在這項研究中，我們提出一個動態演算法針對組合優化中知名覆蓋問題，此研究的動機主要是由供應商選擇問題與管理科學領域的設施位置問題所啟發。我們主要關注集合覆蓋以及在圖論中著名的端點覆蓋問題。我們首次考慮容量限制的動態覆蓋問題。這個問題概括了近年來深入研究的覆蓋問題的動態模型。給定一個動態變化的頂點加權供需網絡G =（V，E），允許新增和刪除邊，目標是設計一個維護近似最小（頂點）覆蓋集的資料結構，可覆蓋所有的元素（邊），同時滿足每個頂點的容量限制。也就是說，當在子集中選取頂點的數量時，可以覆蓋的的邊的數量有一定的限制。而在此研究我們對於端點挑選的數目並沒有限制。我們提出一個確定性的對偶演算法，用於維持在O( log n /ϵ)攤銷更新時間並維持解的近似比率為O(1)，其中 n 是網絡中端點的數量。此外，此演算法可以延伸至另一個更具普遍性的版本，也就是每個邊有固定的需求而且必須分配給相鄰的端點。這是首次對於探索容量覆蓋問題的動態模型的研究。此外，我們提出的確定性對偶演算法其資料結構也可以應用在現實世界。

In this study, we investigate the dynamic version of the well-known covering problem in combinatorial optimization, which is motivated by the supplier selection problem and the facility location problem in the field of operations research and management science. We mainly focus on the set cover and the vertex cover problems. In addition, we first consider the dynamic covering problem with capacity constraints. The problem generalizes the dynamic model of the covering problem, which has been intensively studied in recent years. Given a dynamically changing vertex-weighted supply-demand network G = (V,E), which allows insertions and deletions of edges, the goal is to design a data structure that maintains an approximate minimum (vertex) covering set that can cover all their elements (incident edges) while satisfying the capacity constraint of each vertex. That is, when picking a copy of a vertex v in the cover, the number of v's incident edges that can be covered is up to a given capacity of v. Here a soft capacitated (vertex) cover means that the number of selected copies of a vertex is not bounded.
We obtain a deterministic primal-dual algorithm for maintaining an O(1)- approximate minimum capacitated (vertex) covering in O(𝑙𝑜𝑔 𝑛𝜖⁄) amortized update time, where n is the number of vertices in the network. Moreover, the dynamic algorithm can be extended to a generalization in which every edge is associated with a given demand which has to be assigned to an adjacent vertex. This is the pioneering study exploring dynamic algorithms for the capacitated covering problem. Furthermore, the deterministic data structure designed for the proposed primal-dual algorithm can be straightforwardly implemented for real-world applications.

摘要....................................................... I
Abstract............................................... II
誌謝................................................... III
Contents............................................... IV
List of Figures and Tables............................. V
1 Introduction......................................... 1
1.1 Motivation and background......................... 1
1.2 Problem model..................................... 3
1.3 Overview of our technique......................... 9
2 Level scheme and its key properties................ 10
3 Maintaining an α(β+1)-tight Level Scheme Dynamically 15
3.1 The algorithm: Handling insertion or deletion of an edge 15
3.2 Time complexity................................... 16
3.2.1 Amortized cost of level up...................... 19
3.2.2 Amortized cost of level down.................... 22
3.3 Summary ........................................... 28
4 Approximation with Edge Demand d_e................... 28
4.1 The hyper-graph case.............................. 30
5 Conclusion Remarks.................................. 31
Reference............................................... 33

[1] F. A. Chudak, D. B. Shmoys. Improved approximation algorithms for a capacitated facility location problem. In Proc. the 10th ACM-SIAM Symposium on Discrete Algorithms(SODA), 1999, pp. 875-876.
[2] V. Arya, N. Garg, R. Khandekar, A. Meyerson, K. Munagala, V. Pandit. Local search heuristics for k-median and facility location problems. SIAM J. Comput. 33(3), 2004, pp. 544-562.
