Download presentation

Presentation is loading. Please wait.

Published byTylor Horner Modified over 2 years ago

1
A Better Algorithm for Finding Planar Subgraph Gruia Călinescu Cristina G. Fernandes Ulrich Finkler Howard Karloff

2
Introduction 4/9-approxi. algorithm for maximum planar subgraph problem 2/3-approxi. algorithm for Outer-planar subgraphs( all vertices on the boundary) Maximum planar subgraph problem and its complement ( removing as few edges as possible to leave a planar subgraph) are Max SNP-hard

3
Maximum planar subgraph problem given a graph G, find a planar subgraph of G with the maximum number of edges. NP-complete The simplest algorithm : Spanning tree (assume G is connected.) Maximal planar subgraph : output any planar subgraph to which the addition of any new edges would violate planarty.

4
Motivation Spanning tree of G achieves a performance ratio of 1/3. A connected spanning subgraph of G whose cycles are triangles, besides being planar, has one more edge per triangle than a spanning tree of G has.

5
Definition A triangular cactus is a graph whose cycles if any are triangles and such that all edges appear in some cycle. A triangular structure is a graph whose cycles are triangles.

6
A greedy algorithm A for G with bounded degree. Performance ratio is 7/18. Linear time First, A greedily constructs a maximal triangular cactus in G Second, A extends triangular cactus to triangular structure.

7
Algorithm A Starting with E 1 =Ø, repeatedly (as long as possible) find a triangle T whose vertices are in different components of G[E 1 ], and add the edges of T to E 1. Let S 1 = G[E 1 ]. Starting with E 2 =E 1, repeatedly (as long as possible) find an edge e in G whose endpoints are in different components of G[E 2 ]., and add e to E 2. Let S 2 =G[E 2 ]. Output S 2.

8
Q1 : P-times? Yes! Linear time for bounded-degree graphs.

9
Q2 : Feasible? Yes. S 2 is indeed a triangular structure in G.

10
Q3 : ratio=7/18 ? OBS : ( # of edges of algorithm A ) = ( # of edges of spanning tree) + ( # of triangles in S 1 )

11
Q3 : (def1) H : maximum planar subgraph of G ( # of edges of H )=3n-6- t Where t : missing edges If t=0 then H is a triangulated graph ( # of faces of H ) ≥ 2n-4-2t each missing edges can destroy at most 2 triangles.

12
Q3 : (def2) k components in S 1. p i : ( # of triangles in i th component) p = sum of p i.

13
Q3 : ( 小結論 ) 在 S1 產生後， G 內所有的 triangle ，必有兩點在 S1 裡的同一個 component 內 →H 內所有的 triangle ，必有兩點在 S1 裡的同一個 component 內 (H is a subgraph of G) → H 內任一個 triangle ，存在一個 edge e ， such that e 的兩端點在 S1 裡的某一個 component 內 小結論： ( # triangles in H) ≤ 2*( # of e in H ) 每個 e 可能由 2 個 triangle 共用

14
Q3 : (def3) H’ : subgraph of H induced by edges of H whose endpoints in the same component of S 1. ( # of vertices of i th component of S 1 )=2p i +1 H’ 最多有 edges 2(6p-3k) ≥ 2E(H’) ≥ ( # of triangles in H) ≥ 2n-4- 2t ≥ by 小結論 ≥ by Q3 (def1)

15
Q3 : (end)

16
Q4 : tight ? S : any connected triangular cactus with p triangles. (# 0f v=2p+1) S’ : supergraph of S (f=2n- 4=2(2p+1)-4)=4p-2 G : 在每一個 face 加一個點再補 上邊 (# of v=2p+1+4p-2=6p-1) (# of edges = 3(6p-1)-6)=18p-9

17
Q4 : (tight?) Input : G S1=S S2=S + 每個 face 中的一個紅色邊 E(S2)=E(S)+(4p-2) =3p+4p-2 =7p-2 Ratio=(7p-2)/(18P-9)

18
A better algorithm B Algorithm B (ratio=4/9) Let S 1 = maximum triangular cactus in G. Starting with E 2 =E 1, repeatedly (as long as possible) find an edge e in G whose endpoints are in different components of G[E 2 ]., and add e to E 2. Let S 2 =G[E 2 ]. Output S 2.

19
Q1 : P-time? By [CN85] and [GS85], algorithm for graphic matroid parity runs in time O(m 3/2 nlog 6 n). Maximum triangular cactus can be obtained from graphic matroid parity in time O(n)

20
Q2 : feasible? Yes ! Output of algorithm A is feasible.

21
Q3 : ratio=4/9 (p.1) According to Matching Theory, Lovász and Plummer[LP86], we know The number of triangles in a maximum triangular cactus in G = the mininmum of Ф(P,Q) taken over all valid pairs (P,Q) for G where

22
Q3 : ratio=4/9 (p.2) Theorem 2.3 : then

23
Q4 : tight ? (p.1) G’ : triangular plane graph with n’ vertices ( and 2n’-4 triangles). G : for all faces, add a new vertices in the face and adjacent to all three vertices on the boundary of that face. G has n’+2n’-4 vertices and 3(3n’-4)- 6=9n’-18 edges.

24
Q4 : tight ? (p.2) The following lemma is observed in [LP86,p.440] If S is triangular structure with t triangles in G then there is a matching in G of size t. Any edge in G has at least one endpoint in V’. Therefore a maximum matching in G has at most n’ edges. We conclude that S has at most n’ triangles

25
Q4 : tight ? (p.3) Input : G Output : S 2 has at most n’ triangles.

26
Outerplanar subgraph An outerplanar graph G is a maximal outerplanar graph if no edge can be added without losing outerplanarity. Note: algorithm B produces outerplanar graphs, so it is a approximation algorithm for maximum outerplanar subgraph.

27
Outerplanar graph has at most 2n-3 edges 用國中的觀念 n 多邊形可以割成 (n-2) 個三角 形 ( 內角和 =(n-2)*180) Face=(n-2)+1 By Euler’s formula : n – m + f =2 n-m+ (n-1)=2 2n-3 =m

28
Ratio=2/3 用 algorithm B 的分析 將分母改為 2n-3

29
Tight ? There are outerplanar graph H i with 2i vertices and 3i-2 edges which do not have any triangles.

30
The complexity of the problem MAXIMUM PLANAR SUBGRAPH is Max SNP-hard. NPD is Max SNP-hard.

31
Open problems How large a performance ratio one can achieve is an obvious one. Is there a linear-time approximation algorithm for MAXIMUM PLANAR SUBGRAPH with performance ratio 1/3+ε? Is there any approximation algorithm with a constant performance ratio for NPD?

Similar presentations

OK

Graph Theory Ch6 Planar Graphs. Basic Definitions curve, polygon curve, drawing crossing, planar, planar embedding, and plane graph open set region,

Graph Theory Ch6 Planar Graphs. Basic Definitions curve, polygon curve, drawing crossing, planar, planar embedding, and plane graph open set region,

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on focus group discussion Ppt on history of atomic theory Ppt on diabetes type 1 Ppt on road accidents in south Design ppt online Ppt on history of badminton games Template download ppt on pollution Best ppt on earth day and night Revising vs editing ppt on ipad Free ppt on mobile number portability pdf