## Sharp Thresholds of Graph properties, and the k-sat Problem (1998)

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Venue: | J. Amer. Math. Soc |

Citations: | 161 - 5 self |

### BibTeX

@ARTICLE{Friedgut98sharpthresholds,

author = {Ehud Friedgut},

title = {Sharp Thresholds of Graph properties, and the k-sat Problem},

journal = {J. Amer. Math. Soc},

year = {1998},

volume = {12},

pages = {1017--1054}

}

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### Abstract

Given a monotone graph property P , consider p (P ), the probability that a random graph with edge probability p will have P . The function d p (P )=dp is the key to understanding the threshold behavior of the property P . We show that if d p (P )=dp is small (corresponding to a non-sharp threshold), then there is a list of graphs of bounded size such that P can be approximated by the property of having one of the graphs as a subgraph. One striking consequences of this result is that a coarse threshold for a random graph property can only happen when the value of the critical edge probability is a rational power of n.

### Citations

1842 | On the evolution of random graphs
- Erdős, Rényi
- 1960
(Show Context)
Citation Context ...h. A monotone symmetric family of graphs is a family defined by such a property. One of the first observations made about random graphs by Erdös and Rényi in their seminal work on random graph theory =-=[12]-=- was the existence of threshold phenomena, the fact that for many interesting properties P , the probability of P appearing in G(n, p) exhibits a sharp increase at a certain critical value of the para... |

1789 | Random Graphs
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- 2001
(Show Context)
Citation Context ...(1) is bounded by a constant times C R =R!, asymptotically less than n \Gammak =2 for any fixed B. This follows from the usual way of computing the moments of the random variable XH , see for example =-=[5]-=-. The event (2) can be described by the existence of a subgraph from a list of subgraphs whose length is a function of B, l(B). Any graph in this list can be described by a union of at least B but no ... |

225 | The influence of variables on boolean functions
- Kahn, Kalai, et al.
- 1988
(Show Context)
Citation Context ...H must be balanced, and of bounded expectation. Proof: First we define a set of functions ff e g such as those defined by Talagrand in [28], which are a generalization of similar functions defined in =-=[18]-=-. The idea behind these functions is that they measure I e , the influence of the edge e on f : For every edge e let f e be the function defined by: f e (H) = ae q(f(H) \Gamma f(H \Phi e)) if f(H) = 1... |

138 | Analysis of two simple heuristics for random instances of k-SAT
- Frieze, Suen
- 1996
(Show Context)
Citation Context ...ee [10], [17], but for ks3 was not known. For k = 3 a series of upper and lower bounds have seemed to be slowly converging to the value suggested by simulations (c = 4:2:::) see [19], [11], [9], [8], =-=[16]-=-, [22], [21]. We now show that the existence of a threshold for any given k can be demonstrated by the proof of theorem 1.1. I would like to thank Svante Janson for pointing out the following subtlety... |

132 | Every monotone graph property has a sharp threshold
- Friedgut, Kalai
- 1996
(Show Context)
Citation Context ...g in G(n; p) exhibits a sharp increase at a certain critical value of the parameter p. Bollob'as and Thomason proved the existence of threshold functions for all monotone set properties ([6]), and in =-=[14]-=- it is shown that this behavior is quite general, and that all monotone graph properties exhibit threshold behavior, i.e. the probability of their appearance increases from values very close to 0 to v... |

89 |
On the satisfiability and maximum satisfiability of random 3-CNF formulas
- Broder, Frieze, et al.
- 1993
(Show Context)
Citation Context ... 1, see [10], [17], but for ks3 was not known. For k = 3 a series of upper and lower bounds have seemed to be slowly converging to the value suggested by simulations (c = 4:2:::) see [19], [11], [9], =-=[8]-=-, [16], [22], [21]. We now show that the existence of a threshold for any given k can be demonstrated by the proof of theorem 1.1. I would like to thank Svante Janson for pointing out the following su... |

82 | Approximating the unsatisfiability threshold of random formulas
- Kirousis, Kranakis, et al.
- 1996
(Show Context)
Citation Context ...], but for ks3 was not known. For k = 3 a series of upper and lower bounds have seemed to be slowly converging to the value suggested by simulations (c = 4:2:::) see [19], [11], [9], [8], [16], [22], =-=[21]-=-. We now show that the existence of a threshold for any given k can be demonstrated by the proof of theorem 1.1. I would like to thank Svante Janson for pointing out the following subtlety to me: What... |

