Deterministic vs Non-deterministic Graph Property Testing

Transcription

1 Deteministic vs Non-deteministic Gaph Popety Testing Lio Gishboline Asaf Shapia Abstact A gaph popety P is said to be testable if one can check whethe a gaph is close o fa fom satisfying P using few andom local inspections. Popety P is said to be non-deteministically testable if one can supply a cetificate to the fact that a gaph satisfies P so that once the cetificate is given its coectness can be tested. The notion of non-deteministic testing of gaph popeties was ecently intoduced by Lovász and Vesztegombi [5], who poved that (somewhat supisingly) a gaph popety is testable if and only if it is non-deteministically testable. Thei poof used gaph limits, and so it did not supply any explicit bounds. They thus asked if one can obtain a poof of thei esult which will supply such bounds. We answe thei question positively by poving thei esult using Szemeédi s egulaity lemma. An inteesting aspect of ou poof is that it highlights the fact that the egulaity lemma can be intepeted as saying that all gaphs can be appoximated by finitely many template gaphs. 1 Intoduction We conside popeties of finite gaph, whee a popety of gaphs is simply a family of gaphs closed unde isomophism. The main focus of ou pape is the following notion of efficiently checking if a gaph satisfies popety P o is ɛ-fa fom satisfying it, whee a gaph G is said to be ɛ-fa fom satisfying P if one should add/delete at least ɛn 2 edges to tun G into a gaph satisfying P. Definition 1.1. (Testable popety) A gaph popety P is called testable if thee is an algoithm T P, called a teste, that does the following: given ɛ > 0 and a gaph G, the teste T P samples a set S of q P (ɛ) vetices fom G, checks fo evey i, j S whethe (i, j) E(G) and then accepts/ejects deteministically based on the subgaph of G spanned by S. The success pobability of T P should be at least 2 3. In othe wods, if the input G satisfies P then T P accepts it with pobability at least 2 3, and if G is ɛ-fa fom satisfying P then T P ejects G with pobability at least 2 3. The function q P(ɛ) is called the quey complexity of T P, and does not depend on the size of the input gaph. School of Mathematics, Tel-Aviv Univesity, Tel-Aviv, Isael School of Mathematics, Tel-Aviv Univesity, Tel-Aviv, Isael asafico@tau.ac.il. Suppoted in pat by ISF Gant 224/11 and a Maie-Cuie CIG Gant

2 The usual definition of a popety P being testable, as intoduced in [9], allows fo the algoithm to be adaptive, but as poved in [1, 10] one can tansfom any teste into a teste of the fom stated in Definition 1.1 with a vey mino loss in the quey complexity. Theefoe we do not lose geneality by esticting ouselves to Definition 1.1. The impotant point to obseve about Definition 1.1 is that the algoithm makes its decision solely on the basis of the andom inspections it makes into the input gaph G. In othe wods, the decision of the algoithm is uniquely detemined by the distibution of induced subgaphs of size q P (ɛ) in the input gaph G. Stating with [9], the poblem of chaacteizing the testable gaph popeties eceived a lot of attention and by now thee ae seveal geneal esults of this type, see [2, 6] and the ecent suveys [8, 11] fo moe esults and efeences on gaph popety testing. A dawback of these chaacteizations is that they ae had to state (and use). An altenative clean chaacteization was ecently obtained by Lovász and Vesztegombi [5]. To state this chaacteization we need a bit of notation. A k-coloed gaph on n vetices is a coloing of the edges of K n (the complete gaph on n vetices) using k colos. Thus, a gaph can be thought of as a 2-coloed gaph. A popety of k-coloed gaphs is again just a family of k-coloed gaphs closed unde isomophism, and it is said to be testable 1 if it is testable in the sense of Definition 1.1. A (k, m)-coloing of a gaph G is a coloing of the edges and non-edges of G with the colos {1,..., k}, so that edges ae coloed by {1,..., m} and non-edges ae coloed by {m + 1,..., k}. The following is the notion of non-deteministic testing intoduced in [5]. Definition 1.2. (Non-deteministically testable popety) A gaph popety P is called nondeteministically testable if thee ae integes k, m and a popety Q of k-coloed gaphs so that: 1. A gaph G satisfies P if and only if thee is a (k, m)-coloing of G which satisfies Q. 2. Q is testable. We ae now eady to state the chaacteization of the testable gaph popeties that was obtained by Lovász and Vesztegombi [5]. Theoem 1. ([5]) A popety P is testable if and only if it is non-deteministically testable. Clealy any testable popety is also non-deteministically testable, thus the inteesting pat of the above theoem is that given the fact that a popety is non-deteministically testable, one can constuct a standad teste fo the popety. Quoting [5], one could say that this theoem shows that P=NP fo popety testing in dense gaphs. We efe the eade to [5] fo seveal nice illustations showing how to apply Theoem 1. 1 We define a k-coloed gaph to be ɛ-fa fom satisfying a popety Q of k-coloed gaphs if one should modify the colos of at least ɛn 2 edges in ode to tun G into a k-coloed gaph satisfying P. 2

