S. Janson. Poisson approximation for large deviations. Random Structures and Algorithms, 1(2):221-230, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....i =i ; i = 0; 1; P r[P (1 ) e =2 (3) see, e.g. Theorem A.15 of [2] When X is the sum of many mostly independent indicator random variables, there is some hope that X is also highly concentrated around its mean as a binomial random variable. Recent results ( 9] 7] [8], 12] give su cient conditions for this phenomenon to hold. Below we describe a general framework for applying these results. Suppose Q is a nite set ( in our instances Q is the edge set of a complete graph on n vertices) Let fJ i : i 2 Qg be a set of independent random indicator variables, ....

....P r[X 1 = 1] P r[X k = 1] P r[X 1 = 1] P r[X k = 1] where is over sets of k mutually independent events X i = 1, while is over ordered k tuples of mutually independent events and is over all ordered k tuples of events. It is worth mentioning that Janson ([8]) obtained a similar result which yields exponential estimates of P r[X 0 (1 ) for every 0. However, his proof is a bit more complicated than the one cited above) In particular, we deduce from the above Claim that P r[X 0 5 ] 5 : 5) The inequalities (1) and (5) are ....

S. Janson, Poisson approximation for large deviations, Random Structures and Algorithms, 1 (1990), 221-230.

..... d 1, let M k be defined as J#I n,d k I#I n,d :J#I # j#I J t I . Note that all M k are again suprema of non negative boolean polynomials, but the degree of M k is k d. Lower tails for booleans polynomials are by now well understood thanks to the Janson Suen inequalities [17, 38]. On the other hand, upper tails for such simple polynomials are notoriously more di#cult, see Janson and Rucinski [20] for a survey. We obtain the following general result. Theorem 15. Let Z and M k be defined as above. For all reals q j=1 # d [Z] M d j ] #q) j d ....

S. Janson. Poisson approximation for large deviations. Random Structures and Algoruthms, 1:221--230, 1990.

....for E[jA n j] which will be a closed expression of r and n. Letting this closed expression converge to zero with n, we will get an equation in terms of r that gives the required bound for . To compute an upper bound for the second factor of the sum, we will make use of the Janson s inequality [8], which gives an estimate for the probability of the intersection of dependent events. We give the details in the first subsection of the present section. The computations that will then give a closed expression that is an upper bound for E[jA n j] are carried out in the second subsection. 3.1 ....

....That concludes the proof. 2 Unfortunately, we cannot just multiply the probabilities in the previous lemma to compute Pr[A 2 A n ] because these probabilities are not independent. This is so because two double flips may have variables in common. Fortunately, we can apply Janson s inequality [8] that gives an estimate for the probability of the intersection of dependent events. For a detailed presentation of this theorem we refer to the 2nd edition of Spencer s book [15] In our case we will apply a variation tailored to our needs. Below we follow as closely as possible the notation of ....

S. Janson, "Poisson approximation for large deviations," Random Structures and Algorithms 1, pp 221--230, 1990.

....representing F . Proposition 3.6 ( SBI02] Let # be an h CNF, P be a Res(k) refutation of # , and let # be a partial assignment so that for every line F of P , h(F h. Then # # has a resolution refutation of width kh. Finally, we recall a combinatorial inequality originally proved in [Jan90] and further generalized in [AS00, Section 8.1] Definition 3.7 For propositional variables x 1 , x n and probabilities p 1 , p n [0, 1] denote by a p a random assignment that independently assigns every variable x i to 1 with probability p i , and to 0 with probability ....

S. Janson. Poisson approximation for large deviations. Random Structures and Algorithms, 1:221--230, 1990.

....and for each pair (u; v) of vertices an edge is inserted between u and v with probability p, independently. We write m = and denote the indicator variables of the m edges by X 1 ; Xm . Remark. Janson s inequality. For problems of the type studied in this section, Janson s inequality [10] provides a sharp tool to obtain upper bounds for the lower tail. More precisely, let I denote a family of subsets of the edges and de ne the number of occurrences of elements of I in the random graph by Z = One typical example is when I contains all triples of edges which form a triangle. ....

