A Better Algorithm for Finding Planar Subgraph

1
A Better Algorithm for Finding Planar Subgraph

• Gruia Calinescu
• 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 E1 Ø, repeatedly (as long as
  possible)
• find a triangle T whose vertices are in different
  components of GE1, and add the edges of T to
  E1.
• Let S1 GE1.
• Starting with E2E1, repeatedly (as long as
  possible) find an edge e in G whose endpoints are
  in different components of GE2. , and add e to
  E2.
• Let S2GE2.
• Output S2.

8
Q1 P-times?

• Yes!
• Linear time for bounded-degree graphs.

9
Q2 Feasible?

• Yes.
• S2 is indeed a triangular structure in G.

10
Q3 ratio7/18 ?

• OBS
• ( of edges of algorithm A )
• ( of edges of spanning tree)
• ( of triangles in S1)

11
Q3 (def1)

• H maximum planar subgraph of G
• ( of edges of H )3n-6- t
• Where t missing edges
• If t0 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 S1.
• pi ( of triangles in ith component)
• p sum of pi.

13
Q3 (???)

• ? S1 ???,G????triangle,?????S1?????component?
• ?H????triangle,?????S1?????component?(H is a
  subgraph of G)
• ? H????triangle ,????edge e ,such that e
  ?????S1?????component?
• ???( triangles in H) 2( of e in H )
• ?? e ???2? triangle ??

14
Q3 (def3)

• H subgraph of H induced by edges of H whose
  endpoints in the same component of S1.
• ( of vertices of ith component of S1 )2pi1
• H ???
  edges
• 2(6p-3k) 2E(H) ( of triangles in H)
  2n-4-2t
• by ???
• by Q3 (def1)

15
Q3 (end)
16
Q4 tight ?

• S any connected triangular cactus with p
  triangles. ( 0f v2p1)
• S supergraph of S (f2n-42(2p1)-4)4p-2
• G ????face????????
  ( of v2p14p-26p-1)
• ( of edges 3(6p-1)-6)18p-9

17
Q4 (tight?)

• Input G
• S1S
• S2S ??face???????
• E(S2)E(S)(4p-2)
• 3p4p-2
• 7p-2
• Ratio(7p-2)/(18P-9)

18
A better algorithm B

• Algorithm B (ratio4/9)
• Let S1 maximum triangular cactus in G.
• Starting with E2E1, repeatedly (as long as
  possible) find an edge e in G whose endpoints are
  in different components of GE2. , and add e to
  E2.
• Let S2GE2.
• Output S2.

19
Q1 P-time?

• By CN85 and GS85 , algorithm for graphic
  matroid parity runs in time O(m3/2nlog6n).
• 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 ratio4/9 (p.1)

• According to Matching Theory, Lovász and
  PlummerLP86, we know
• The number of triangles in a maximum
  triangular cactus in G
• the mininmum of ?(P,Q) taken over all valid
  pairs (P,Q) for G
• where

22
Q3 ratio4/9 (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 n2n-4 vertices and 3(3n-4)-69n-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 S2 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

• ??????n???????(n-2)????(???(n-2)180)
• Face(n-2)1
• By Eulers formula n m f 2
• n-m (n-1)2
• 2n-3
  m

28
Ratio2/3

• ?algorithm B ???
• ?????2n-3

29
Tight ?

• There are outerplanar graph Hi 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 1/3e?
• Is there any approximation algorithm with a
  constant performance ratio for NPD?