3 The poof of Theoem 1 in [5] used the machiney of gaph limits. Hence, the poof was not explicit, that is, given the fact that a popety P is non-deteministically testable (in the sense of Definition 1.2), it poved the existence of a standad testing algoithm fo P (in the sense of Definition 1.1) but it did not supply any uppe bound fo the quey complexity of the new teste (i.e. the function q P (ɛ) in Definition 1.1). Lovász and Vesztegombi [5] thus asked if Theoem 1 can be poved using explicit aguments that will give an effective bound. Ou main esult in this pape gives a positive answe to thei question. Ou new poof of Theoem 1 uses seveal tools elated to Szemeédi s egulaity lemma [12]. In Section 2 we give the necessay backgound fo applying this lemma, state some pevious esults as well as some peliminay lemmas that will be used in ou new poof of Theoem 1. As the poofs of these technical lemmas ae somewhat outine we diffe them to Section 4. The poof of Theoem 1 appeas in Section 3. As ou poof applies the egulaity lemma, although the bounds it supplies fo q P (ɛ) ae explicit, they ae athe weak ones, given by Towe-type function of ɛ. Theefoe, we will not keep tack of the exact dependence of q P (ɛ) on ɛ. Finally, as we mentioned in the abstact, we believe that ou poof gives a nice illustation of the fact that the egulaity lemma implies that all gaph can be appoximated using only a finitely many template gaphs. In fact, this intuition is the main idea behind the poof. 2 Tools and Peliminay Lemmas As mentioned ealie, ou poof of Theoem 1 will apply vaious tools elated to Szemeédi s egulaity lemma [12]. We will stat with the basic definitions, then state some pevious esults that we will use (Theoem 2 and Lemmas 2.4 and 2.8) and then state some technical lemmas that we will need fo the poof (Lemmas 2.13, 2.14 and 2.16). The poofs of these technical lemmas appea in Section 4. Hee, and thoughout the pape, when we wite x = y ±z we mean that y z x y +z. Given two disjoint vetex sets U, V we use E(U, V ) to denote the set of edges connecting U to V and set d(u, V ) = E(U, V ) / U V to be the density of the bipatite gaph between U and V. The basic notion of a egula pai is the following. Definition 2.1. (Regula pai) Suppose U, V ae disjoint vetex sets in a gaph and let γ (0, 1). The pai (U, V ) is said to be γ-egula if fo evey two subsets U U, V V satisfying U γ U, V γ V the inequality d(u, V ) d(u, V ) γ holds. A γ-egula pai can/should be thought of as behaving almost like a andom bipatite gaph of the same density. A patition V 1,..., V of the vetex set of a gaph is called an equipatition if V i V j 1 fo evey 1 i < j. The ode of a patition V 1,..., V is the numbe of pats in it (i.e. the intege ). Definition 2.2. (Regula equipatition) An equipatition V 1,..., V of the vetices of a gaph is γ-egula if all but at most γ ( 2) of the pais (Vi, V j ) ae γ-egula. 3

4 We now define a gaph popety of having a γ-egula equipatition with a pedefined set of densities. Definition 2.3. (Regulaity instance) A gaph egulaity instance R is given by a egulaity paamete γ, an intege (the ode of R), a set of densities η i,j whee 1 i < j, and a set of non-egula pais R of size at most γ ( 2). A gaph G is said to satisfy R if G has an equipatition V 1,..., V such that fo evey (i, j) / R the pai (V i, V j ) is γ-egula and satisfies d(v i, V j ) = η i,j. A key element in the poof of Theoem 1 is the following esult, which follows fom the main esults of [2, 3]. It allows us to test how close a gaph is to satisfying a given egulaity instance. Theoem 2. ([2, 3]) Let R be a gaph egulaity instance, let ɛ 2 > ɛ 1 > 0 and let p < 1 2. Then thee is a teste T = T 2 (R, ɛ 1, ɛ 2, p) that distinguishes gaphs that ae ɛ 1 -close to satisfying R fom gaphs that ae ɛ 2 -fa fom satisfying R, with success pobability at least 1 p. Futhemoe, the quey-complexity of T depends only on R, ɛ 1, ɛ 2 and p (and not on the input gaph) and can be expessed as an explicit function of these paametes. We note that the aguments used in [2, 3] to pove the above esult all elied heavily on the egulaity lemma. dependence on the input paametes. Theefoe, the bounds they give have a vey poo (yet explicit) Towe-type The second esult we will need is Coollay 3.8 fom [2]. Lemma 2.4. ([2]) Let R be a gaph egulaity instance of ode, egulaity paamete γ, densities η i,j and a set of non-egula pais R. Suppose that a gaph G has an equipatition V 1,..., V such that fo evey (i, j) / R the pai (V i, V j ) is γ-egula 2 and satisfies d(v i, V j ) = η i,j ± γ2 ɛ 50. Then G is ɛ-close to satisfying R. We now tun to conside k-coloed gaphs. We fist genealize the definitions of a egula pai, egula equipatition and egulaity instance, to the moe geneal setting of k-coloed gaphs. We stat with the following notation: Suppose U, V ae two disjoint vetex sets in a k-coloed gaph. We use 3 d l (U, V ) to denote the density of edges of colo l between U and V, that is d l (U, V ) = E l (U, V ) / A B, whee E l (U, V ) is the set of edges with colo l that connect U to V. In case thee is moe than one gaph, we use d l G (U, V ) to denote the density of edges coloed by l between U, V in the k-coloed gaph G. Definition 2.5. (Regula pai in a k-coloed gaph) Suppose U, V ae disjoint vetex sets in a k-coloed gaph. The pai (U, V ) is γ-egula if fo evey two subsets U U, V V satisfying U γ U, V γ V, and fo evey 1 l k, the inequality d l (U, V ) d l (U, V ) γ holds. ( ) 2 Actually, Coollay 3.8 in [2] only needs to assume that (V i, V j) is γ + γ2 ɛ -egula. 3 Hee, and thoughout the pape, we always use l as a supescipt and neve as an exponent. So x l should ead x supescipt l not x to the powe l. 50 4