73 |
Inequalities with applications to percolation and reliability
- Berg, Kesten
- 1985
(Show Context)
Citation Context ... j (E(x i x j ) \Gamma E(x i )E(x j )) : For i 6= j let x i ffi x j be the random variable indicating the event that there exist edge disjoint extensions of the corresponding sets. The BK inequality (=-=[4]-=-) implies E(x i ffi x j )sE(x i )E(x j ): (See also [24] for a more general inequality.) Let x i \Pi x j = x i x j \Gamma x i ffi x j . We have var(X)sX Var(x i ) + X i6=j E(x i \Pi x j ) = X i6=j E(x... |

66 | Probabilistic analysis of two heuristics for the 3-satisfiability problem
- Chao, Franco
- 1986
(Show Context)
Citation Context ...t c = 1, see [10], [17], but for ks3 was not known. For k = 3 a series of upper and lower bounds have seemed to be slowly converging to the value suggested by simulations (c = 4:2:::) see [19], [11], =-=[9]-=-, [8], [16], [22], [21]. We now show that the existence of a threshold for any given k can be demonstrated by the proof of theorem 1.1. I would like to thank Svante Janson for pointing out the followi... |

63 |
Boolean functions with low average sensitivity depend on few coordinates
- Friedgut
- 1998
(Show Context)
Citation Context ...ugh p, i.e. p bounded from above by a negative power of n. The question of understanding coarse thresholds for non-symmetric properties at values of p that are bounded from 0 is also interesting, see =-=[13]-=-. Example: Connectivity has a sharp threshold since the critical p is approximatelyslog(n)=n where as ffis1=n. On the other hand the property of having a triangle in the graph has a coarse threshold s... |

47 |
Probabilistic characteristics of graphs with large connectivity
- Margulis
- 1974
(Show Context)
Citation Context ...f and with dµ(A)/dp. Lemma 2.1 (Russo, Margulis). dµ(A)/dp =1/p � A h(a)dµp where h(a) is |{a ′ |a ′ �∈ A, dist(a, a ′ )=1}|. (Heredist(a, a ′ ) is the Hamming distance.) For proofs of this lemma see =-=[23]-=-, [25]. An equivalent statement in different notation is: Lemma 2.2. dµ(A)/dp = I. (Notice that I is a function of p.) Remark. These two lemmas imply an easy converse of Theorem 1.1: if �A� is small, ... |

46 | Evidence for a Satisfiability Threshold for Random 3CNF Formulas
- Larrabee, Tsuji
- 1992
(Show Context)
Citation Context ...[17]), but for k ≥ 3 was not known. For k = 3 a series of upper and lower bounds have seemed to be slowly converging to the value suggested by simulations (c =4.2...); see [19], [11], [9], [8], [16], =-=[22]-=-, [21]. We now show that the existence of a threshold for any given k can be demonstrated by the proof of Theorem 1.1. I would like to thank Svante Janson for pointing out the following subtlety to me... |

33 |
Threshold functions, Combinatorica 7
- Bollobás, Thomason
- 1987
(Show Context)
Citation Context ...of P appearing in G(n; p) exhibits a sharp increase at a certain critical value of the parameter p. Bollob'as and Thomason proved the existence of threshold functions for all monotone set properties (=-=[6]-=-), and in [14] it is shown that this behavior is quite general, and that all monotone graph properties exhibit threshold behavior, i.e. the probability of their appearance increases from values very c... |

33 |
On the critical percolation probabilities
- Russo
- 1981
(Show Context)
Citation Context ...with dµ(A)/dp. Lemma 2.1 (Russo, Margulis). dµ(A)/dp =1/p � A h(a)dµp where h(a) is |{a ′ |a ′ �∈ A, dist(a, a ′ )=1}|. (Heredist(a, a ′ ) is the Hamming distance.) For proofs of this lemma see [23], =-=[25]-=-. An equivalent statement in different notation is: Lemma 2.2. dµ(A)/dp = I. (Notice that I is a function of p.) Remark. These two lemmas imply an easy converse of Theorem 1.1: if �A� is small, then s... |

21 | An approximate zero-one law - Russo - 1982 |

10 | Threshold functions for H-factors
- Alon, Yuster
- 1993
(Show Context)
Citation Context ...can not make such a difference. 2 Remark: A similar proof works for the case of "H-factors", the property of having a covering of the vertices of G(n; p) by disjoint copies of some fixed gra=-=ph H. See [3]-=- for this problem. However in this case, as pointed out to me by Noga Alon, it is not enough to use the fact that o( p E) edges (where E is the expected number of edges) do not make a difference. Here... |

