JOURNAL OF MATHEMATICAL PHYSICS 46, 113302 共2005兲

Random multi-overlap structures for optimization problems Luca De Sanctisa兲 Department of Condensed Matter, ICTP, Strada Costiera 11, Trieste, Italy 共Received 5 August 2005; accepted 22 September 2005; published online 28 November 2005兲

We extend to the K-SAT and p-XOR-SAT optimization problems the results recently achieved, by introducing the concept of random multi-overlap structure, for the Viana-Bray model of diluted mean field spin glass. More precisely we can prove a generalized bound and an extended variational principle for the free energy per site in the thermodynamic limit. Moreover a trial function implementing ultrametric breaking of replica symmetry is exhibited. The ultrametric structure exhibits the same factorization property as the optimal structures for the Viana-Bray model and the Sherrington-Kirkpatrick nondiluted model. © 2005 American Institute of Physics. 关DOI: 10.1063/1.2121267兴

I. INTRODUCTION

In the case of nondiluted spin glasses, Aizenman, Sims, and Starr1 introduced the idea of random overlap structure 共ROSt兲 to express in a very elegant manner the free energy of the model as an infimum over a rich probability space, to exhibit an optimal structure 共the so-called Boltzmann one兲, to write down a general trial function through which one can formulate various ansatzs for the free energy of the model. It was also described how to formulate in particular the Parisi ansatz within this formalism. In Refs. 2 and 3 we extended those results to the case of diluted spin glass 共Viana-Bray model兲. Here we extend the same results to optimization problems, the K-SAT and the p-XOR-SAT. The latter is the simple extension to p-body interactions of the Viana-Bray model, which is the diluted version of the famous model of Sherrington and Kirkpatrick 共SK兲 of mean field spin glass. We also prove that the optimal structures must enjoy a certain factorization property, known as invariance with respect to the cavity step, that was first found by Guerra in Ref. 5 for the SK model, and that turned out to be valid also for the Viana-Bray dilute spin glass model.2 The ultrametric ansatz we propose verifies such a property. Many of the calculations in the present paper are quite simple and standard, and as a general reference with many details the reader can take for instance Refs. 4 and 7.

II. MODEL, NOTATIONS, DEFINITIONS

Consider configurations of Ising spins ␴: i → ␴i = ± 1, i = 1 , . . . , N. Let P␨ be a Poisson random variable of mean ␨, and 兵i␯␮其 be independent identically distributed random variables, uniformly distributed over points 兵1 , . . . , N其. If 兵J␯␮其 are independent identically distributed copies of a symmetric random variable J = ± 1, then the Hamiltonian of random K-SAT is

a兲

Electronic mail: [email protected]

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46, 113302-1

© 2005 American Institute of Physics

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113302-2

J. Math. Phys. 46, 113302 共2005兲

Luca De Sanctis P␣N

H=−

1

1

兺 共1 + J␯1␴i 兲 ¯ 2 共1 + J␯K␴i 兲. ␯=1 2 1 ␯

K ␯

Here ␣ 艌 0 is the degree of connectivity and K is assumed to be even. We do not consider the presence of an external field, but all the results trivially extend to this case as well. By ␻ we mean the Bolztmann-Gibbs average

␻共O兲 = ZN−1 兺 O共␴兲exp共− ␤H兲,

ZN = 兺 exp共− ␤H兲.

兵␴其

兵␴其

We will denote by E the average over all the other 共quenched兲 random variables, and the free energy f N per site and its thermodynamic limit are defined by − ␤fN =

1 E ln ZN, N

f = lim f N . N→⬁

The existence of the above-mentioned limit has been proven in Ref. 4 共see also Ref. 2 for the proof in the framework of overlap structures兲. We will use the notation ⍀ for the product of the needed number of independent copies 共replicas兲 of ␻ and 具·典 for the composition of an E-type average over some quenched variables and some sort of Boltzmann-Gibbs average over the spin variables, which will be clear from the context. The multi-overlaps are defined 共using replicas兲 by N

q1,¯,n =

1 兺 ␴共1兲 ¯ ␴共n兲 i . N i=1 i

˜ n其 , ␰兲 where Definition 1: A random multi-overlap structure 共RaMOSt兲 R is a triple 共⌺ , 兵q 1. 2. 3.