5 Definition 2.6. (Regula equipatition in a k-coloed gaph) An equipatition V 1,..., V of the vetices of a k-coloed gaph is γ-egula if all but at most γ ( 2) of the pais (Vi, V j ) ae γ-egula. Definition 2.7. (k-coloed egulaity instance) A k-coloed egulaity instance R is given by a egulaity paamete γ, an intege (the ode of R), a set of densities η l i,j whee 1 i < j and 1 l k, and a set of non-egula pais R of size at most γ ( 2). A k-coloed gaph G is said to satisfy R if G has an equipatition V 1,..., V such that fo evey (i, j) / R the pai (V i, V j ) is γ-egula and satisfies d l (V i, V j ) = η l i,j fo evey 1 l k. The Regulaity Lemma fo k-coloed gaphs states that evey k-coloed gaph has a γ-egula equipatition whose ode can be bounded by a function of γ and k. It can be fomulated in tems of egulaity instances in the following way. Lemma 2.8. (Regulaity lemma fo k-coloed gaphs) Fo evey γ > 0 and integes t and k, thee exists T = T 2.8 (γ, t, k) so that evey k-coloed gaph satisfies some k-coloed egulaity instance of ode at least t and at most T, and egulaity paamete γ. Note that the usual egulaity lemma is the special case of the k-coloed egulaity lemma with k = 2. The poof of the k-coloed vesion equies a mino adaptation of the poof of the standad egulaity lemma. See [4] fo the details. Having descibed the known esults that will be used in the poof of Theoem 1, we now tun to state the additional technical lemmas we shall ely on. We stat with a lemma that allows one to appoximate the numbe of copies of small k-coloed gaphs in a k-coloed gaph which satisfies a given egulaity instance. Definition 2.9. (IC(B, W, σ)) Suppose B is a k-coloed gaph on vetices [q] in which (i, j) is coloed by c(i, j). Suppose W = {ηi,j l : 1 i < j q, 1 l k} ae densities and σ : [q] [q] is a pemutation. Define: IC(B, W, σ) = i<j η c(σ(i),σ(j)) i,j Definition (IC(B, W )) Suppose B and W ae as in the pevious definition. Define: 1 IC(B, W ) = IC(B, W, σ) Aut(B) whee Aut(B) is the numbe of automophisms of B, that is, the numbe of injections φ : V (B) V (B) that peseve the colo of the edges. Definition (IC(B, R)) Let R be a k-coloed egulaity instance of ode and densities {η l i,j : 1 i < j, 1 l k}. Let B be a k-coloed gaph on the vetex set [q]. Fo evey A [] of size q put W (A) = {ηi,j l : i, j A, 1 l k}. Define: IC(B, R) = ( 1 IC(B, W (A)) q) σ A [], A =q 5

6 Remak It is easy to see that IC(B, W, σ), IC(B, W ) and IC(B, R) ae quantities in [0, 1]. To undestand Definition 2.9, conside a andom k-coloed gaph whose vetices ae V 1... V q. Suppose that the pobability that the colo of (v i, v j ) is l is η l i,j (v i V i, v j V j ). Suppose also that V 1 =... V q = n. Let B be a fixed k-coloed gaph on the vetices [q] and let σ be a pemutation of [q]. What is the expected numbe of q-tuples v 1 V 1,..., v q V q which span a copy of B such that v i plays the ole of σ(i)? It is easy to see that this numbe is IC(B, W, σ)n q whee we set W = {ηi,j l : 1 i < j q, 1 l k}. We show (in Lemma 4.2) that fo evey δ, if all pais (V i, V j ) ae γ-egula fo some small enough γ, then the numbe of such q-tuples v 1,..., v q is (IC(B, W, σ) ± δ)n q. This fact demonstates the almost andom behavio of egula patitions. The expession IC(B, W ) (in Definition 2.10) is used to appoximate the total numbe of q-tuples v 1 V 1,..., v q V q which span a copy of B. The expession IC(B, R) (in Definition 2.11) is used to appoximate the numbe of copies of B in a gaph that satisfies the egulaity instance R. The most geneal esult of this sot is the following lemma. Lemma Fo any δ > 0 and integes k and q thee ae γ = γ 2.13 (δ, q, k) and t = t 2.13 (δ, q, k) with the following popety: Fo any k-coloed egulaity instance R of ode at least t and egulaity paamete at most γ, and fo any family B of k-coloed gaphs on q vetices, the numbe of copies of k-coloed gaphs B B in any k-coloed gaph on n vetices satisfying R is ( ) (n ) IC(B, R) ± δ q B B The poof of Lemma 2.13 appeas in Subsection 4.1. The second lemma we will need is the following. Lemma Fo evey δ and integes q and k thee is λ = λ 2.14 (δ, q, k) such that the following holds: Let R and M be k-coloed egulaity instances of ode, and densities {ηi,j l : 1 i < j, 1 l k} and {µ l i,j : 1 i < j, 1 l k} espectively. Let B be a family of k-coloed gaphs of ode q. Suppose that µ l i,j = ηl i,j ± λ. Then IC(B, R) IC(B, M) δ. (1) B B B B The poof of Lemma 2.14 appeas in Subsection 4.2. The last ingedient we will need is the following lemma whose poof appeas in Subsection

