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