9 | Perfect Matchings in Random s-uniform Hypergraphs
- Frieze, Janson
- 1995
(Show Context)
Citation Context ...s that of the existence of a disjoint covering of the vertices by k edges. What is currently known about the value of the critical p for this property is log(n)=n 2sp csn \Gamma4=3 : 39 (See [27] and =-=[15]-=-). The question of showing that p csn \Gamma(2\Gammao(1)) is considered to be one of the challenging problems in random (hyper)graph theory. However we may now deduce the sharpness of the threshold: B... |

9 |
A threshold for perfect matchings in random d-pure hypergraphs
- Schmidt, Shamir
- 1983
(Show Context)
Citation Context ... of interest is that of the existence of a disjoint covering of the vertices by k edges. What is currently known about the value of the critical p for this property is log(n)/n 2 ≤ pc ≤ n −4/3 . (See =-=[27]-=- and [15].) The question of showing that pc ≤ n −(2−o(1)) is considered to be one of the challenging problems in random (hyper)graph theory. However we may now deduce the sharpness of the threshold: B... |

7 |
On Russo's approximate 0-1 law, Annals of Probability 22
- Talagrand
- 1994
(Show Context)
Citation Context ...now define an orthonormal basis, with respect to µ, for the space of real functions on G(n, p). The use of these functions and their nice properties in a similar setting is introduced by Talagrand in =-=[28]-=-. These functions will be indexed by all subgraphs of the complete graph on n vertices. Let E denote the set of all edges of the complete graph. Define U∅ to be identically equal to 1. For any edge e ... |

6 |
A threshold for satisfiability
- Goerdt
- 1996
(Show Context)
Citation Context ...i.e. that the following is true: There exists a constant c such that for any ffl ? 0, f((c \Gamma ffl)n) ! 1; f((c + ffl)n) ! 0. This was shown to be true for k = 2 with the constant c = 1, see [10], =-=[17]-=-, but for ks3 was not known. For k = 3 a series of upper and lower bounds have seemed to be slowly converging to the value suggested by simulations (c = 4:2:::) see [19], [11], [9], [8], [16], [22], [... |

4 |
Fernandez de la Vega, On Random
- Maftouhi, W
- 1995
(Show Context)
Citation Context ...onstant c = 1, see [10], [17], but for ks3 was not known. For k = 3 a series of upper and lower bounds have seemed to be slowly converging to the value suggested by simulations (c = 4:2:::) see [19], =-=[11]-=-, [9], [8], [16], [22], [21]. We now show that the existence of a threshold for any given k can be demonstrated by the proof of theorem 1.1. I would like to thank Svante Janson for pointing out the fo... |

4 |
On the Satis ability and Maximum Satis ability of Random 3-CNF Formulas
- Broder, Frieze, et al.
- 1993
(Show Context)
Citation Context ...nt c =1,see [10], [17], but for k 3 was not known. For k =3aseries of upper and lower bounds have seemed to be slowly converging to the value suggested by simulations (c =4:2:::) see [19], [11], [9], =-=[8]-=-, [16], [22], [21]. We nowshow that the existence of a threshold for any given k can be demonstrated by the proof of theorem 1.1. Iwould like to thank Svante Janson for pointing out the following subt... |

4 |
Approximating the unsatis - ability threshold of random formulas
- Kirouris, Kranakis, et al.
(Show Context)
Citation Context ... [17], but for k 3 was not known. For k =3aseries of upper and lower bounds have seemed to be slowly converging to the value suggested by simulations (c =4:2:::) see [19], [11], [9], [8], [16], [22], =-=[21]-=-. We nowshow that the existence of a threshold for any given k can be demonstrated by the proof of theorem 1.1. Iwould like to thank Svante Janson for pointing out the following subtlety to me: What I... |

3 |
A Threshold for k-Colorability, To appear, Random Structures and Algorithms
- Achlioptas, Friedgut
- 1999
(Show Context)
Citation Context ...hreshold would be sharp. This type of proof seems to be easy for some "non-local" properties such as connectivity or having a perfect matching. ffl k-colorability for k ? 2. In a paper in pr=-=eparation [1]-=- it is shown by similar techniques that the property of being non-k-colorable for a fixed k larger than 2 has a sharp threshold. The crux of the proof there is to show that if G(n; p) is non-k-colorab... |

