Abstract
We consider the problem of scheduling complex-valued demands over a discretized time horizon. Given a set of users, each user is associated with a set of demands representing different user’s preferences. A demand is represented by a complex number, a time interval, and a utility value obtained if the demand is satisfied. At each time slot, the magnitude of the total selected demands should not exceed a given capacity. This naturally captures the supply constraints in alternating current (AC) electric systems. In this paper, we consider maximizing the aggregate user utility subject to power supply limits over a time horizon. We present approximation algorithms characterized by the maximum angle \(\phi \) between any two complex-valued demands. More precisely, a PTAS is presented for the case \(\phi \in [0,\tfrac{\pi }{2}]\), a bi-criteria FPTAS for \(\phi \in [0,{\pi } \text{- } \delta ]\) for any polynomially small \(\delta \), assuming the number of time slots in the discretized time horizon is a constant. Furthermore, if the number of time slots is polynomial, we present a reduction to the real-valued unsplittable flow on a path problem with only a constant approximation ratio. Finally, we present a practical greedy algorithm for the single time slot case with an approximation ratio of \(\tfrac{1}{2}\cos \frac{\phi }{2}\), while the running time is \({O}(n\log n)\), which can be implemented efficiently in practice.
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Notes
- 1.
The complete work is provided in a technical report [7].
- 2.
This assumption is mainly needed for the dynamic program that is invoked by Algorithm 2.
References
Anagnostopoulos, A., Grandoni, F., Leonardi, S., Wiese, A.: Amazing 2+ \(\varepsilon \) approximation for unsplittable flow on a path. In: Proceedings of SODA, pp. 26–41. SIAM (2014)
Bansal, N., Chakrabarti, A., Epstein, A., Schieber, B.: A quasi-PTAS for unsplittable flow on line graphs. In: Proceedings of STOC, STOC 2006, pp. 721–729. ACM (2006)
Chakaravarthy, V.T., Choudhury, A.R., Gupta, S., Roy, S., Sabharwal, Y.: Improved algorithms for resource allocation under varying capacity. In: Schulz, A.S., Wagner, D. (eds.) ESA 2014. LNCS, vol. 8737, pp. 222–234. Springer, Heidelberg (2014)
Chakaravarthy, V.T., Pandit, V., Sabharwal, Y., Seetharam, D.P.: Varying bandwidth resource allocation problem with bag constraints. In: Parallel & Distributed Processing (IPDPS), pp. 1–10. IEEE (2010)
Chakrabarti, A., Chekuri, C., Gupta, A., Kumar, A.: Approximation algorithms for the unsplittable flow problem. Algorithmica 47(1), 53–78 (2007)
Chau, C.K., Elbassioni, K., Khonji, M.: Truthful mechanisms for combinatorial AC electric power allocation. In: Proceedings of AAMAS (2014). http://arxiv.org/abs/1403.3907
Chau, C.K., Elbassioni, K., Khonji, M.: Truthful mechanisms for combinatorial allocation of electric power in alternating current electric systems for smart grid. ACM Trans. Econ. Comput. (2016). http://arxiv.org/abs/1507.01762
Chekuri, C., Mydlarz, M., Shepherd, F.B.: Multicommodity demand flow in a tree and packing integer programs. ACM Trans. Algorithms (TALG) 3(3), 27 (2007)
Darmann, A., Pferschy, U., Schauer, J.: Resource allocation with time intervals. Theor. Comput. Sci. 411(49), 4217–4234 (2010)
Elbassioni, K., Garg, N., Gupta, D., Kumar, A., Narula, V., Pal, A.: Approximation algorithms for the unsplittable flow problem on paths and trees. In: LIPIcs-Leibniz International Proceedings in Informatics, vol. 18 (2012)
Elbassioni, K., Nguyen, T.T.: Approximation schemes for multi-objective optimization with quadratic constraints of fixed CP-rank. In: Walsh, T. (ed.) ADT 2015. LNCS, vol. 9346, pp. 273–287. Springer, Heidelberg (2015)
Fang, X., Misra, S., Xue, G., Yang, D.: Smart grid the new and improved power grid: a survey. IEEE Commun. Surv. Tutorials 14(4), 944–980 (2012)
Grainger, J., Stevenson, W.: Power System Analysis. McGraw-Hill, New York City (1994)
Grandoni, F., Ingala, S., Uniyal, S.: Improved approximation algorithms for unsplittable flow on a path with time windows. In: Sanità , L., et al. (eds.) WAOA 2015. LNCS, vol. 9499, pp. 13–24. Springer, Heidelberg (2015). doi:10.1007/978-3-319-28684-6_2
Karapetyan, A., Khonji, M., Chau, C.K., Elbassioni, K., Zeineldin, H.: Efficient algorithm for scalable event-based demand response management in microgrids. Technical report, Masdar Institute (2015)
Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Heidelberg (2010)
Khonji, M., Karapetyan, A., Elbassioni, K., Chau, C.K.: Complex-demand scheduling problem with application in smart grid. Technical report, Masdar Institute (2016). http://arxiv.org/abs/1603.01786
Khonji, M., Chau, C.K., Elbassioni, K.: Optimal power flow with inelastic demands for demand response in radial distribution networks. Technical report, Masdar Institute (2016). http://arxiv.org/abs/1507.01762
Khonji, M., Chau, C.K., Elbassioni, K.M.: Inapproximability of power allocation with inelastic demands in AC electric systems and networks. In: ICCCN, pp. 1–6 (2014)
Nemirovski, A.S., Todd, M.J.: Interior-point methods for optimization. Acta Numerica 17(1), 191–234 (2008)
Spieksma, F.C.: On the approximability of an interval scheduling problem. J. Sched. 2(5), 215–227 (1999)
Woeginger, G.J.: When does a dynamic programming formulation guarantee the existence of a fully polynomial time approximation scheme (fptas)? INFORMS J. Comput. 12(1), 57–74 (2000)
Yu, L., Chau, C.K.: Complex-demand knapsack problems and incentives in AC power systems. In: Proceedings of AAMAS, pp. 973–980. Richland, SC (2013)
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Khonji, M., Karapetyan, A., Elbassioni, K., Chau, CK. (2016). Complex-Demand Scheduling Problem with Application in Smart Grid. In: Dinh, T., Thai, M. (eds) Computing and Combinatorics . COCOON 2016. Lecture Notes in Computer Science(), vol 9797. Springer, Cham. https://doi.org/10.1007/978-3-319-42634-1_40
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