A 2-Approximation algorithm for finding an optimum 3-Vertex-Connected Spanning Subgraph.
Post on 21-Dec-2015
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The problem
• Having sites, the problem is to build roads between them, so you'll be able to travel from each city to the other.
• You have to build the roads in a way that even if sites are destroyed, the other cities will be still connected.
• You’ll need to build it as cheap as possible when the cost of each road (if possible to build) is known.
N
1k
Definitions
• Graph , Directed graph• degree of v• Spanning subgraph of graph• Weighted Graph• Weight of an edge set
( ) ( ) ( )G G GN v and d v N v
EeewEW :)()(
( , )G V E
G
Definitions cont.
• A graph is connected if for any two vertices of there is a path connecting them.
• A subset is a vertex cut of if is disconnected.
• If then is called a k-cut
• A side of a cut is the vertex set of a connected component of .
C V
GG
G \G C
C k C
C\G C
• Graph is k-vertex-connected if either it is a complete graph of vertices or if it has at least vertices and contains no with .
• Alternatively it can be said that is k-vertex-connected graph if for every set of vertices , and G\V’ is connected.
• Connectivity of G, defined to be the maximum k, for which G is k-connected.
V V 1V k
Definitions cont.
( )G
( , )G V Ek
1k h cut h k
G
• k-connected subgraph problem is an NP-hard.
• This is our motivation to find approximation algorithms.
• Approximation Algorithm is called if it is a polynomial time algorithm and produces a solution of weight no more than times the weight of the optimum solution.
- approximation
Known Approximation
• Ravi and Williamson [1997]:
• For an arbitrary k, it achieves where
For k=2 achieves 3-approximation k=3 achieves -approximation
k=4 achieves -approximation
1 1 1( ) 1 .. log
2 3H k k
k
233164
2 ( )H k approximation
More results
Improved results for particular cases:
• In case of edge weights satisfying triangle inequality, Kuller-Raghavachari [1996] suggested an –approximation, algorithm for an arbitrary k.
2( 1)2 kn
More result (cont.)
• Cheyiyan-Thurimella [1996]: -approximation for finding minimum size k-connected spanning subgraph, for an arbitrary k, meaning finding the k-connected spanning subgraph with minimal number of edges.
11 k
• Kuller-Raghavachari [1996]: achieves a for .
• Result was improved by Penn and Shasha-Krupnik [1997] to for .
• Penn and Shasha-Krupnik [1997] also introduced a for .
12 n approximation
More result cont.
2k
2 approximation2k
3 approximation 3k
• Today we show an improved result: a 2-approximation algorithm for finding a minimim weight 3-connected subgraph, introduced by Auletta, Dinitz ,Nutov and Parente [1999]
• Path are internally disjoint paths if no two of them have an internal vertex in common.
• Menger’s therom: For any graph G and its vertices s,t holds: The minimal size if a cut separating t from s equals the maximum number of vertex-disjoint paths between s and t.
More Definitions
Definition:
Graph is k-out-connected from vertex r if there exists k internally vertex-disjoint simple paths to every other vertex
• If there are 2 vertices with k internally disjoint simple paths between them, then for every implies . h cut h k
( , )G V E
Corollary:
A graph is a k-out-connected from vertex r if it has no with separating r from some other vertex
( , )G V Eh cut h k
• Conclusion: In k-out-connected graph from vertex r, any with , if exists, must contain .
h cut h kr
• What is the motivation to use k-out-connected graph algorithm for the problem of finding minimum weight k-connected subgraph?
• There is known algorithm by Frank and Tardos [1996] that find in a directed graph a minimum weight k-out-connected subdigraph in polynomial time.
Lemma 1:
Let be a k-out-connected graph from , and
let be an of with .
Then and for any side holds:
. ( ) 1h k S N r
h cutG r
C G h kr C \S G C
• Corollary:Let be a k-out-connected graph from a vertex r of degree . Then is .In particular if then is .
• Conclusion: For such a vertex r (of degree k) k-connected graph and k-out-connected graph from r are equal.
2 1k connected kG
G 2,3k G
k connected
r
• Graph G=(V,E) is given and its weight function, .
• D(G)= weight digraph obtained from (G,w):
Each undirected edge is replaced by 2 directed edge with same weight as the undirected edge.U(D) underlying graph of digraph D, where for each directed edge replace it by undirected edge .
( )( ), D GD G w
( , ), ,u v u v V
( , ), ,u v u v V
w
( , ), ( , )u v v u
( , )u v
• If is then it is from .
• If is undirected graph and from then is also a from .
G
k connectedk out connected r
G
k out connected r ( )D G
k out connected r
• Theorem [Halin]: Any minimally graph has a vertex of degree k.
• Corollary: It follows from Halin’s theorem that in any graph , exists a minimum weight subgraph with a vertex of degree . We denotes this vertex by .
k connected Gk connected
*vk
k connected
Out Connected Subgraph Algorithm
• Input: A weighted graph , and an integer k.
• Output: a subgraph of and a vertex such that is k-out-connected from and if exists.
GG
( , )G w ( , )G V E
G rr
( )Gd r k
• Set undefined,
• For every vertex do:(1) Set
(2) Find a minimum weight k-out-connected from r subdigraph ofif such exists.(3) If the degree of r in is k and
then set:
,G r , 2 ( ) 1w M w G r V
( )( )
( )r
w e M if e is incident to rw e
w e otherwise
rD ( , )rD G w
rU D
rw U D w
, ,r rG U D r r w w U D
Remarks:
• The algorithm finds k-out-connected subdigraphs with a minimal outdegree in r.
• From all those subdigraph it chooses the subdigraph with the minimal weight.
• Lemma: For any integer and for any weighted graph G that contains a spanning subgraph which is k-out-connected from a vertex of degree k, the algorithm outputs such a subgraph of weight at most twice the minimal possible.The complexity is .
1k
2 3( )O k n m
• Theorem: For any and any weighted k-connected graph G, algorithm outputs a spanning subgraph of G of weight at most in time where is the weight of the optimal subgraph of
2k
2 1k connected *2w 2 3( )O k n m
Theorem: For OCSA is a 2-approximation algorithm for the minimum weight k-connected subgraph problem with complexity 3 5O mn O n
*wk connected G
2,3k
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