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The Complete Partition Problem involves finding a complete partition with a maximum k in a given graph G. This problem is well-studied and related to various graph properties and notions such as harmonious coloring and graph contraction to clique. The paper presents results on hardness and approximation, including upper bounds, hardness of approximation, and rare ratios in approximation. The text discusses a related but computable function, expansion techniques, and iterative greedy algorithms for finding expanding sets in graphs.
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Approximating complete partitions Guy Kortsarz Joint work with J. Radhakrishnan and S.Sivasubramanian
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
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/2 + 1. Figure 2: cp(G)= n/2 + 1
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….
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 (nlog 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 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
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.
The Method • We imitate the complete bipartite graph. But we do so with subsets: Figure 3: A complete bipartite graph of subsets
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
tk t1 c2 c1 ct Expanding sets imply large complete partition • First step: Partition V \Ci into random equal parts. Figure 5
Claim • With constant probability, all Ci will have neighbors in all but fraction of the subsets.
Second Step • Randomly group theCiinto supersets • Every superset is a union of • With a constant probability every superset has a neighbor in everyTi
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 )
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
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 Gn,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 (log1/4n) lower bound. Kortsarz and Shende.
