V. Rodl and L. Thoma, Asymptotic packing and the random greedy algorithm, Random Structures & Algorithms 8 (1996), 161-178  Home/Search   Document Not in Database   Summary   Related Articles   Check

....P i = P i 1 # e i and S i = S i 1 e i , else keep P i = P i 1 and S i = S i 1 . That is, consider the edges in random sequential order and add each to the packing if you can. We conjecture that E[ S # ] meets the bounds of our Theorem. This author [6] and, independently, V. Rodl and L. Thoma [4] have shown that E[ P # ] # N k 1 or equivalently that E[ S # ] o(N) Viewed in this light we are now looking at a second order term, just how close to a perfect packing can we get. Unfortunately, this natural algorithm has eluded more refined analysis. We feel it would be most interesting ....

V. Rodl and L. Thoma, Asymptotic packing and the random greedy algorithm, Random Structures & Algorithms 8 (1996), 161-178

....implication for Theorem 1.3 from [36] His technique shows, moreover, that a random greedy algorithm almost surely finds an efficient cover for each relevant hypergraph. An alternate proof of this last fact as it relates to Pippenger s theorem (Theorem 1.2) was then given by Rodl and L. Thoma [30]. Their proof is based on the Rodl nibble. Acknowledgements Here seems an appropriate place to repeat the acknowledgement given to the author s research advisor within the dissertation [21] Most sincere thanks go to Jeff Kahn for his endless patience, professional guidance and generous, ....

V. Rodl and L. Thoma, Asymptotic packing and the random greedy algorithm, Random Struct. Alg., to appear.

....adding each one in its turn to the matching iff it does not intersect any of the edges picked to the matching before. We conjecture that the expected number of uncovered vertices by the matching this algorithm produces is O(ND Gamma1= k Gamma1) ln N) O(1) This remains open. See [12] [11] for some related results. When applied to the hypergraph described in the proof of Corollary 4.1, the above algorithm is the following procedure for constructing a large partial S(k Gamma 1; k; n) Let K 1 ; K 2 ; Km be a random order of all the m = Gamma n k Delta k subsets of f1; ....

V. Rodl and L. Thoma, Asymptotic packing and the random greedy algorithm, Random Structures & Algorithms (to appear)

....0 j = l 1 so that codeg(B; B 0 ) n Gamma l Gamma 1 k Gamma l Gamma 1 = o(D) Our thanks to Nati Linial for noting that Pippenger s Theorem is the natural setting for the branching process approach. Indeed, our original proof was only for the Erdos Hanani case. V. Rodl and L. Thoma [3] recently gave a different proof of our main result. They show that the random greedy algorithm gives a packing in some sense close in distribution to that given by the Rodl nibble argument of [2] 2 The Continuous Model We turn to a continuous time branching process that will appropriately ....

V. Rodl and L. Thoma, Asymptotic packing and the random greedy algorithm, preprint, 1994

....and S i = S i Gamma1 Gamma e i , else keep P i = P i Gamma1 and S i = S i Gamma1 . That is, consider the edges in random sequential order and add each to the packing if you can. We conjecture that E[jS j] meets the bounds of our Theorem. This author [6] and, independently, V. Rodl and L. Thoma [4] have shown that E[jP j] N k 1 or equivalently that E[jS j] o(N ) Viewed in this light we are now looking at a second order term, just how close to a perfect packing can we get. Unfortunately, this natural algorithm has eluded more refined analysis. We feel it would be most interesting ....

V. Rodl and L. Thoma, Asymptotic packing and the random greedy algorithm, Random Structures & Algorithms 8 (1996), 161-178

....at random from among all triples which do not share a pair with any previously picked triple, until there are no more candidate triples. It is perhaps mildly surprising that such a simple random greedy procedure almost always results in a collection of triples which cover almost all of the pairs [13, 12]. In this paper we obtain significantly tighter bounds on the number of uncovered pairs. In particular, we show that the number of uncovered pairs is almost always no more than n 7=4 o(1) where o(1) is a function going to 0 as n goes to infinity. This problem is expressed nicely in the ....

.... of asymptotically good partial designs as conjectured by Erdos and Hanani [4] and, later, the existence of packings and colourings in hypergraphs [5, 10, 7, 1, 9] The nibble algorithm isn t exactly the same as the random greedy algorithm but there are enough similarities that Rodl and Thoma [12] were able to use it to show that the random greedy algorithm almost always produces a partial triple system with only o(n 2 ) uncovered pairs. Spencer [13] proved the same using completely different techniques. By almost always we mean that with probability 1 Gamma o(1) the algorithm ....

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V. Rodl and L. Thoma, Asymptotic packing and the random greedy algorithm, Random Structures and Algorithms 8 (1996), 161--177.

....A more sophisticated analysis, given by the author in [7] shows that, in fact, ffi need not be unreasonably small to guarantee such more than nearly perfect packings. Unfortunately, total independence of the edge selections seems in this case to be essential. Rodl and Thoma show in [15] that the nibble algorithm proceeds in essentially the same manner as the random greedy algorithm (pick edges for the packing one at a time, uniformly at random from those edges not incident with any previously picked edge) This suggests that the algorithm given here is, in some sense, a ....

V. Rodl and L. Thoma, Asymptotic packing and the random greedy algorithm, Random Structures and Algorithms 8 (1996), 161--177.

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