[3] A. Aggarwal, A. Louis, M. Bansal, N. Garg, N. Gupta, S. Gupta, and S. Jain. A 3-approximation algorithm for the facility location problem with uniform
capacities. Math. Program., Vol. 141, Issue 1, 2013, pp. 527-547.
[4] F. A. Chudak, D.P. Williamson. Improved approximation algorithms for capacitated facility location problems. Math. Program. 102(2), 2005, pp. 207-222.
[5] A. Andersson and M. Thorup. Dynamic ordered sets with exponential search trees. Journal of the ACM (JACM), Vol. 54, Issue 3, No. 13 , 2007.
[6] S. Baswana, M. Gupta, and S. Sen. Fully dynamic maximal matching in O(log n) update time. SIAM J. Comput. 44(2015), no. 1, pp. 88-113.
[7] S. Bhattacharya, D. Chakrabarty, and M. Henzinger. Fully dynamic approximate maximum matching and minimum vertex cover in O(log3n) worst case
update time. In Proc. the 28th ACM-SIAM Symposium on Discrete Algorithms (SODA), Barcelona, Spain, 2017, pp. 470-489.
[8] S. Bhattacharya, D. Chakrabarty, and M. Henzinger. Deterministic fully dynamic approximate vertex cover and fractional matching in O(1) amortized update time. In Proc. the 19th Conference on Integer Programming and Combinatorial Optimization (IPCO), Waterloo, Canada, 2017.
[9] M. Bansal, N. Garg, and N. Gupta. A 5-Approximation for capacitated facility location. In Proc. the 20th European Symposium on Algorithms(ESA), 2012, pp. 133-144.
[10] S. Bhattacharya, M. Henzinger, and G. F. Italiano. Deterministic fully dynamic data structures for vertex cover and matching. In Proc. the 26th ACM-SIAM Symposium on Discrete Algorithms (SODA), Philadelphia, USA, 2015, pp. 785-804.
[11] S. Bhattacharya, M. Henzinger, and G. F. Italiano. Design of dynamic algorithms via primal-dual method. In Proc. the 42nd International Colloquiumon Automata, Languages, and Programming (ICALP), Heidelberg, Germany 2015, pp. 206-218.
[12] D. B. Shmoys, E. Tardos, K. Aardal. Approximation algorithms for facility location problems. In: Proceedings of 29th ACM Symposium on Theory of Computing, 1997, pp. 265-274.
[13] R. Chen, S. AhmadBeygi, D.R. Beil, A. Cohn, and A. Sinha. Solving truckload procurement auctions over an exponential number of bundles. Forthcoming in Transportation Science, 2009.
[14] M. Charikar, S. Guha. Improved combinatorial algorithms for facility location and k-median problems. In Proc. the 40th IEEE Symposium of Foundations of Computer Science(FOCS), 1999, pp. 378-388.
[15] C. Demetrescu and G. F. Italiano. A new approach to dynamic all pairs shortest paths. Journal of the ACM (JACM), Vol. 51, Issue 6, 2004, pp. 968-992.
[16] L. de Boer, E. Labro, and P. Morlacchi. A review of methods supporting supplier selection. European Journal of Purchasing and Supply Management, 2001, pp. 75-89.
[17] D. R. Beil. Supplier Selection, in Wiley Encyclopedia of Operations Research and Management Science, no. July, J. J. Cochran, Ed. John Wiley and Sons, Inc, 2010, pp. 1-13.
[18] S. Guha, R. Hassin, S. Khuller, and E. Or. Capacitated vertex covering. Journal of Algorithms, Vol. 48, Issue 1, August 2003, pp. 257-270.
[19] A. Gupta, R. Krishnaswamy, A. Kumar, and D. Panigrahi. Online and dynamic algorithms for set cover. In Proc. the 49th ACM Symposium on Theory of Computing (STOC), Montreal, Canada, 2017.
[20] D. Ghawai and G.P. Scheider. New approaches to online procurement. In Proc. the Academy of Information and Management Sciences, vol. 8(2), 2004, pp. 25-28.
[21] F. Hedderich, R. Giesecke, and D. Ohmsen. Identifying and evaluating Chinese suppliers: China sourcing practices of german manufacturing companies. Practix, vol. 9, 2006, pp. 1-8.
