Algorithmica

, Volume 80, Issue 7, pp 2134–2159

Optimal Composition Ordering Problems for Piecewise Linear Functions

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

Abstract

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.

Keywords

Function composition Time-dependent scheduling Ordering problem

Notes

Acknowledgements

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.

References

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)
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)
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)
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)
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)
9. 9.
Ferguson, T.S.: Who solved the secretary problem? Stat. Sci. 4(3), 282–289 (1989)
10. 10.
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)
11. 11.
Gawiejnowicz, S.: Scheduling deteriorating jobs subject to job or machine availability constraints. EJOR 180(1), 472–478 (2007)
12. 12.
Gawiejnowicz, S.: Time-Dependent Scheduling. Springer, Berlin (2008)
13. 13.
Gawiejnowicz, S., Pankowska, L.: Scheduling jobs with varying processing times. Inf. Process. Lett. 54(3), 175–178 (1995)
14. 14.
Gupta, J.N., Gupta, S.K.: Single facility scheduling with nonlinear processing times. Comput. Ind. Eng. 14(4), 387–393 (1988)
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)
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)
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)
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)
21. 21.
Wajs, W.: Polynomial algorithm for dynamic sequencing problem. Archiwum Automatyki i Telemechaniki 31(3), 209–213 (1986)

Authors and Affiliations

• Yasushi Kawase
• 1
• Kazuhisa Makino
• 2
• Kento Seimi
• 3
1. 1.Tokyo Institute of TechnologyTokyoJapan
2. 2.Kyoto UniversityKyotoJapan
3. 3.TokyoJapan