Approximation Algorithm for Graph Augmentation

Samir Khuller

Ramakrishna Thurimella

報告人：蕭志宣　鄭智懷

Outline

Introduction Related Work 2-approximation

Related Work (History)

Tarjan solve 2 edge-connected augmentation problem in linear time (1976).

But the graph must be complete graph.

Related Work (History)

Somebody modified Tarjan’s algorithm which solves triconnected subgraph in linear time.

In a paper’s conference, it holds for k-connected.

K connected problem

Minimum subgraph augmentation

weighted unweighted weighted unweighted

Minimum k-connected NP-h

ard

NP-hard NP-h

ard

NP-hard

NP-hard NP-h

ardNP-h

ard

Related Work - Guideline

Related Word - Approximation Edge connectivity augment

1993Samir Khuller, Ramakrishna Thurimella 2-approximation

2003Anna Galluccio and Guido Proietti faster 2-approximation

Related Word - Approximation Vertex connectivity subgraph

1994(2-connected)Samir Khuller, Uzi Vishkin 5/3-approximation

1994(2-connected)Garg, Santosh and Singla 3/2-approximation

2001(2-connected)S. Vempala and A. Vetta 4/3-approximation

Related Word - Approximation Edge connectivity subgraph

1994(2-connected)Samir Khuller, Uzi Vishkin 3/2-approximation1995(k-connected)Samir Khuller, Balaji Raghavachari 1.85-approximation

2003(2-connected)Raja Jothi Balaji Raghavachari Subramanian Varadarajan 5/4-approximation

2001(2-connected)J. Cheriyan, A. SebS, Z. Szigeti 17/12-approximation

2001(2-connected)S. Vempala and A. Vetta 4/3-approximation

2003(k-connected)Harold N. Gabow 1.61-approximation

Related Word – Special Case

符合三角不等式

1995 (k vertex connectivity)Samir Khuller, Balaji Raghavachari some approximation with k

NP-hard

Related Word – Special Case

已知道至少有 6k2 個 vertices 求 k vertex connectivity

O(pn=)-approximation algorithm for any > 0 andk (1 - )n

Related Word – Special Case

已知 G 是 planar graph

1998 (2 edge connected augment problem)Sergej Fialko, Petra Mutzel 5/3-approximation

2004(2 edge,2 vertex subgraph)Artur Czumaj, Michelangelo Grigni, Papa Sissokho, Hairong Zhao PTAS

NP-hard

Related Word – Special Case

已知 G 是 bipartite graph

1998(k-connectivity augment problem)Jørgen Bang-Jensen, Harold N. Gabow, Tibor Jordán, Zoltán Szigeti Polynomial time solvable

Related Word – Special Case

Augment problem 已知 tree 是 depth first search tree

2003(2 edge connected augment problem)Anna Galluccio and Guido Proietti polynomial solvable

Related Word - Randomized

1998András A. Benczúr, David R. Karger

K-connectivity

K-edge connected

K-vertex connected

Graph Augmentation

Input:

G0=(V,E0), a set Feasible of m weighted edges on V

Output:

A subset Aug of edges whose addition make G0 2-connected

The minimum branching

A branching of a directed graph G rooted at a vertex r is a spanning tree of G such that each vertex except r has indegree exactly one and r has indegree zero

The minimum weight branching is a branching with the least weight.

r

1

2

6

3

5

3 6

4

2

Algorithm

Step1: pick an arbitrary leaf r and root the tree G0 at r, and directing all tree edges toward the root r. Set all tree edges weight to 0.

(undirected tree G0 directed tree T)

Step1

r

Algorithm

Step2: Consider the edges that belong to G=(V,E) but not belong to G0, for each such edge (u,v) do

If (u,v) is a back edge

add one directed edge to Ed

If (u,v) is a cross edge

add two directed edges to Ed

Step2

r

Algorithm

Step3: find a minimum weight branching in Gd rooted at r. For each edge in Ed picked , add corresponding edge in E-E0 to Aug.

Step4: Output Aug.

Lemma 1 & Lemma 2

If G is two-edge connected, then directed graph GD is strongly connected.

If G is two-edge connected, then the edge connectivity of G0 U Aug is at least 2.

(G0+ Aug is two-edge connected)

Lemma 3

The weight of Aug is less than twice the optimal augmentation. That is, the algorithm is a 2-approximation algorithm for augmentation problem.

Time complexity

O(m+nlogn) (for finding the minimum weight branching)