7 Definition (Chopping) Let R be a gaph egulaity instance of ode, egulaity paamete γ, densities µ i,j and a set R of non-egula pais. A (k, m)-chopping of R is any k-coloed egulaity instance R of ode, egulaity paamete 2γ, non-egula set R = R and densities η l i,j that satisfy m k ηi,j l = µ i,j and ηi,j l = 1 µ i,j l=m+1 Lemma Fo evey γ > 0 and integes t and k, thee is n 2.16 (γ, t, k) such that the following holds: Suppose R is a gaph egulaity instance of ode at most t and egulaity paamete γ, that R is a (k, m)-chopping of R and that G is a gaph satisfying R with at least n 2.16 (γ, t, k) vetices. Then G has a (k, m)-coloing that satisfies R. 3 The New Poof of Theoem 1 Conside any ɛ > 0. Let popety Q and integes k and m be those fom Definition 1.2, that is, so that Q is a popety of k-coloed gaphs and so that a gaph satisfies P if and only if it has a (k, m)-coloing satisfying Q. Suppose Q can be tested by a teste T Q as in Definition 1.1. Let q = q Q ( ɛ 2 ) be the quey-complexity of T Q, i.e. the numbe of vetices that T Q samples when testing if a k-coloed gaph satisfies Q o is ɛ 2-fa fom satisfying it. Let B be the set of all k-coloed gaphs B on q vetices, such that when T Q samples a k-coloed gaph isomophic to B, it accepts the input. Put t = t 2.13 (1/12, q, k), γ = γ 2.13 (1/12, q, k), T = T 2.8 (γ/2k, t, k) and η = min { ) 2 λ 2.14 (1/12, q, k), ɛ ( γ 2 200m }. Let I be the set of all k-coloed egulaity instances of ode at least t and at most T, egulaity paamete γ and densities fom the set {0, η, 2η, 3η,..., 1}. Obseve that all the above constants, as well as I, depend only on ɛ, k and the popeties P and Q. We now aive at the citical definition of the poof: Definition 3.1. (Good egulaity instance) A gaph egulaity instance R with egulaity measue γ/2 is consideed good if it has a (k, m)-chopping R that satisfies: 1. R I. 2. B B IC(B, R ) 1 2. We say that R is a witness to the fact that R is good. We set GOOD to be the family of good egulaity instances. 7

8 Suppose fist that the input gaph has less than n 2.16 ( γ 2, T, k) vetices. In this case the algoithm can just ask about all edges of G and then check if G satisfies popety P. Since γ, T and k ae all functions of ɛ, P and Q, we get that so is the quey complexity in this case. Hence fom this point on we will assume that n n 2.16 ( γ 2, T, k). The following ae the key obsevations we will need fo the poof. Claim 3.2. If G satisfies P, then G is ɛ 4-close to satisfying some R GOOD. Claim 3.3. If G is ɛ-fa fom satisfying P, then G is ɛ 2-fa fom satisfying any R GOOD. Let us fist complete the poof based on these claims. We descibe a andomized algoithm that distinguishes between gaphs satisfying P and gaphs that ae ɛ-fa fom satisfying P, with success pobability at least 2 3, and by making a numbe of queies that can be bounded by a function of 1 ɛ. Put p = 3 GOOD. Let G be a gaph on at least n 2.16( γ 2, T, k) vetices. In ode to test G fo popety P we do the following: Fo evey R GOOD use T 13 (R, ɛ 4, ɛ 2, p) (ecall Theoem 2) to test whethe G is ɛ 4 -close to satisfying R o ɛ 2-fa fom satisfying it. If one of these tests accepts, then accept the input G, othewise eject it. If G satisfies P then by Claim 3.2 it is ɛ 4-close to some R GOOD, and ou teste accepts it with pobability at least 1 p 2 3. If G is ɛ-fa fom satisfying P then by Claim 3.3 it is ɛ 2-fa fom satisfying any R GOOD. Ou teste accepts G with pobability at most GOOD p = 1 3 and so it ejects with pobability at least 2 3, as equied. Finally, since all the paametes involved ae given by explicit functions of ɛ and the popeties P and Q, we get via Theoem 2 that the numbe of queies made by the teste can be bounded by an explicit function of ɛ. We now complete the poof of Theoem 1 by poving Claims 3.2 and 3.3. Poof (of Claim 3.2): Suppose G satisfies P. Then thee exists some (k, m)-coloing of G that satisfies Q. Denote this k-coloed gaph by H. By Lemma 2.8, H satisfies some k-coloed egulaity instance R 1 of ode t T, egulaity paamete γ 2k and densities {η l i,j : 1 i < j, 1 l k}. Since H satisfies Q, we infe that T Q must accept H with pobability at least 2 3. This means that when sampling q vetices fom H, the pobability to get a k-coloed gaph isomophic to one of the elements of B is at least 2 3. By the choice of γ and t via Lemma 2.13 we get that this pobability is B B IC(B, R 1 ) ± Theefoe IC(B, R 1) (2) B B Let V 1,..., V be an equipatition of H which coesponds to R 1. We claim that V 1,..., V is also a γ 2 -egula equipatition of G. To see this let (i, j) / R 1. Fo evey x V i, y V j, the edge (x, y) is in G if and only if (x, y) is coloed in H by a colo l {1,..., m}. 8 Theefoe