3 |
The in uence of variables on Boolean functions
- Kahn, Kalai, et al.
- 1988
(Show Context)
Citation Context ...ions H must be balanced, and of bounded expectation. Proof: First we de ne a set of functions ffeg such as those de ned by Talagrand in [28], which are a generalization of similar functions de ned in =-=[18]-=-. The idea behind these functions is that they measure Ie, the in uence of the edge e on f: For every edge e let fe be the function de ned by: fe(H) = q(f(H) ; f(H e)) if f(H) =1 p(f(H) ; f(H e)) if f... |

2 |
Influence of variables in product spaces under group symmetries
- Bourgain, Kalai
- 1997
(Show Context)
Citation Context ...r, i.e. the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval. More precise analysis of the size of the threshold interval is done in =-=[7]-=-. This threshold behavior which occurs in various settings which arise in combinatorics and computer science, is an instance of the phenomenon of phase transitions which is the subject of much interes... |

2 |
K.Palem, P.Spirakis Tail Bounds for Occupancy and the Satis ability Threshold Conjecture
- Kamath, Motwani
- 1994
(Show Context)
Citation Context ... the constant c = 1, see [10], [17], but for ks3 was not known. For k = 3 a series of upper and lower bounds have seemed to be slowly converging to the value suggested by simulations (c = 4:2:::) see =-=[19]-=-, [11], [9], [8], [16], [22], [21]. We now show that the existence of a threshold for any given k can be demonstrated by the proof of theorem 1.1. I would like to thank Svante Janson for pointing out ... |

1 |
Minimal Non-2-colorable Hypergraphs and Minimal Unsatisfiable Formulas
- Aharoni, Linial
- 1986
(Show Context)
Citation Context ...s described in theorem 5.2. Assume F is such a formula. Obviously if F has a sub formula R that itself is a tautology then P r(T jF ) = 1s1 \Gamma ffl, however, an unpublished result of M. Tarsi (see =-=[2]-=-) states that if such a formula R, uses r variables it must have at least r +1 clauses. The expected number of formulas of such an isomorphism type in a random formula is therefore at most n r p r+1 .... |

1 |
gets some
- Chvatal, Reed, et al.
(Show Context)
Citation Context ...ability of such event by f(M ). It is obvious that f is a monotone decreasing function of M . It is known that for any given k there are constants c 1 ; c 2 such that f(c 1 n) ! 1; f(c 2 n) ! 0: (see =-=[10]-=-.) Computer simulations suggest that f exhibits a threshold behavior, i.e. that the following is true: There exists a constant c such that for any ffl ? 0, f((c \Gamma ffl)n) ! 1; f((c + ffl)n) ! 0. T... |

1 |
On the evolution of random graphs, Mat Kutat'o Int
- Erdos, R'enyi
(Show Context)
Citation Context .... A monotone symmetric family of graphs is a family defined by such a property. One of the first observations made about random graphs by Erdos and R'enyi in their seminal work on random graph theory =-=[12]-=- was the existence of threshold phenomena, the fact that for many interesting properties P , the probability of P appearing in G(n; p) exhibits a sharp increase at a certain critical value of the para... |

1 |
A Threshold for k-Colorability, Random Structures and Algorithms 14
- Achlioptas, Friedgut
- 1999
(Show Context)
Citation Context ...the threshold would be sharp. This type of proof seems to be easy for some “non-local” properties such as connectivity or having a perfect matching. • k-colorability for k>2.In a paper in preparation =-=[1]-=- it is shown by similar techniques that the property of being non-k-colorable for a fixed k larger than 2 has a sharp threshold. The crux of the proof there is to show that if G(n, p) is non-k-colorab... |

1 |
Analysis of Two Simple Heuristics on aRandom Instance of k-sat
- Frieze, Suen
- 1992
(Show Context)
Citation Context ...=1,see [10], [17], but for k 3 was not known. For k =3aseries of upper and lower bounds have seemed to be slowly converging to the value suggested by simulations (c =4:2:::) see [19], [11], [9], [8], =-=[16]-=-, [22], [21]. We nowshow that the existence of a threshold for any given k can be demonstrated by the proof of theorem 1.1. Iwould like to thank Svante Janson for pointing out the following subtlety t... |

1 |
A threshold for satis ability
- Goerdt
- 1992
(Show Context)
Citation Context ...threshold behavior, i.e. that the following is true: There exists a constant c such that for any > 0, f((c ; )n) ! 1�f((c + )n) ! 0. This was shown to be true for k =2with the constant c =1,see [10], =-=[17]-=-, but for k 3 was not known. For k =3aseries of upper and lower bounds have seemed to be slowly converging to the value suggested by simulations (c =4:2:::) see [19], [11], [9], [8], [16], [22], [21].... |