Approximating complete partitions Guy Kortsarz Joint work with J. Radhakrishnan and...
-
date post
21-Dec-2015 -
Category
Documents
-
view
215 -
download
0
Transcript of Approximating complete partitions Guy Kortsarz Joint work with J. Radhakrishnan and...
Approximating Approximating complete partitionscomplete partitions
Guy Kortsarz
Joint work with
J. Radhakrishnan and S.Sivasubramanian
Problem DefinitionsProblem Definitions
A disjoint partition of the vertices
of a graph is complete if every share an edge
The Complete partition problem: Given a graph G Find a complete partition with maximum k
Let cp(G) denote the optimum number of Ci
jiCCCV jii
k
i ,0,1
jiCC ji ,,
ExampleExample
In the following graph, the optimum is 4.
Figure 1: cp(G) = 4
Another ExampleAnother Example
In an equal sides complete bipartite graph,
cp(G)= n/2 + 1.
Figure 2: cp(G)= n/2 + 1
Previous Work:Previous Work:
Related to the Achromatic Number. But in AN Ci have to be independent sets.
Many previous results on AN. See the surveys [Edwards ’97], [Hughes & MacGillivray ’97].
CP: Defined by Gupta (1969) Well studied. For example: [Sampathkumar & Bhave ’76], [Bhave ’79], [Bollobás, Reed &Thomason ’84],
[Kostochka ’82], [Yegnanarayanan 2002], [Balasubramanian 2003]
Was defined in the context of homomorphism. Related to many known graph properties an dnotions:
Harmonious coloring, Graph contraction to clique, r – reductions….
Hardness and ApproximationHardness and Approximation
NP – hardness results:
Interval & co – graphs [Bodlaender ’89] Trees [Cairnie & Edwards ’97]
Approximable by +1 on forests
[Cairnie & Edwards ’97]
An approximation for d – regular
graphs [Halldórsson 2004])log( nO
Our ResultsOur Results
1. Upper Bound: Algorithm that finds a complete
partition with parts.
ratio approximation.
2. First hardness of approximation: For some constant c < 1 – no approximation ratio of
unless NP RTIME (nlog log n)
)log(
)(
n
Gcp
)log( nO
nc log
Rare ratios in approximationRare ratios in approximation
The first log n, < 1 constant, threshold. Congestion minimization:
UB: log n/ log log n. Raghavan, Thompson, 87 LB: log log n. Chuzhoy, Naor, 2004
Domatic number: (log n) for maximization problem. Feige, Halldórsson, Kortsarz, Srinivasan
Non-Symmetric k – center: (log* n ). UB: log* n, Panigrahy and Vishwanathan.
Also: log* n by Archer LB: Chuzhoy, Guha, Halperin, Khanna, Kortsarz,
Krauthgamer and Naor, 2004
Rare ratios cont.Rare ratios cont.
Polylogarithmic ratio:
Multiplicative. Group Steiner on trees. UB: O( log 2 n). Garg, Konjevod, Ravi LB: ( log 2 - n) for every constant . Halperin and Krauthgamer.
Additive. Minimum time radio broadcast. opt + O( log 2 n) (for small radius graphs). Bar- Yehuda, Goldreich, Itai ’91. Kowalski and
Pelc 2004. LB: opt + o( log 2 n) is hard to compute. Elkin, Kortsarz, 2004
A related but computable functionA related but computable function
( G ): Maximize d so that there exists a subgraph with at least d2 / 2 edges and d.
Computable in polynomial time. Edmonds and Johnson 1970.
Given a cp ( G ) parts partition, select one edge per pair. Delete edges inside the subsets. Maximum degree cp(G) – 1 per vertex and at least cp(G)(cp(G) – 1) / 2
Thus, (G) cp(G) – 1 In Gn,1/2 , (G) = ( n ) but cp(G) =
There exists a (polynomially computable) complete partition
with parts.
)log/( nnO
)log/)(( nG
The MethodThe Method We imitate the complete bipartite graph. But we do so with
subsets:
FFigure 3: A complete bipartite graph of subsetsigure 3: A complete bipartite graph of subsets
How do we find such subsetsHow do we find such subsets
A collection T of disjoint sets Ci
is t expanding if:There are at least t Ci in the
collection.Every Ci has at least t neighbors
outside i Ci
Figure 4: Expanding subsetsFigure 4: Expanding subsets
Expanding sets imply large complete Expanding sets imply large complete partitionpartition
First step: Partition V \ Ci into random equal parts.
Figure 5
)log/( ntk
c1
c2
ct
t1
tk
Claim
With constant probability, all Ci
will have neighbors in all but
fraction of the subsets.
))log(exp( t
Second StepSecond Step
Randomly group the Ci into supersets
Every superset is a union of
With a constant probability every superset has a neighbor in every Ti
iC
iCt )log(
iC
Large Large implies large expansion implies large expansion
Iterative greedy algorithm: Start with a degree at most and ( 2)
edges bipartite graph
When construction Ci+1 add a new vertex to Ci+1 only if it has at least half its neighbors
outside ij = 1 N(Cj )
Figure 6Figure 6
SummarySummary
Let t be the maximum expansion possible.
We show t = ( (G) ).
Hence the algorithm overview is: Find a (G) partition Use the greedy algorithm to get an expanding
collection {Ci} of size t = ( (G) ) = (cp (G) )
Randomly partition V \ iCi into
Randomly group the Ci into superset each containing
parts)))(log(/)(()log/( GcpGcptt
iCGcpGcpt )))(log(/)(()log(
Remarks on the lower boundRemarks on the lower bound
Based on the Feige, Halldórsson, Kortsarz and Srinivasan result for set-cover packing. Every NPC problem can be mapped into a set-cover instance with n elements and subsets of size d so that: A yes instance is mapped into a set cover
instance that can be covered with n/d pairwise disjoint sets
For a no instance, the sets are essentially random subsets of size d and so n·log(n)/d subsets are required to cover all elements
Remarks on the lower bound cont.Remarks on the lower bound cont.
But needs additional and complicated analysis
At a very high level, the comes from this: given Gn,1/2, what size of subsets do we need in order for partition to be complete?
nlog
Further RemarksFurther Remarks
Standard methods of derandomization give a deterministic algorithm .
A simple algorithm gives 1/2 ratio; Better for bounded degree graphs.
In the domatic number case the constant in the ratio is known (equals 1!). Here there is a gap.
Our lower bound gives
inapproximability for the Achromatic number problem on bipartite graph.
The best previous result (log1/4n) lower bound.
Kortsarz and Shende.
)log( n