1. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (1) Augmenting undirected node-connectivity by one László A. Végh STOC 2010 Accepted Paper June 7, 2010 ★★

2. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (2) Introduction (1/4) Undirected graph -connected ( -node-connected): – Still connected after deletion of any set of at most nodes.

3. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (3) Introduction (2/4) In this problem, the input graph is already -connected. Find an edge set with minimum number of edges, such that is -connected.

4. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (4) Introduction (3/4) Similar problems & Previous results: Node-connectivityEdge-connectivity Undirected Jackson & Jordán (2005) László A. Végh (2009) Watanabe & Nakamura (1987) Directed Frank & Jordán (1995) András Frank (1992) Solved! NotSolved! Solved!

5. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (5) Introduction (4/4) Special cases: – – Minimum degree is at least – There exists a set with so that has at least connected components

6. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (6) Solving the Problem If we can calculate in polynomial time: – the minimum number of edges whose addition makes -connected. Then we can find the edge set by finding edges one by one inductively. – Try to add each edge to and re-calculate the value of. How to find exactly?

7. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (7) NXNX X1X1 X2X2 X3X3 X4X4 X5X5 pieces separator Definitions (1/N) A subpartition of with is called a clump if and for any. := number of edges between and : small clump : large clump G = (V,E) is (k-1)-connected

8. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (8) Definitions (2/N) An edge connects if, lies on different piece of Two clumps are said to be independent if there is no edge connecting both. Two clumps are said to be dependent if they are not independent. G = (V,E) is (k-1)-connected X, Y are clumps. Y1Y1 Y2Y2 X1X1 X2X2 uv Y1Y1 Y2Y2 X1X1 X2X2 st

9. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (9) Definitions (3/N) A bush is a set of pairwise different small clumps, so that each edge in connects at most two of them. A shrub is a set consisting of pairwise independent (possibly large) clumps. For a bush, define For a shrub, define G = (V,E) is (k-1)-connected clumps, bushes, shrubs Observation If is -connected, then must contain at least edges connecting clump.

10. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (10) Definitions (4/N) A grove is a set consisting of some (possibly zero) bushes and one (possibly empty) shrub, so that: 1)two clumps in different bushes are independent. 2)A clump in a bush is independent from all clumps in the shrub. Define G = (V,E) is (k-1)-connected clumps, bushes, shrubs, groves

11. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (11) Main Theorem Define. Let be a -connected graph with. Then. 仿照有向圖版本的證明，定義出一套偏序關係，最 後從限制更多的集合一步步推導出來 … G = (V,E) is (k-1)-connected clumps, bushes, shrubs, groves

12. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (12) Example

13. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (13) Directed Version (1/2) In a digraph, an ordered pair is called an one-way pair if and there is no arc in from to. tailhead

14. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (14) Directed Version (2/2) Theorem: For a -connected digraph with, the minimum number of new arcs whose addition results in a -connected digraph equals the maximum number of pairwise independent one-way pairs. Can define a natural partial order on one-way pairs. A subset is called cross-free if any two non-independent pairs in are comparable with respect to. Such maximal is called a skeleton.

15. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (15) Definition (5/N) A clump is called basic if all pieces is connected. The clump is derived from the basic clump if each piece of is union of some pieces of. : all clumps derived from : all small clumps derived from : the set of all basic clumps. For, denotes the union of the sets with. G = (V,E) is (k-1)-connected

16. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (16) Claim 1. (1) Two clumps X and Y are derived from the same basic clump if and only if N X = N Y (2) If two basic clumps X and Y have a piece in common, then X = Y. – Since G is (k-1)-connected, – All vertex in N X is adjacent to this piece, same as N Y. – So N X = N Y. – By (1) we have X = Y. G = (V,E) is (k-1)-connected

17. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (17) Definition (6/N) An edge set covers if after deleting from the graph is still connected. We say covers/connects if covers/connects all clumps in. Claim 2: for, an edge set, we have covers covers connects. G = (V,E) is (k-1)-connected X, Y clumps

18. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (18) Definition (7/N) Two clumps and are nested if or for some and, for all and for all. G = (V,E) is (k-1)-connected X, Y clumps dominant piece of X w.r.t Ydominant piece of Y w.r.t X

19. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (19) Lemma 3. Assume is a large basic clump, and is an arbitrary basic clump. If and are dependent then and are nested. G = (V,E) is (k-1)-connected X, Y clumps

20. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (20) Claim 4. Claim 4-1: For the basic clumps X and Y, implies for some. Claim 4-2: Let X and Y be two different clumps both basic or both small. If for some then X and Y are nested with being the dominant piece of Y w.r.t X. Y1 X1 Y2 X2 uv

21. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (21) Lemma 3. Assume is a large basic clump, and is an arbitrary basic clump. If and are dependent then and are nested. G = (V,E) is (k-1)-connected X, Y clumps

22. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (22) Definition (8/N) Two clumps are crossing: dependent but not nested. A subset is crossing if for any two dependent clumps X, Y, is a subset of A subset is cross-free if it contains no crossing clumps. => any two dependent clumps are nested.

23. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (23) Definition (9/N) For a crossing system and a clump, define the set of clumps in independent from or nested with. Similarly, for a subset, is the set of clumps in not crossing any clump in. A cross-free is called a skeleton of if it is maximal cross-free in, that is,.

24. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (24) Main Theorem Define. Let be a -connected graph with. Then. Let denote max over groves consisting of a shrub and bushes of clumps in Let be the min #edges covering. Theorem: For a crossing system, G = (V,E) is (k-1)-connected clumps, bushes, shrubs, groves

25. 黃上恩 ( 大學部專題 ) Augmenting undirected node-connectivity by one 2010/06/07 (25) Finally… Thanks for your attention!