Approximating complete partitions

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

2. Problem Definitions • A disjoint partition of the vertices of a graph iscomplete if every share an edge The Complete partition problem: • Givena graphG • Find a complete partition with maximumk Let cp(G) denote the optimum number ofCi

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

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

5. Previous Work: • Related to the Achromatic Number. But inAN 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….

6. 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]

7. 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 (nlog log n)

8. 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

9. Rare ratios cont. • Polylogarithmic ratio: • Multiplicative.Group Steiner on trees. • UB: O( log 2n). Garg, Konjevod, Ravi • LB:  ( log 2 - n) for every constant . Halperin and Krauthgamer. • Additive. Minimum time radio broadcast. • opt + O( log 2n) (for small radius graphs). Bar- Yehuda, Goldreich, Itai ’91. Kowalski and Pelc 2004. • LB: opt + o( log 2n) is hard to compute. Elkin, Kortsarz, 2004

10. A 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.

11. The Method • We imitate the complete bipartite graph. But we do so with subsets: Figure 3: A complete bipartite graph of subsets

12. How do we find such subsets • A collection T of disjoint sets Ci is t expanding if: • There are at least tCi in the collection. • Every Cihas at least t neighbors outside iCi

13. Figure 4: Expanding subsets

14. tk t1 c2 c1 ct Expanding sets imply large complete partition • First step: Partition V \Ci into random equal parts. Figure 5

15. Claim • With constant probability, all Ci will have neighbors in all but fraction of the subsets.

16. Second Step • Randomly group theCiinto supersets • Every superset is a union of • With a constant probability every superset has a neighbor in everyTi

17. Large  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 )

18. Figure 6

19. Summary Lettbe the maximum expansion possible. We showt =  ( (G) ). Hence the algorithm overview is: • Find a (G)partition • Use the greedy algorithm to get an expandingcollection{Ci}of sizet =  ( (G) ) =  (cp (G) ) • Randomly partitionV\iCiinto • Randomly group the Ciinto superset each containing

20. 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

21. 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?

22. 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 (log1/4n) lower bound. Kortsarz and Shende.