9 d G (V i, V j ) = m dl H (V i, V j ). Let U V i, V V j such that U γ 2 V i, V γ 2 V j. As we assume that (V i, V j ) is a γ 2k -egula pai in H, we have dl H (U, V ) d l H (V i, V j ) γ 2k fo evey 1 l k. By the tiangle inequality, we have d G (U, V ) d G (V i, V j ) m d l H(U, V ) d l H(V i, V j ) mγ 2k γ 2. We thus infe that G satisfies a egulaity instance R 1 with ode, egulaity paamete γ 2, a set of iegula pais R 1 and densities {η i,j : 1 i < j } whee η i,j = m ηl i,j. Let R 2 be the k-coloed egulaity instance that is obtained fom R 1 by eplacing each of the densities ηi,j l with the closest intege multiples of η. Obseve that we thus change each density by at most η. As we chose η λ 2.14 ( 1 12, q, k), we get fom Lemma 2.14 and (2) that IC(B, R 1) (3) B B IC(B, R 2) B B Denote the densities of R 2 by µl i,j. Let R 2 be the gaph egulaity instance of ode, egulaity paamete γ 2, densities {µ i,j : 1 i < j }, whee µ i,j = m µl i,j, and a set of iegula pais R 2. By Definition 2.15 R 2 is a (k, m)-chopping of R 2. Futhemoe R 2 I and we get fom (3) that B B IC(B, R 2 ) 1 2. By Definition 3.1 R 2 is a witness to the fact that R 2 is good. Finally, ecalling that η ɛ( γ 2 ) 2 200m, we get that fo evey i < j we have m m η i,j µ i,j = ηi,j l µ l m i,j ηi,j l µ l i,j mη ɛ ( γ ) In othe wods, the densities of R 1 and R 2 diffe by at most ɛ( γ 2 ) 2. We now get via Lemma 2.4 that G is ɛ 4 -close to satisfying R 2. Poof (of Claim 3.3): We will pove that if G is ɛ 2-close to satisfying some R GOOD then G is ɛ-close to satisfying P. Suppose that G is ɛ 2 -close to a gaph G that satisfies some R GOOD. By Definition 3.1 R has a (k, m)-chopping R such that B B IC(B, R ) 1 2. By Lemma 2.16 G has a (k, m)-coloing satisfying R. Call this k-coloed gaph H. By Lemma 2.13 the pobability to get a k-coloed gaph isomophic to an element of B when sampling q vetices fom H is B B IC(B, R ) ± Theefoe this pobability is at least 12. If H was ɛ 2-fa fom satisfying Q this pobability would have to be at most 1 3, because T Q would have to eject H with pobability at least 2 3. So we infe that H is ɛ 2 -close to satisfying Q. This means that H can be tuned into a k-coloed gaph H that satisfies Q by changing the colos of at most ɛ 2 n2 edges. Constuct a gaph G by doing the following: Fo evey x, y V (H ), put an edge between x and y if (x, y) is coloed by a colo l {1,..., m} in H. Fist, G satisfies P because H is a (k, m)-coloing of G which satisfies Q. Futhemoe, we claim that G is ɛ 2 -close to G. Indeed, obseve that the numbe of edge modifications we pefomed is exactly the numbe of pais (x, y) 9 200

10 so that in one of the gaphs H, H the colo of (x, y) belonged to the set {1,..., m} while in the othe it belonged to {m + 1,..., k}. This numbe is clealy bounded fom above by the numbe of modifications made when tuning H to H. Since H and H diffe in at most ɛ 2 n2 edges the same thus holds fo G and G, implying that G is ɛ 2 -close to G. Since G is assumed to be ɛ 2-close to G, we infe that G is ɛ-close to G. Since G satisfies P the poof is complete. 4 Poofs of Auxiliay Lemmas 4.1 Poof of Lemma 2.13 We will need the following folkloe esult stating the a q-tuple of vetex sets that ae paiwise egula have the coect numbe of copies of K q (the complete gaph on q vetices). A detailed poof can be found in [7]. Lemma 4.1. Fo evey δ > 0 and q thee exists γ = γ 4.1 (δ, q) such that the following holds: Suppose V 1,..., V q ae disjoint vetex sets in a gaph, V 1 =... = V q = n, and all pais (V i, V j ) ae γ -egula. Put IC(K q ; V 1,..., V q ) = d(v i, V j ). Then the numbe of q-tuples v 1 V 1,..., v q V q that span a copy of K q is i<j (IC(K q ; V 1,..., V q ) ± δ)n q As a fist step towads poving Lemma 2.13 we pove a vaiant of Lemma 4.1 fo k-coloed gaphs with espect to IC(B, W, σ). We will then obtain simila lemmas with espect to IC(B, W ) and IC(B, R) (ecall Definitions 2.9, 2.10 and 2.11) and then deive fom them the poof of Lemma Lemma 4.2. Fo evey δ > 0 and q thee exists γ = γ 4.2 (δ, q) such that the following holds: Suppose V 1,..., V q ae disjoint vetex sets in a k-coloed gaph, V 1 =... = V q = n, and all pais (V i, V j ) ae γ-egula. Put W = {d l (V i, V j ) : 1 i < j q, 1 l k}. Then fo evey k-coloed gaph B on the vetices [q], and fo any pemutation σ : [q] [q], the numbe of q-tuples v 1 V 1,..., v q V q which span copy a of B with v i playing the ole of σ(i) is (IC(B, W, σ) ± δ)n q Poof: While the poof of Lemma 4.1 can be adapted to the moe geneal setting of Lemma 4.2 it will be easie to educe Lemma 4.2 to Lemma 4.1. Set γ = γ 4.2 (δ, q) = γ 4.1 (δ, q) and suppose (V i, V j ) is γ-egula fo evey 1 i < j q. We call a q-tuple v 1 V 1,..., v q V q pope, if v 1,..., v q span a copy of B with v i playing the ole of σ(i). We denote by c(i, j) the colo of the edge (i, j) in B. Let E i,j be the set of edges connecting a vetex in V i to a vetex in V j whose colo is c(σ(i), σ(j)). If v 1 V 1,..., v q V q is pope, then the colo of (v i, v j ) is c(σ(i), σ(j)). We see that the edges in E i,j ae the only edges between V i and V j 10