S. Janson. Poisson approximation for large deviations. Random Structures and Algorithms, 1:221-230, 2000.

....in their respective independent random experiments, the probability of which is Xel Xe2 Xe X. 1 Xel Xep Garg, Konjevod and Ravi [12] further show that for each group g, the probability that a vertex of g is included in T is f(1 log Igl) For this, they use an inequality of Janson [13]. By a similar argument, and using a generalization of the same inequality, we show that for each group, the probability that at least alof its requirement is satisfied by T is Ft(1 log THEOREM 3.1. Let T be the tree arising from the random experiment described above. Then there exists a constant ....

....be subsets of f, and denote by Bi the event that Ai C R. Write i j ifBi and Bj are not independent. Define A = i Pr[Bi N Bj] the sum is over ordered pairs) Let X = i Xi, where Xi is an indicator variable for the event Bi, and let E[X] Ei Pr[Bi] THEOREM 3.2. Janson s inequality [13]. With the notation as above, Pr[X (1 ) e 2u (2 ) In our application, Ft = E(T) and p, x, Xp( where p(e) is the parent or predecessor of e on the path from r. The subsets Ai are edge sets of paths from r to leaves belonging to a fixed group g, and X is the number of vertices ....

S. Janson. Poisson approximations for large devia- tions. Randoms Structures and Applications, 1:221230, 1990.

....range over all triangles these conditions hold and G(n; p) is trianglefree with probability exp( Gammac =6) as known to Erdos and R enyi. Sweeping generalizations of this are given in [13] where the first proof of Janson s Inequality may be found. Other proofs and generalizations are given in [14][2] Applying Janson to Pr[ G(n; 5) k] we let A S = S] S ranging over the k sets of vertices. Then ffl 0, f(k) Delta is the expected number of edge overlapping k cliques, calculation gives domination by cliques overlapping in a single edge and Delta (2k ) The Poisson ....

S. Janson. Poisson approximation for large deviations. Random Stuctures & Algorithms 1:221-230, 1990

....for f r;s (n) To establish f r;s (n) m we have to show that there exists a graph G of order n not containing a copy of K s such that every set of m vertices of G contains a K r . The existence of such G is shown by using the Lovasz Local Lemma ( 6] 2] Ch. 5) and Janson s inequality ([8]; 2] Ch. 8) Consider a random graph G(n; p) a graph on n vertices in which all edges are chosen independently and with probability p, where the value of p = p(n) will be chosen later. For a set S of s vertices let AS be an event G[S] K s . Obviously, P r(AS ) p ( s 2 ) For a ....

S. Janson, Poisson approximation for large deviations, Random Structures and Algorithms 1 (1990), 221-230.

....a better approximation guarantee when the maximum requirement of a group is large compared to the number of groups. 3 This is achieved via a more careful rounding procedure at every phase. Randomization is a key ingredient in the rounding process of all our algorithms, and a tail bound of Janson [12] helps much in our analyses. In establishing the performance guarantee of our rst algorithm we make no attempt at optimizing constants. We are more careful in treating the constants in our second and third algorithm, but we do not attempt a complete optimization. The rest of this paper is ....

.... their respective independent random experiments, the probability of which is x e1 1 x e2 x e1 x e x ep = x e : Garg, Konjevod and Ravi [11] show that for each group g, the probability that a vertex of g is included in T is = log jgj) For this, they use an inequality of Janson [12]. By a similar argument, and using a generalization of the same inequality, it follows that for each group, the probability that at least half of its requirement is satis ed by T is = log jgj) Theorem 3.1. Let T be the tree arising from the random experiment described above. Then there exists ....

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S. Janson. Poisson approximations for large deviations. Random Structures & Algorithms, 1:221-230, 1990.

....the process until exhaustion. To make this approach work one needs a competent bound on the probability that a random graph contains at least one independent set of size r. Such bounds were first obtained by B. Bollobas [B] using martingales. A recent powerful correlation inequality of Janson [J] is both simpler and more powerful than the martingale approach (compare [A S] p. 87 and p. 148) It is of some interest to note that Theorem 4.1.1 does as well as Janson s inequality. We fix an integer r. For e = i, j) # E 0 , we denote by N(G, e) the number of independent sets of size r that ....

