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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)