11 that can paticipate in a pope q-tuple. Define a q-patite gaph S with vetex sets V 1,..., V q, in which the edges between V i and V j ae E i,j. A q-tuple v 1 V 1,..., v q V q is pope if and only if it spans a copy of K q in S. So in ode to pove Lemma 4.2 it is enough to show that the numbe of copies of K q in S is (IC(B, W, σ) ± δ)n q. By Lemma 4.1, the numbe of copies of K q in S is (IC(K q ; V 1,..., V q ) ± δ)n q whee IC(K q ; V 1,..., V q ) = d S (V i, V j ). So to complete the poof it is enough fo us to show that IC(B, W, σ) = IC(K q ; V 1,..., V q ). Indeed, we have IC(B, W, σ) = d c(σ(i),σ(j)) (V i, V j ) = E i,j V i<j i<j i V j = d S (V i, V j ) = IC(K q ; V 1,..., V q ) i<j i<j Lemma 4.3. Fo evey δ > 0 and evey q thee exists γ = γ 4.3 (δ, q) such that the following holds: Suppose that V 1,..., V q ae disjoint vetex sets of size n each, and all pais (V i, V j ) ae γ-egula. Put W = {d l (V i, V j ) : 1 i < j q, 1 l k}. Then fo evey k-coloed gaph B on the vetices [q], the numbe of copies of B which have pecisely one vetex in each of the sets V 1,..., V q is (IC(B, W ) ± δ)n q Poof: Set γ 4.3 (δ, q) = γ 4.2 ( δ q!, q). Let V 1,..., V q be as in the statement, and let B be any k-coloed gaph. By Claim 4.2 fo any pemutation σ : [q] [q], ( the numbe of copies ) of B spanned by v 1 V 1,..., v q V q such that v i plays the ole of σ(i) is IC(B, W, σ) ± δ q! n q. If we sum ove all pemutations σ : [q] [q], we get ( ) IC(B, W, σ) ± δ q! n q. In this summation, we count evey σ copy of B exactly Aut(B) times. Thus, by dividing by Aut(B), we get that the numbe of copies of B is 1 Aut(B) ( σ ( ) IC(B, W, σ) ± δ )n q q! = ( ) 1 IC(B, W, σ) ± δ Aut(B) = (IC(B, W ) ± δ)n q σ n q Lemma 4.4. Fo evey δ > 0 and q thee ae γ = γ 4.4 (δ, q) and t = t 4.4 (δ, q) such that the following holds: Suppose that R is a k-coloed egulaity instance of ode at least t and egulaity paamete at most γ. Then fo evey k-coloed gaph B on q vetices, the numbe of copies of B in any n-vetex k-coloed gaph satisfying R is ( ) n (IC(B, R) ± δ) q 11

12 Poof: Put and 4q 2 t = t 4.4 (δ, q) = δ { ( )} δ δ γ = γ 4.4 (δ, q) = min 4q 2, γ 4.3 4, q Let R be a k-coloed egulaity instance as in the statement, let G be an n-vetex k-coloed gaph satisfying R and let B be any k-coloed gaph on q vetices. Let V 1,..., V be an equipatition of V (G) satisfying R. Let C be the collection of all q-tuples that have at most one vetex in each of the sets V i. By a union bound, the numbe of q-tuples that have moe than one vetex in one of the sets V i is at most ( n ) 2 ( n 2 q 2 ) q2 ( n q ) q2 t ( ) n 1 ( ) n q 4 δ. q So C ( 1 δ 4) ( n q). Theefoe the lemma will follow fom showing that the numbe of q-tuples belonging to C which span a copy of B is ( IC(B, R) ± 3 4 δ) C. Given A = {x 1,..., x q } {1,..., } let N(A) denote the numbe of q-tuples v 1 V x1,..., v q V xq which span a copy of B. We say that A is good if all the pais (V xi, V xj ) (1 i < j q) ae γ-egula. Othewise A is called bad. If A is good we get fom ou choice of γ via Lemma 4.3 that ( N(A) = IC(B, W (A)) ± 1 ) (n ) q 4 δ. whee we set W (A) = {d l (V i, V j ) : i, j A, 1 l k}. We can thus estimate the numbe of q-tuples belonging to C which span a copy of B by N(A) = (( IC(B, W (A)) ± 1 ) (n ) ) q 4 δ + N(A) = A is good A is bad A [], A =q A [], A =q (( IC(B, W (A)) ± 1 ) (n ) ) q 4 δ + ( ( n ) q ) N(A) IC(B, W (A)) = A is bad ( IC(B, R) ± 1 ) ( ) (n ) q 4 δ ( ( n ) q ) + N(A) IC(B, W (A)) q A is bad Since (V i, V j ) is γ-egula fo evey (i, j) / R thee ae at most γ ( 2) pais (Vi, V j ) which ae not γ-egula. Theefoe the numbe of bad sets A {1,..., } is at most ( )( ) ( ) 2 γ γq 2 1 ( ) 2 q 2 q 4 δ q (4) 12

