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Order of elements

Order of elements. Yuzhong Qu Nanjing University. Topological ordering. Directed Acyclic Graph (DAG). Kahn, Arthur B. (1962), Topological sorting of large networks, Communications of the ACM 5 (11): 558–562.

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Order of elements

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  1. Order of elements Yuzhong Qu Nanjing University

  2. Topological ordering • Directed Acyclic Graph (DAG) Kahn, Arthur B. (1962), Topological sorting of large networks, Communications of the ACM 5 (11): 558–562. Tarjan, Robert E. (1976), Edge-disjoint spanning trees and depth-first search, Actanformatica 6 (2): 171–185

  3. Depth-First Search (DFS) Undirected graph: No cross edges DAG: No backward edges

  4. 竞赛图（Tournament） 竞赛图：底图为Kn的有向图 A B C 4个选手参加的循环赛，比赛结果图

  5. 竞赛图与有向哈密尔顿通路 • 竞赛图含有向哈密尔顿通路（归纳证明）

  6. 循环赛排名 A B 一条有向Hamilton通路（排名） CA B D E F F C 另一条有向Hamilton通路（排名） A B D E F C E D C 从第一名变成了最后一名

  7. 循环赛排名 A B 按照得胜的竞赛场次(得分)排名： A(胜4)B,C(胜3)D, E(胜2)F(胜1) F C 问题：很难说B,C并列第二名是否公平，毕竟C战胜的对手比B战胜的对手的总得分更高(9比5)。 E D

  8. 循环赛排名 建立对应与每个对手得分的向量 s1= (a1, b2, c3, d4, e5, f6) 然后逐次求第k级的得分向量sk, 每个选手的第k级得分是其战胜的对手在第k-1级得分的总和。 A B F C 对应于左图所示的竞赛结果，得分向量： s1=(4,3,3,2,2,1) s2=(8,5,9,3,4,3) s3=(15,10,16,7,12,9) s4=(38,28,32,21,25,16) s5=(90,62,87,41,48,32) ...... E D 当竞赛图满足某种条件下，这个序列收敛于一个固定的排列，这可以作为排名：A C B E D F。

  9. 循环赛排名（FAS） A’: Feedback Arc Set (FAS) (V, A-A’) becomes acyclic (DAG) Minimum Feedback Arc set {FC, EC} C A B D E F A B  F C  E D

  10. It’s difficult to decide who is the winner • References • [BTT1989] Bartholdi, J. and C. Tovey, and M. Trick, Voting schemes for which it can be difficult to tell who won the election, Social Choice and Welfare 6 (1989) 157-166. • [Younger1963] D. Younger. Minimum Feedback Arc Sets for a Directed Graph. IEEE Transactions on Circuit Theory, Vol. 10, No. 2. (1963), pp. 238-245 • [Kemeny1959] J. G. Kemeny. Mathematics without numbers. Daedalus, 88:571-591, 1959.

  11. Adjacent matrix representation for digraph

  12. FAS and MAS • Weighted digraph • Minimum Feedback Arc Set (最小反馈弧集) • Maximum Acyclic subgraph (最大无环子图)

  13. MST (undirected graph) • Weighted graph • Minimum spanning tree (Prim, Kruskal) • Maximum spanning tree (-wij)

  14. Approximation algorithms for FAS • Maximal (may not be Maximum) • Fast VS approximation ratio

  15. Approximation algorithms for FAS • Heuristics [ELS1993] • Ordering vertices by degree (outdegree-indegree) • Result • O(m) worst-case time on a digraph m arcs • Approximation ratio bounded by O((n)) [ELS1993] Eades, P.; Lin, X.; Smyth, W. F. (1993), A fast and effective heuristic for the feedback arc set problem, Information Processing Letters 47: 319–323.

  16. Approximation algorithms for FAS • Heuristics [CF2003] • Breaking cycles by removing arcs: • Arcs with small weight in cycles VS arcs belonging to a large number of cycles //the weight of all arcs in C is decreased • Result • O(mn) worst-case time on a digraph with n vertices and m arcs • Approximation ratio bounded by the length of a longest simple cycle of the digraph. [CF2003] Camil Demetrescu, Irene Finocchi: Combinatorial algorithms for feedback problems in directed graphs. Inf. Process. Lett. 86(3): 129-136 (2003)

  17. An Example

  18. Maximum Acyclic Subgraph (MAS) • References • [BW1997] B. Berger and P. W. Shor. Tight bounds for the Maximum Acyclic Subgraph problem. J. Algorithms, 25(1):1–18, 1997. • [HR1994] R. Hassin and S. Rubinstein. Approximations for the Maximum Acyclic Subgraph Problem. Information Processing Letters, 51:133-140, 1994.

  19. Rank aggregation • a special case of the weighted feedback arc set problem Fig. 4 (Kemeny, 1959)

  20. Partial Ranking • Reference • [Ailon2010] NirAilon: Aggregation of Partial Rankings, p-Ratings and Top-m Lists. Algorithmica 57(2): 284-300 (2010) • [FKMSV2006] Ronald Fagin, Ravi Kumar, Mohammad Mahdian, D. Sivakumar, Erik Vee: Comparing Partial Rankings. SIAM J. Discrete Math. 20(3): 628-648 (2006) • [BFB2009] Mukul S. Bansal, David Fernandez-Baca. Computing distances between partial rankings. Inf. Process. Lett. 109(4): 238-241 (2009) • [DKNS2001] Cynthia Dwork, Ravi Kumar, MoniNaor, D. Sivakumar: Rank aggregation methods for the Web. WWW 2001: 613-622

  21. Other references • [ACN2008] Nir Ailon, Moses Charikar, Alantha Newman: Aggregating inconsistent information: Ranking and clustering. J. ACM 55(5) (2008) • Frans Schalekamp, Anke van Zuylen: Rank Aggregation: Together We're Strong. ALENEX 2009: 38-51

  22. Acknowledgement http://ws.nju.edu.cn Websoft Research Group

  23. Related NP-hard problem • Minimum path cover problem • Longest path problem

  24. 有向无环图的极小路径覆盖（Minimum path cover of a directed acyclic graph） 有向无环图的路径覆盖 2 5 1 4 3

  25. X2 X3 X4 X5 Y2 Y3 Y4 Y5 有向无环图的路径覆盖（二部图匹配） G’ X1 Y1 R L

  26. X2 X3 X4 X5 Y2 Y3 Y4 Y5 有向无环图的路径覆盖（二部图匹配） G’ X1 Y1 R L

  27. 由二部图的匹配生成路径覆盖 • 覆盖路径的个数=顶点个数-匹配对个数 2 5 1 4 3

  28. 由二部图的匹配生成路径覆盖

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