[22] M. T. J. Holm, K. de. Lichtenberg. Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. Journal of the ACM (JACM) Vol. 48 Issue 4, 2001, pp. 723-760.
[23] Z. Ivkovic and E. L. Lloyd. Fully dynamic maintenance of vertex cover. In Proc. the 19th International Workshop on Graph-theoretic Concepts in Computer Science (WG), London, UK, 1994, pp 99-111.
[24] K. Jain, M. Mahdian, A. Saberi. A new greedy approach for facility location problems. In Proc. the 34th ACM Symposium on Theory of Computing(STOC), 2002, pp. 731-740.
[25] K. Jain, V. V. Vazirani. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation. J. ACM 48(2), 2001, pp. 274-296.
[26] M. Korupolu, C. Plaxton, R. Rajaraman. Analysis of a local search heuristic for facility location problems. J. Algorithms 37(1), 2000, pp. 146-188.
[27] R. Levi, D.B. Shmoys, C. Swamy. LP-based approximation algorithms for capacitated facility location. In Proc. the 10th Conference on Integer Programming and Combinatorial Optimization(IPCO), 2004, pp. 206-218.
[28] J. Lynn Lunsford and Paul Glader. Boeings nuts-and-bolts problem; Shortage of fasteners tests ability to nish dreamliners. Wall Street Journal, page A8, June 19, 2007.
[29] L.M. Ellram. Total cost modeling in purchasing. Center for Advanced Purchasing Studies, 1994.
[30] M. K. S. Bhutta. Supplier selection problem: Methodology literature review Journal of International Technology and Information Management Vol. 12, Issue 2, 2003, pp. 53-72.
[31] M. Mahdian, M. Pal. Universal facility location. In Proc. the 11th European Symposium on Algorithms(ESA), 2003, pp. 409-421.
[32] M. Mahdian, Y. Ye, J. Zhang. Approximation algorithms for metric facility location problems. SIAM J. Comput. 36(2), 2006, pp. 411-432.
[33] O. Neiman and S. Solomon. Simple deterministic algorithms for fully dynamic maximal matching. In Proc. the 45th ACM Symposium on Theory of Computing (STOC), Palo Alto, USA, 2013, pp. 745-754.
[34] K. Onak and R. Rubinfeld. Maintaining a large matching and a small vertex cover. In Proc. the 42nd ACM Symposium on Theory of Computing (STOC), Cambridge, USA, 2010, pp. 457-464.
[35] D. Peleg and S. Solomon. Dynamic (1+ϵ)-approximate matchings: a density-sensitive approach. In Proc. the 27th ACM-SIAM Symposium on Discrete Algorithms (SODA), Virginia, USA, 2015, pp. 712-729.
[36] M. Pal, E. Tardos, T. Wexler. Facility location with nonuniform hard capacities. In Proc. the 42th Symposium of Foundations of Computer Science(FOCS), 2001, pp. 329-338.
[37] A.P. Kontis, V.Vrysagotis, Supplier selection problem: a literature review of multi criteria approaches based on DEA. Advances in Management and Applied Economics, vol. 1, no. 2, 2011, pp. 207-219.
[38] R. Myers. Food ghts. CFO Magazine, June, 2007.
[39] S. Solomon. Fully dynamic maximal matching in constant update time. In Proc. the 57th Symposium on Foundations of Computer Science (FOCS), New Jersey, USA, 2016, pp. 325-334.
[40] W. Thanaraksakul and B. Phruksaphanrat. Supplier evaluation framework based on balanced scorecard with integrated corporate social responsibility perspective. In Proc. the International Multi Conference of Engineers and Computer Scientists Vol 2, Hong Kong, March 18-20, 2009.
[41] U.S. Census Bureau. Statistics for industry groups and industries: 2005. Technical Report M05(AS)-1, U.S. Census Bureau, November 2006. Annual Survey of Manufactures.
[42] J. Zhang, B. Chen, Y. Ye. A multi-exchange local search algorithm for the capacitated facility location problem. Math. Oper. Res. 30(2), 2005, pp. 389-403.