S. Janson, Poisson approximation for large deviations, Random Structures and Algorithms 1, 221-290.

....each pair (u, v) of vertices an edge is inserted between u and v with probability p, independently. We write m = # n 2 # , and denote the indicator variables of the m edges by X 1 , Xm . Remark. Janson s inequality. For problems of the type studied in this section, Janson s inequality [8] provides a sharp tool to obtain upper bounds for the lower tail. More precisely, let I denote a family of subsets of the edges and define the number of occurrences of elements of I in the random graph by Z = # ##I # i## X i . One typical example is when I contains all triples of edges ....

S. Janson. Poisson approximation for large deviations. Random Structures and Algoruthms, 1:221--230, 2000.

....G and H is obtained from H 0 by adding to it less than a(H 0 ) edges between C 1 and C 2 then H 2 G. If H 2 G then n Gamma1=a(H) is a sharp threshold function for the existence of an H factor. The proofs are presented in the next section. They rely on the Janson inequalities (cf. 6] and [8]) and on a method used by Alon and Furedi in [2] The last section contains some concluding remarks and open problems. 2 The proofs We begin by establishing the first statement in Theorem 1.1 which is not difficult and, in fact, holds even when H is not a member of F . Lemma 2.1 Let H be any ....

S. Janson, Poisson approximation for large deviations, Random Structures and Algorithms 1 (1990), 221-230.

....problem. Then an extension of randomized rounding is employed to get the solution subtree. The bound on the cost of the tree follows from the rounding process. On the other hand, to show that the solution tree actually covers all the groups with reasonable probability, we use Janson s inequality [19]. As a corollary to the performance guarantee, we also get an upper bound on the integrality gap of our linear programming relaxation. Our algorithm works with similar performance bounds when applied to the errand scheduling problem of [29] also known as the generalized TSP [10, 13, 28] to the ....

....e is included in T if and only if all the edges in the path from r to e, say e 0 ; e 1 ; e p = e are picked in their respective independent random trials. This event happens with probability x e0 Q p i=1 (x e i =x e i 1 ) x e : To analyze this experiment, we use Janson s inequality ([19], see also [1] p. 95) which can be stated as follows: let be a universal set, and R determined by the experiment in which each element r 2 is independently included in R with probability p r . In what follows I will denote a nite index set. Let A = fA i j i 2 Ig be a family of subsets of ....

S. Janson. Poisson approximations for large deviations. Randoms Structures and Applications, 1:221-230, 1990.

....of vertices of Q n is connected by a path of weight at most 20. We prove Theorem 1 by considering a related question about random subgraphs of the cube, where the edges of the subgraph will be all edges with weight below a certain threshold. The main tool we shall use is Janson s inequality from [9]. Lemma 2. Let R be a random subset of a xed groundset [N ] f1; 2; Ng obtained by selecting each element i independently with probability p i . Let S 1 ; S s be subsets of [N ] and let X(R) count the number of S i for which S i R. Let = E X(R) and = X (i;j) S i S j ....

S. Janson, Poisson approximation for large deviations, Random Structures and Algorithms 1 (1990), 221-229. 7

....E[jA 2] n j] which will be a closed expression of r and n. Letting this closed expression converge to zero with n, we will get an equation in terms of r that gives the required bound for . To compute an upper bound for the second factor of the sum, we will make use of the Janson s inequality [7], which gives an estimate for the probability of the intersection of dependent events. We give the details in the first subsection of the present section. The computations that will then give a closed expression that is an upper bound for E[jA 2] n j] are carried out in the second subsection. ....

....proof. 2 Unfortunately, we cannot just multiply the probabilities in the previous lemma to compute Pr[A 2 A 2] n j A 2 A 1 n ] because these probabilities are not independent. This is so because two 7 double flips may have variables in common. Fortunately, we can apply Janson s inequality [7] that gives an estimate for the probability of the intersection of dependent events. For a detailed presentation of this theorem we refer to the 2nd edition of Spencer s book [14] In our case we will apply a variation tailored to our needs. Below we follow as closely as possible the notation of ....