13 Using the facts that 0 IC(B, W (A)) 1 and 0 N(A) ( ) n q fo evey A [], and the bound on the numbe of bad sets, we get that ( ( n ) q ) N(A) IC(B, W (A)) 1 ( ) (n ) q 2 δ q A is bad By plugging the above inequality in (4) we get that the numbe of q-tuples belonging to C which span a copy of B is ( IC(B, R) ± 3 4 δ) ( ( n ) q. ( q) Obseve that C = ( n ) q. q) Theefoe the numbe of those q-tuples is ( IC(B, R) ± 3 4 δ) C, as equied. Poof (of Lemma 2.13): Put t = t 2.13 (δ, q, k) = t 4.4 (k (q 2) δ, q), γ = γ2.13 (δ, q, k) = γ 4.4 (k (q 2) δ, q). Let R be a egulaity instance of ode at least t and egulaity paamete at most γ and let G be a k-coloed gaph satisfying R. Let B B. By ou choice of γ and t via Lemma 4.4, the numbe of copies of B in G is (IC(B, R) ± k (q 2) δ) ( n q). Clealy B k ( q 2), so the numbe of copies of gaphs B B in G is ( ( ) ( )) ( ) n (n ) IC(B, R) ± k (q 2) δ = IC(B, R) ± δ q q B B B B 4.2 Poof of Lemma 2.14 We will deive Lemma 2.14 fom the following lemma. Lemma 4.5. Fo evey δ and q thee is λ = λ 4.5 (δ, q) such that the following holds: Let R and M be k-coloed egulaity instances of ode, and densities {ηi,j l : 1 i < j, 1 l k} and {µ l i,j : 1 i < j, 1 l k} espectively. Let B be a k-coloed gaph on the vetices [q]. Suppose that µ l i,j = ηl i,j ± λ. Then IC(B, R) IC(B, M) δ Poof: Put λ = λ 4.5 = (δ, q) = δ 2 (q 2) q!. Let R, M be k-coloed egulaity instances as in the statement. Let A = {x 1,..., x q } {1,..., }, and put W R (A) = {η l x i,x j : 1 i < j q, 1 l k} and W M (A) = {µ l x i,x j : 1 i < j q, 1 l k}. 13

14 Denote the colo of (i, j) in B by c(i, j). Let σ : [q] [q] be a pemutation. By Definition 2.9 we have IC(B, W R (A), σ) IC(B, W M (A), σ) = 1 i<j q η c(σ(i),σ(j)) x i,x j 1 i<j q 1 i<j q η c(σ(i),σ(j)) x i,x j ( λ) ηx c(σ(i),σ(j)) i,x j ± 1 i<j q µ c(σ(i),σ(j)) x i,x j Opening the paentheses in the above poduct gives 2 (q 2) 1 summands, all of which ae multiples of ±λ. Theefoe IC(B, W R (A), σ) IC(B, W M (A), σ) λ2 (q 2). By Definition 2.10, the tiangle inequality and ou choice of γ we have IC(B, W R (A)) IC(B, W M (A)) = = 1 (IC(B, W R (A), σ) IC(B, W M (A), σ)) Aut(B) σ q!λ2 (q 2) = δ. By Definition 2.11 and the tiangle inequality we have IC(B, R) IC(B, M) = 1 ( (IC(B, W R (A)) IC(B, W M (A))) q) δ, A {1,...,}, A =q as needed. Poof (of Lemma 2.14): Put λ = λ 2.14 (δ, q, k) = λ 4.5 (k (q 2) δ, q). By the choice of λ via Lemma 4.5 we get that IC(B, R) IC(B, M) k (q 2) δ fo evey B B. Since B k ( 2) q the tiangle inequality thus gives (1). 4.3 Poof of Lemma 2.16 We will deive Lemma 2.16 fom the following lemma. Lemma 4.6. Let U, V be disjoint vetex sets in a gaph satisfying U = V = n n 4.6 (γ, k). Suppose the bipatite gaph (U, V ) is γ-egula with d(u, V ) = µ. Let {η l : 1 l k} be nonnegative eals satisfying m ηl = µ and k l=m+1 ηl = 1 µ. Then thee is a (k, m)-coloing of (U, V ) such that the esulting k-coloed gaph is 2γ-egula and satisfies d l (U, V ) = η l fo evey 1 l k. Fo the poof of Lemma 4.6 we need the following standad Chenoff-type inequality: Lemma 4.7. Suppose X 1,..., X m ae independent Boolean andom vaiables and P(X i = 1) = p i. Let E = m i=1 p i. Then P( m i=1 X i E δm) 2e 2δ2m. 14

