{a_ix+b_i,d_i,x\}\) for some constants \(a_i\,(\ge 0)\), \(b_i\) and \(d_i\). As a corollary, we show that the two-valued free-order secretary problem can be solved in polynomial time. We finally prove that there exists no constant-factor approximation algorithm for the problems, even if \(f_i\)’s are monotone, piecewise linear functions with at most two pieces, unless P \(=\) NP.


Function composition Time-dependent scheduling Ordering problem 



The first author is supported by JSPS KAKENHI Grant Numbers 26887014 and JP16K16005. The second author is supported by supported by JSPS KAKENHI Grant Numbers JP24106002, JP25280004, JP26280001, and JST CREST Grant Number JPMJCR1402, Japan.


  1. 1.
    Babaioff, M., Immorlica, N., Kempe, D., Kleinberg, R.: A knapsack secretary problem with applications. In: Proceedings of APPROX/RANDOM 2007, pp. 6–28 (2007)Google Scholar
  2. 2.
    Babaioff, M., Immorlica, N., Kleinberg, R.: Matroids, secretary problems, and online mechanisms. in: Proceedings of SODA 2007, pp. 434–443 (2007)Google Scholar
  3. 3.
    Cai, J.Y., Cai, P., Zhu, Y.: On a scheduling problem of time deteriorating jobs. J. Complex. 14(2), 190–209 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cheng, T.C.E., Ding, Q.: The complexity of scheduling starting time dependent tasks with release times. Inf. Process. Lett. 65(2), 75–79 (1998)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cheng, T.C.E., Ding, Q., Kovalyov, M.Y., Bachman, A., Janiak, A.: Scheduling jobs with piecewise linear decreasing processing times. Naval Res. Logist. 50(6), 531–554 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cheng, T.C.E., Ding, Q., Lin, B.M.T.: A concise survey of scheduling with time-dependent processing times. EJOR 152(1), 1–13 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dean, B., Goemans, M., Vondrák, J.: Adaptivity and approximation for stochastic packing problems. In: Proceedings of SODA 2005, pp. 395–404 (2005)Google Scholar
  8. 8.
    Dean, B., Goemans, M., Vondrák, J.: Approximating the stochastic knapsack problem: the benefit of adaptivity. Math. Oper. Res. 33(4), 945–964 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ferguson, T.S.: Who solved the secretary problem? Stat. Sci. 4(3), 282–289 (1989)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)MATHGoogle Scholar
  11. 11.
    Gawiejnowicz, S.: Scheduling deteriorating jobs subject to job or machine availability constraints. EJOR 180(1), 472–478 (2007)CrossRefMATHGoogle Scholar
  12. 12.
    Gawiejnowicz, S.: Time-Dependent Scheduling. Springer, Berlin (2008)MATHGoogle Scholar
  13. 13.
    Gawiejnowicz, S., Pankowska, L.: Scheduling jobs with varying processing times. Inf. Process. Lett. 54(3), 175–178 (1995)CrossRefMATHGoogle Scholar
  14. 14.
    Gupta, J.N., Gupta, S.K.: Single facility scheduling with nonlinear processing times. Comput. Ind. Eng. 14(4), 387–393 (1988)CrossRefGoogle Scholar
  15. 15.
    Ho, K.I.J., Leung, J.Y.T., Wei, W.D.: Complexity of scheduling tasks with time-dependent execution times. Inf. Process. Lett. 48(6), 315–320 (1993)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kawase, Y., Makino, K., Seimi, K.: Optimal composition ordering problems for piecewise linear functions. In: Proceedings of ISAAC 2016, pp. 42:1–42:13 (2016)Google Scholar
  17. 17.
    Melnikov, O.I., Shafransky, Y.M.: Parametric problem of scheduling theory. Cybernetics 15, 352–357 (1980)CrossRefGoogle Scholar
  18. 18.
    Ng, C.T., Barketau, M., Cheng, T.C.E., Kovalyov, M.Y.: “Product partition” and related problems of scheduling and systems reliability: computational complexity and approximation. EJOR 207, 601–604 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Oveis Gharan, S., Vondrák, J.: On variants of the matroid secretary problem. In: Proceedings of ESA 2011, pp. 335–346 (2011)Google Scholar
  20. 20.
    Tanaev, V.S., Gordon, V.S., Shafransky, Y.M.: Scheduling Theory: Single-Stage Systems. Kluwer Academic Publishers, Dordrecht (1994)CrossRefMATHGoogle Scholar
  21. 21.
    Wajs, W.: Polynomial algorithm for dynamic sequencing problem. Archiwum Automatyki i Telemechaniki 31(3), 209–213 (1986)MATHGoogle Scholar

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Authors and Affiliations

  1. 1.Tokyo Institute of TechnologyTokyoJapan
  2. 2.Kyoto UniversityKyotoJapan
  3. 3.TokyoJapan

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