B. Bollobas, P. Erdos, Cliques in random graphs, Mathematical Proceedings of the Cambridge Philosophical Society, 80 (1976), 419-427.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in this latter problem see [2, 14, 17, 22] Luczak [24] proved that asymptotically c k 2k ln k by showing the existence of suitably large, disjoint independent sets in G n;p=ck=n . Algorithmically, substantially less progress has been made. One of the rst heuristics to be analyzed [18] [4], GIC, forms color classes by repeatedly removing greedily chosen independent sets until the remaining graph has no component with more than one cycle, at which point it is easy to 3 color it . Shamir and Upfal [32] and Fernandez de la Vega [13] showed that for any 0, there exists k k( ....

B. Bollobas, P. Erdos, Cliques in random graphs, Mathematical Proceedings of the Cambridge Philosophical Society, 80 (1976), 419-427.

....for classes G(n; 0:5) and G(n; 0:7) only. Additional experiments with RB were conducted in order to test the rate of learning of the method on class G(n; 0:5) The results are presented in plots. All experiments were performed on a Sun Ultra 2, Model 2200. 5.1 Selecting the target. It is known ([4], 5] 8] that the expected size of the maximum independent set in G(n; 0:5) is 2 log 2 n. Thus, the size of the maximal independent set increases by one when the number of vertices increases roughly by a factor of p 2. For every target value t and for every q (0 q 1) we de ne n(t; q) to ....

B. Bollobas and P. Erdos, Cliques in Random Graphs, Mathematical Proceedings of the Cambridge Philosophical Society, Vol 80, (1976) pp. 419-427.

....in this latter problem see [2, 14, 17, 22] Luczak [24] proved that asymptotically c k 2k ln k by showing the existence of suitably large, disjoint independent sets in G n;p=ck=n . Algorithmically, substantially less progress has been made. One of the first heuristics to be analyzed [18] [4], GIC, forms color classes by repeatedly removing greedily chosen independent sets until the remaining graph has no component with more than one cycle, at which point it is easy to 3 color it . Shamir and Upfal [32] and Fernandez de la Vega [13] showed that for any ffl 0, there exists k k(ffl) ....

B. Bollob'as, P. Erdos, Cliques in random graphs, Mathematical Proceedings of the Cambridge Philosophical Society, 80 (1976), 419--427.

....edge pair. This family of graphs has been studied extensively. Given parameters n and p, let X k be a stochastic variable denoting the number of stable sets of size k as follows: X k = n k (1 0 p) k(k01) 2 : 7) If Z n;p denotes the maximum size of a clique in a random graph, then (see [6, 28]) for the threshold function z(n; p) 2 log 1= 10p) n 0 2 log 1= 10p) log 1= 10p) n 2 log 1= 10p) e=2) 1 and any ffl 0, the following holds true: lim n 1 P robfbz(n; p)c 0 1 0 ffl Z n;p bz(n; p)c fflg = 1: 5.1.2 DIMACS Benchmark Problems The second class of graph is the DIMACS ....

B. Bollob'as and P. Erdos, Cliques in Random Graphs, Mathematical Proceedings of the Cambridge Philosophical Society, Vol 80, (1976) pp. 419--427.

.... test graph G 2 64=k=0:5 (not shown here) disregarding the embedded k clique and looking only at the 64 node 0:5 random graph, one may estimate (very roughly) the maximum clique size to be around 8, using the threshold function 2 log 1 p n Gamma 2 log 1 p log 1 p n 2 log 1 p e=2 1 [3]. The actual size of the embedded clique corresponding to normalized clique size 0:2 is 13. The height at 0:2 for the multi phase simple case is 0:6 which corresponds to a clique size of 8. Hence, quite plausibly, the algorithm is getting stuck in local minima created by the underlying p random ....

B. Bollob'as and P. Erdos. Cliques in Random Graphs. Mathematical Proceedings of the Cambridge Philosophical Society, 80:419-427, 1976.

....= 100 0.04s 0.07s 0.27s 1.67s 21s 838s 200 0.64s 2.83s 23s 452s 24250s 300 4.61s 26s 551s 19090s 500 42s 665 31723s 700 290s 6435s 1000 5400s Table 3 3.3. Random, brock and gen graphs. The asymptotics of the expected sizes of maximum cliques in random graphs have been studied by many researchers ([3], 6] 9] 12] Let Z n;p denote the size of the largest clique in a random graph with n vertices and edge probability p. Matula proved in [12] that for every given integer n and p 0, k X j=0 Gamma n k Delta Gamma n k Delta Gamma n k Delta p Gamma j(j Gamma1) 2 P rob(Z n;p ....

B. Bollob'as and P. Erdos, Cliques in Random Graphs, Mathematical Proceedings of the Cambridge Philosophical Society, Vol 80, (1976) pp. 419--427.

....presented here are concerned with the class G(n; 0:5) of random graphs with n vertices and edge probability 0:5. The objectives of our experiments were to test the efficiency of database learning and the efficiency of the algorithms resulting from learning. Selecting the target. It is known ([2], 3] 6] 8] that the expected size of the maximum independent set in G(n; 0:5) is 2 log 2 n. Thus, the size of the maximal independent set increases by one when the size of n increases, roughly, by a factor of p 2. Therefore, testing should be done on graphs whose sizes grow ....

B. Bollob'as and P. Erdos, Cliques in Random Graphs, Mathematical Proceedings of the Cambridge Philosophical Society, Vol 80, (1976) pp. 419--427.

....for classes G(n; 0:5) and G(n; 0:7) only. Additional experiments with RB were conducted in order to test the rate of learning of the method on class G(n; 0:5) The results are presented in plots. All experiments were performed on a Sun Ultra 2, Model 2200. 5.1 Selecting the target. It is known ([4], 5] 8] that the expected size of the maximum independent set in G(n; 0:5) is 2 log 2 n. Thus, the size of the maximal independent set increases by one when the number of vertices increases roughly by a factor of p 2. For every target value t and for every q (0 q 1) we define n(t; q) ....

B. Bollob'as and P. Erdos, Cliques in Random Graphs, Mathematical Proceedings of the Cambridge Philosophical Society, Vol 80, (1976) pp. 419--427.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC