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 weightedunweighted augmentation weightedunweighted Minimum k-connected NP-hard

Related Work - Guideline

Related Word - Approximation Edge connectivity augment 1993 Samir Khuller, Ramakrishna Thurimella  2-approximation 2003 Anna 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-approximation 1995(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 6k 2 個 vertices 求 k vertex connectivity O(pn=)-approximation algorithm for any > 0 and k (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) J ø rgen Bang-Jensen, Harold N. Gabow, Tibor Jord á n, Zolt á n 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 1998 Andr á s A. Bencz ú r, David R. Karger

K-connectivity K-edge connected K-vertex connected

Graph Augmentation Input: G 0 =(V,E 0 ), a set Feasible of m weighted edges on V Output: A subset Aug of edges whose addition make G 0 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

Algorithm Step1: pick an arbitrary leaf r and root the tree G 0 at r, and directing all tree edges toward the root r. Set all tree edges weight to 0. (undirected tree G 0 directed tree T)

Step1 r

Algorithm Step2: Consider the edges that belong to G=(V,E) but not belong to G 0, for each such edge (u,v) do If (u,v) is a back edge add one directed edge to E d If (u,v) is a cross edge add two directed edges to E d

Step2 r

Algorithm Step3: find a minimum weight branching in G d rooted at r. For each edge in E d picked, add corresponding edge in E-E 0 to Aug. Step4: Output Aug.

Lemma 1 & Lemma 2 If G is two-edge connected, then directed graph G D is strongly connected. If G is two-edge connected, then the edge connectivity of G 0 U Aug is at least 2. (G 0 + 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)