黃上恩 ( 大學部專題 ) augmenting undirected node-connectivity by one 2010/06/07 (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. VéghSTOC 2010 Accepted Paper

June 7, 2010

★★


Introduction (1/4)

• Undirected graph• -connected ( -node-connected):– Still connected after deletion of any set of at most

nodes.


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.


Introduction (3/4)

• Similar problems & Previous results:

Node-connectivity Edge-connectivity

UndirectedJackson & Jordán

(2005)László A. Végh

(2009)

Watanabe & Nakamura(1987)

Directed Frank & Jordán(1995)

András Frank(1992)

Solved

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Solved

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Solved

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NotSolved!

Solved

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Introduction (4/4)

• Special cases:– – – – – – Minimum degree is at least – There exists a set with so that

has at least connected components


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?


NX

X1

X2

X3

X4

X5

pieces

separator

Definitions (1/N)

• A subpartition of withis called a clump if and for any .

• := number of edges between and

• : small clump• : large clump

G = (V,E) is (k-1)-connected


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)-connectedX, Y are clumps.

Y1 Y2

X1 X2

u v

Y1 Y2

X1 X2s t


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)-connectedclumps, bushes, shrubs

Observation

If is -connected, then must contain at least edges connecting clump .


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)-connectedclumps, bushes, shrubs, groves


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…



Example


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 .

tail head


Directed Version (2/2)

• Theorem: For a -connected digraphwith , 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.


Definition (5/N)

• A clump is called basic if all pieces is connected.

• The clump is derived from the basic clumpif 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 .



Claim 1.

• (1) Two clumps X and Y are derived from the same basic clump if and only if NX = NY

• (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 NX is adjacent to

this piece, same as NY.

– So NX = NY.– By (1) we have X = Y.



Definition (6/N)

• An edge set covers if after deletingfrom 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)-connectedX, Y clumps


Definition (7/N)

• Two clumps and arenested if or for some and , for all and for all .


dominant piece of X w.r.t Y dominant piece of Y w.r.t X


Lemma 3.

• Assume is a large basic clump, and is an arbitrary basic clump. If and are dependent then and are nested.



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 somethen X and Y are nested with being the dominant piece of Y w.r.t X.

Y1

X1

Y2

X2u v


Lemma 3.

• Assume is a large basic clump, and is an arbitrary basic clump. If and are dependent then and are nested.



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.


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, .


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 ,



Finally…

Thanks for your attention!

黃上恩 ( 大學部專題 ) augmenting undirected node-connectivity by one 2010/06/07 (1)...

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