S. Janson, "Poisson approximation for large deviations," Random Structures and Algorithms 1, pp 221--230, 1990.

....Y , when the expectation of Y is small (at most polylog(n) The case when the expectation of Y is large was studied in two other papers [KV, Vu1] and will be briefly discussed here. Readers familiar with probabilistic combinatorics would immediately realize that the famous result of Janson [Jan] provides a strong bound for the lower tail for functions of the above type. We write I i # I j if the two monomials share a common atom variable. Janson s inequality. With Y as above and # = # I i #I j E(I i I j ) the following holds P r(Y # (1 #)E(Y ) # e (#E(Y ) 2 2(E(Y ....

Janson, S. Poisson approximation for large deviations, Random Structures and Algorithms 1, 221-230 (1990). CONCENTRATION 21

.... the probability of which is x e1 1 Delta x e2 x e1 Delta Delta Delta x e x ep = x e : Garg, Konjevod and Ravi [12] further show that for each group g, the probability that a vertex of g is included in T is Omega Gamma = log jgj) For this, they use an inequality of Janson [13]. By a similar argument, and using a generalization of the same inequality, we show that for each group, the probability that at least half of its requirement is satisfied by T is Omega Gamma = log jgj) Theorem 3.1. Let T be the tree arising from the random experiment described above. Then ....

....by B i the event that A i R. Write i j if B i and B j are not independent. Define Delta = P ij Pr[B i B j ] the sum is over ordered pairs) Let X = P i X i , where X i is an indicator variable for the event B i , and let = E[X ] P i Pr[B i ] Theorem 3.2. Janson s inequality [13]. With the notation as above, Pr Theta X (1 Gamma fl) e Gammafl 2 = 2 Delta ) In our application, Omega = E(T 0 ) and p e = x e =x p(e) where p(e) is the parent or predecessor of e on the path from r. The subsets A i are edge sets of paths from r to leaves belonging to a ....

S. Janson. Poisson approximations for large deviations. Randoms Structures and Applications, 1:221-- 230, 1990.

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S. Janson, Poisson approximation for large deviations. Random Struct. Alg. 1 (1990), 221-230.

....graphs. The asymptotic normality of XG for a wide range of p (as long as pn 1 mG and n p) # #) was proven in [18] Next, it was shown that the lower tail of the distribution of XG decays exponentially in the expectation of the least expected subgraph of G, see [10] P(XG = 0) and [9] (the general case) Namely, let #H : n which is roughly the expected number of copies of H in G(n, p) Then, for all # (0, 1] with c # 0 depending on G and #, XG #)EXG # min . 1.1) Date: October 31, 2002. Research of the second author supported by KBN grant 2 ....

S. Janson, Poisson approximation for large deviations. Random Struct. Alg. 1 (1990), 221--230.

....a (binomial) random subset #p of #, where p = p i : i # #) p i = P(# i = 1) we write # p if p i = p for all i] and X is the number of elements of S that are contained in this random subset. For the lower tail of the distribution of X , the following analogue of the Cherno# bound holds [7], 8, Theorem 2.14] Theorem 0. Let X = # A#S I A as above, and let # = EX = # A E I A and # = ## A#B #=# E(IA I B ) Then, with #(x) 1 x) log(1 x) x, for 0 # t # #, P(X # # t) # exp # #( t #)# 2 # # # exp # t 2 2 # # . It follows from the FKG ....

S. Janson, Poisson approximation for large deviations. Random Struct. Alg. 1 (1990), 221--230.

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S. Janson. Poisson approximation for large deviations. Random Structures and Algorithms, 1(2):221-230, 1990.

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S. Janson. Poisson approximation for large deviations. Random Structures and Algorithms, 1(2):221--230, 1990.

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S. Janson. Poisson approximation for large deviations. Random Structures and Algorithms, 1:221-230, 1990.

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S. Janson. Poisson approximation for large deviations. Random Structures and Algoruthms, 1:221--230, 1990.

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S. Janson, Poisson approximation for large deviations, Random Structures and Algorithms 1 (1990), 221-229.

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