ABSTRACT
The rate capacity region of an uplink Gaussian channel is a generalized symmetric polymatroid. Practical applications impose additional lower and upper bounds on the rate allocations, which are represented by box constraints. A fundamental scheduling problem over an uplink Gaussian channel is to seek a rate allocation maximizing the weighted sum-rate (MWSR) subject to the box constraints. The best-known algorithm for this problem has time complexity O (n5 lnO(1) n). In this paper, we take a polymatroidal approach to developing a quadratic-time greedy algorithm and a linearithmic-time divide-and-conquer algorithm. A key ingredient of these two algorithms is a linear-time algorithm for minimizing the difference between a generalized symmetric rank function and a modular function after a linearithmic-time ordering.
- J. G. Andrews, Interference cancellation for cellular systems: a contemporary overview, IEEE Trans. Wireless Commun. 12(2): 19--29, 2005. Google ScholarDigital Library
- T. M. Cover and J. A. Thomas, Elements of information theory. John Wiley & Sons, 2012.Google ScholarDigital Library
- J. Edmonds, Submodular functions, matroids and certain polyhedra. In Combinatorial structures and their applications, eds. R. Guy, H. Hanani, N. Sauer and J. Schonheim, Pages 69--87, 1970.Google Scholar
- W. H. Cunningham, On submodular function minimization. Combinatorica 5(3): 185--192, 1985. Google ScholarDigital Library
- A. Federgruen and H. Groenevelt, The greedy procedure for resource allocation problems: Necessary and sufficient conditions for optimality. Operations Research 34: 909--918, 1986. Google ScholarDigital Library
- A. Federgruen and H. Groenevelt, Characterization and optimization of achievable performance in general queueing systems, Operations Research 36(5): 733--741, 1988. Google ScholarDigital Library
- L. Fleischer and S. Iwata, A push-relabel framework for submodular function minimization and applications to parametric optimization. Discrete Applied Mathematics 131(2): 311--322, 2003. Google ScholarDigital Library
- A. Frank and E. Tardos, Generalized polymatroids and submodular flows, Mathematical Programming 42: 489--563, 1988. Google ScholarDigital Library
- S. Fujishige, Submodular Functions and Optimization, 2nd ed. Annals of Discrete Mathematics vol. 58, Elsevier, Amsterdam, 2005.Google Scholar
- H. Groenevelt, Two algorithms for maximizing a separable concave function over a polymatroid feasible region. European Journal of Operational Research 54(2): 227--236, 1991.Google ScholarCross Ref
- M. Grötschel, L. Lovász, and A. Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1: 169--197, 1981.Google ScholarCross Ref
- S. Iwata, Submodular function minimization. Mathematical Programming 112(1): 45--64, 2008. Google ScholarDigital Library
- S. Iwata and L. Fleischer and S. Fujishige. A combinatorial strongly polynomial algorithm for minimizing submodular functions. Proc. of 32nd Symposium on Theory of Computing (STOC'00), pp. 97--106, 2000. Google ScholarDigital Library
- S. Iwata and J. B. Orlin, A simple combinatorial algorithm for submodular function minimization. In Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'09), pages 1230--1237, 2009. Google ScholarDigital Library
- E. L. Lawler and C. U. Martel, Computing maximal "polymatroidal" network flows. Mathematical of Operation Research 7: 334--347, 1982. Google ScholarDigital Library
- Y. T. Lee, A. Sidford, and S. C. Wong, A faster cutting plane method and its implications for combinatorial and convex optimization. In Proceedings of the 56th Annual Symposium on Foundations of Computer Science (FOCS'15), pages 1049--1065, 2015. Google ScholarDigital Library
- Y. Liu, Z. Qin, M. Elkashlan, Z. Ding, A. Nallanathan, and L. Hanzo, Nonorthogonal Multiple Access for 5G and Beyond, Proceedings of the IEEE 105(12): 2347--2381, 2017.Google ScholarCross Ref
- J.B. Orlin, A faster strongly polynomial time algorithm for submodular function minimization. Mathematical Programming 118(2): 237--251, 2009. Google ScholarDigital Library
- P. Patel and J. Holtzman, Analysis of a simple successive interference cancellation scheme in a DS/CDMA system, IEEE J. Sel. Areas Commun. 12(5): 796--807, 1994. Google ScholarDigital Library
- A. Schrijver, A combinatorial algorithm minimizing submodular functions in strongly polynomial time. Journal of Combinatorial Theory, Series B, 80(2): 346--355, 2000. Google ScholarDigital Library
- A. Schrijver, Combinatorial optimization: polyhedra and efficiency, Springer, 2003.Google Scholar
- N. V. Shakhlevich, A. Shioura, and V. A. Strusevich, Single machine scheduling with controllable processing times by submodular optimization. Int. J. Found. Comput. Sci. 20: 247--269, 2009.Google ScholarCross Ref
- J. G. Shanthikumar and D. D. Yao, Multiclass queueing systems: polymatroidal structure and optimal scheduling control, Operations Research 40(S2): 293--299, 1992.Google Scholar
- A. Shioura, N. V. Shakhlevich, and V. A. Strusevic, Decomposition algorithms for submodular optimization with applications to parallel machine scheduling with controllable processing times. Math. Program., Ser. A 153: 495--534, 2015. Google ScholarDigital Library
- A. Shioura, N. V. Shakhlevich, and V. A. Strusevich, Application of submodular optimization to single machine scheduling with controllable processing times subject to release dates and deadlines, INFORMS Journal on Computing 28: 148--161, 2016. Google ScholarDigital Library
- A. Shioura, N. V. Shakhlevich, and V. A. Strusevich, Preemptive models of scheduling with controllable processing times and of scheduling with imprecise computation: A review of solution approaches, European Journal of Operational Research 266(3): 795--818, 2018.Google ScholarCross Ref
- D. M. Topkis. Minimizing a submodular function on a lattice. Operations Research 26(2): 305-- 321, 1978. Google ScholarDigital Library
- D. M. Topkis. Supermodularity and complementarity, Princeton University Press, 2011.Google Scholar
- D. Tse and S. V. Hanly. Multiaccess fading channels - part I: polymatroid structure, optimal resource allocation and throughput capacities. IEEE Transactions on Information Theory 44(7): 2796--2994, 1998. Google ScholarDigital Library
- D. Tse and P. Viswanath, Fundamentals of wireless communication. Cambridge university press, 2005. Google ScholarDigital Library
Index Terms
- MWSR over an Uplink Gaussian Channel with Box Constraints: A Polymatroidal Approach
Recommendations
Superposition Coding with Unequal Error Protection for the Uplink of Overloaded DS-CDMA System
The overloaded CDMA schemes exploited in direct sequence CDMA (DS-CDMA) systems are mainly to accommodate a greater number of users than the available spreading factor N . In this paper, a superposition coding CDMA (SPC-CDMA) with unequal error ...
Cache-aided MISO-NOMA concept
Coded caching, non-orthogonal multiple access (NOMA), and massive multiple-input multiple-output (MIMO) are challenging and attractive technologies for rapidly emerging mobile networks, offering numerous significant advantages. The literature shows that ...
Successive interference cancelers for multimedia multicode DS-CDMA systems over frequency-selective fading channels
Noncoherent and coherent multicode direct-sequence code-division multiple access (DS-CDMA) systems with successive interference cancellation (SIC) for multimedia reverse links over frequency-selective fading channels are studied. Followed by a RAKE ...
Comments