Approximating complete partitions Guy Kortsarz Joint work with J. Radhakrishnan and S.Sivasubramanian

Problem 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 C i

Example In the following graph, the optimum is 4. Figure 1: cp(G) = 4

Another Example In an equal sides complete bipartite graph, cp(G)= n/ Figure 2: cp(G)= n/2 + 1

Previous Work: Related to the Achromatic Number. But in AN C i 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 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]

Our 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 (n log log n )

Rare 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. 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 LB: opt + o( log 2 n) is hard to compute. Elkin, Kortsarz, 2004

A related but computable function  ( G ) : Maximize d so that there exists a subgraph with at least d 2 / 2 edges and   d. Computable in polynomial time. Edmonds and Johnson 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 G n,1/2,  (G) =  ( n ) but cp(G) = There exists a (polynomially computable) complete partition with parts.

The Method We imitate the complete bipartite graph. But we do so with subsets: F igure 3: A complete bipartite graph of subsets

How do we find such subsets A collection T of disjoint sets C i is t expanding if: There are at least t C i in the collection. Every C i has at least t neighbors outside  i C i

Figure 4: Expanding subsets

Expanding sets imply large complete partition First step: Partition V \  C i into random equal parts. Figure 5 c1c1 c2c2 ctct t1t1 tktk

Claim With constant probability, all C i will have neighbors in all but fraction of the subsets.

Second Step Randomly group the C i into supersets Every superset is a union of With a constant probability every superset has a neighbor in every T i

Large  implies large expansion Iterative greedy algorithm: Start with a degree at most  and  (  2 ) edges bipartite graph When construction C i+1 add a new vertex to C i+1 only if it has at least half its neighbors outside  i j = 1 N( C j )

Figure 6

Summary 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 {C i } of size t =  (  (G) ) =  (cp (G) ) Randomly partition V \  i C i into Randomly group the C i into superset each containing

Remarks 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. But needs additional and complicated analysis At a very high level, the comes from this: given G n,1/2, what size of subsets do we need in order for partition to be complete?

Further 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  (log 1/4 n) lower bound. Kortsarz and Shende.