15 Poof (of Lemma 4.6): We will show that the edges between U and V can be coloed with colos 1,..., m in a way that satisfies the equiements. The same agument can be used to colo the nonedges with colos m + 1,..., k. Fist assume that µ γ. If this is the case, just colo any η l n 2 of the edges between U and V with colo l, fo evey 1 l m. This way we made sue that d l (U, V ) = η l. Let U U, V V with U, V 2γn. Befoe the coloing we had d(u, V ) = µ ± γ. Theefoe afte the coloing we have 0 d l (U, V ) µ + γ, so d l (U, V ) η l µ + γ 2γ. Assume fom this point on that µ γ. Fo evey edge e E(U, V ) oll a die with sides 1,..., m so that pobability of side l is ηl µ. If the die falls on side l then colo e with colo l. Then the expected numbe of edges of colo l is η l n 2. By Lemma 4.7, the pobability that the numbe of edges coloed by l deviates fom its expectation by moe than n 3 2 is at most 2e 2n/µ 2e 2n. If n n 4.6 (γ, k) then this pobability is less than 1 4k. This means, by a union bound, that with pobability at least 3 4 the numbe of edges coloed by l is ηl n 2 ± n 3 2 fo evey 1 l m. Claim 4.8. With pobability at least 3 4 all sets U U, V V such that U, V 2γn satisfy d l (U, V ) = η l ± 3γ 2 Poof: Let U U, V V such that U, V 2γn. The density of edges between U and V befoe the coloing is µ±γ. Theefoe the expected density of edges with colo l is ηl µ (µ±γ) = ηl ±γ. So it is enough to show that with pobability at least 3 4, thee ae no sets U, V and colo 1 l m such that the density of edges of colo l between U and V deviates fom its expectation by moe than γ 2. By Lemma 4.7, the pobability that the density of edges of colo l between U and V deviates fom its expected value by moe than γ 2 is at most γ 2e 2( 2 )2 U V /d(u,v ) 2e γ2 U V /2. We assumed that U, V 2γn, so this pobability is at most 2e γ4 n 2 /2. The numbe of choices of sets U, V as above is at most 2 2n, and the numbe of colos is at most k, so by a union bound we get: The pobability that thee ae sets U, V and a colo l, such that the density of edges between U, V with colo l deviates fom its expectation by moe than γ 2 is at most k22n 2e γ4 n 2 /2. This expession is less than 1 4 if n is lage enough, namely n n 4.6(γ, k). least 1 2 Getting back to the poof of Lemma 4.6 we see that so fa we poved that with pobability at the following two conditions hold: 1. d l (U, V ) = η l ± n 1 2 fo evey 1 l m. 2. d l (U, V ) = η l ± 3γ 2 fo evey 1 l m and evey two sets U U, V V of size at least 2γn. Now take a coloing that satisfies conditions 1 and 2. Let us wite d l (U, V ) = η l + ɛ l whee ɛ l n 1 2. Obseve that m ( η l + ɛ l) = m d l (U, V ) = µ = 15 m η l

16 Theefoe m ɛ l = 0. We can change the colos of at most mn 3 2 kn 3 2 edges to make sue that d l (U, V ) is exactly η l. Fo evey U U, V V of size at least 2γn this final change changes d l (U, V ) by at most k (2γ) 2 n 1 2 d l (U, V ) = η l ± 2γ as equied. which is less than γ 2 if n n 4.6(γ, k). So in the end we have Remak 4.9. In fact, we could have poved the following stonge lemma: Let U, V be disjoint sets in a gaph, U = V = n n 4.6 (γ, ɛ, k). Suppose U, V is γ-egula and d(a, B) = µ, and let {η l : 1 l k} be nonnegative numbes satisfying m ηl = µ and k l=m+1 ηl = 1 µ. Then thee is a (k, m)-coloing of U, V such that the esulting k-coloed gaph is γ(1 + ɛ)-egula and satisfies d l (U, V ) = η l. The choice of 2γ in Lemma 4.6 and in Definition 2.15 is fo convenience. Poof (of Lemma 2.16): Put n 2.16 (γ, t, k) = t n 4.6 (γ, k). Let R be a gaph egulaity instance of ode t and egulaity paamete γ. Let R be a (k, m)-chopping of R and let {η l i,j : 1 i < j, 1 l k} be the densities of R. Let G be a gaph with at least n 2.16 (γ, t, k) vetices that satisfies R. Let V 1,..., V be a γ-egula equipatition of V (G) that coesponds to R. Fo evey 1 i we have V i n 2.16(γ,t,k) n 4.6 (γ, k). If (i, j) / R apply Lemma 4.6 fo V i, V j and {ηi,j 1,..., ηk i,j }. Colo the est of the edges and non-edges abitaily. The esulting k-coloed gaph satisfies R. Refeences [1] N. Alon, E. Fische, M. Kivelevich and M. Szegedy, Efficient testing of lage gaphs, Combinatoica 20 (2000), [2] N. Alon, E. Fische, I. Newman and A. Shapia, A combinatoial chaactaization of testable gaph popeties - it is all about egulaity, SIAM J. on Computing 39 (2009), [3] E. Fische and I. Newman, Testing vesus estimation of gaph popeties, SIAM J. on Computing 37 (2007), [4] J. Komlós and M. Simonovits, Szemeédi s Regulaity Lemma and its applications in gaph theoy. In: Combinatoics, Paul Edös is Eighty, Vol II (D. Miklós, V. T. Sós, T. Szönyi eds.), János Bolyai Math. Soc., Budapest (1996), [5] L. Lovász and K. Vesztegombi, Nondeteministic gaph popety testing, manuscipt, [6] L. Lovász, B. Szegedy: Testing popeties of gaphs and functions, Isael J. Math. 178 (2010), [7] E. Fische, The difficulty of testing fo isomophism against a gaph that is given in advance, SIAM Jounal on Computing 34 (2005),

17 [8] O. Goldeich (ed), Popety Testing: Cuent Reseach and Sueveys, LNCS 6390, Spinge, [9] O. Goldeich, S. Goldwasse and D. Ron, Popety testing and its connection to leaning and appoximation, J. ACM 45 (1998), [10] O. Goldeich and L. Tevisan, Thee theoems egading testing gaph popeties, Random Stuctues and Algoithms 23 (2003), [11] R. Rubinfeld and A. Shapia, Sublinea time algoithms, SIAM J. on Discete Math 25 (2011), [12] E. Szemeédi, Regula patitions of gaphs, In: Poc. Colloque Inte. CNRS (J. C. Bemond, J. C. Founie, M. Las Vegnas and D. Sotteau, eds.), 1978,