⌺ is a discrete space; ␰: ⌺ → R+ is a system of random weights; ˜qr ,¯,r : ⌺r ⫻ ¯ ⫻ ⌺r → 关0 , 1兴, l 苸 N, 兩q ˜ 兩 艋 1 is a positive definite multi-overlap kernel. 1 2l 1 l

III. THE STRUCTURE OF THE MODEL

In order to understand what is the underlying structure of the model, it is well known that it is useful to compute the derivative of the free energy with respect to the somewhat basic parameter. In the case of nondiluted spin glasses such a parameter is the strength of the couplings 共and it is equivalent to differentiating with respect to the inverse temperature兲. In the case of diluted spin glasses such a parameter is the connectivity. It is very easy to show 共see, e.g., Ref. 4兲 by pretty standard calculation that

冉

共− 1兲n+1 e−␤ − 1 d 1 E ln 兺 exp共− ␤H兲 = 兺 d␣ N n 2K ␴ n⬎0

冊

n

具共1 + Qn共q兲兲K典,

共1兲

where 关n/2兴

Qn共q兲 =

1,n

兺 兺 l=1 r ⬍¯⬍r 1

qr1,. . .,r2l . 2l

The fundamental quantities governing the model are therefore the multi-overlaps, like for diluted spin glasses.2,3 That is why we use RaMOSt in this context as well. The main difference is that here the function 1 + Qn takes the place of the mere multi-overlaps. As should be clear from Refs. 1–3 we must therefore introduce also two random variables ˜ 共␥ , ␣ ;˜J兲 and H ˆ 共␥ , ␣ ; Jˆ兲 such that H .

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J. Math. Phys. 46, 113302 共2005兲

RaMOSts for Optimization Problems

冉

n+1 d e −␤ − 1 ˜ 兲 = K 兺 共− 1兲 E ln 兺 ␰␥ exp共− ␤H . d␣ n 2K−1 n⬎0 ␥

冊

n

˜ 兲兲K−1典, 具共1 + Qn共q

冉

n+1 e −␤ − 1 d 1 ˆ 兲 = 共K − 1兲 兺 共− 1兲 E ln 兺 ␰␥ exp共− ␤H d␣ N n 2K ␥ n⬎0

冊

共2兲

n

˜ 兲兲K典. 具共1 + Qn共q

共3兲

Finally, we introduce as expected the following trial function:

共

兲

N ˜ 1 Hi 2 共1 + Ji␴i兲 兺␴,␥␰␥ exp − ␤兺i=1 1 ˜ ˆ GN共R,H,H兲 = E ln , N ˆ兲 兺 ␰ exp共− ␤H

␥ ␥

˜ are independent copies of H ˜ . We will construct explicitly H ˜ and H ˆ in the next sections. where H i . . Lastly, let us define N

˜ =兺H ˜ 1 共1 + J ␴ 兲. H i i i 2 i=1

IV. GENERAL THEOREMS

Let us state the extension to the K-SAT model of the results presented in Refs. 1, 5, and 2. The first result is a general bound for the free energy per spin. Theorem 1 (Generalized Bound): − ␤ f N 艋 inf GN . R

Proof: Consider the interpolating Hamiltonian ˜ 共1 − t兲 + H ˆ 共t兲, H␥共t兲 = H共t兲 + H

t 苸 关0,1兴,

where t is understood to multiply the connectivity, and consider also R共t兲 =

兺␥,␴␰␥ exp共− ␤H␥共t兲兲 1 E ln . N ˆ 兲 兺␥␰␥ exp共− ␤H ␥

Now observe that R共1兲 = −␤ f N, R共0兲 = GN, and compute the t-derivative of R共t兲 using the expressions in Sec. III,

冉

1 e −␤ − 1 d R共t兲 = − ␣ 兺 dt 2K n⬎0 n

冊

n

˜ 兲兲K−1 + 共K − 1兲共1 + Qn共q ˜ 兲兲K典. ⫻ 具共1 + Qn共q兲兲K − K共1 + Qn共q兲兲共1 + Qn共q Therefore the derivative above is nonpositive since the function xK − Kxy K−1 + 共K − 1兲y K of x and y is non-negative. This completes the proof of the theorem. 䊐 The second result is the explicit construction of a RaMOSt which provides the proof of the existence of such structures and a reversed bound to the one in the previous theorem. This RaMOSt is called Boltzmann, and it is equivalent to the existence of the thermodynamic limit for the free energy per site.6 Definition 2 (Boltzmann RaMOSt): The Boltzmann RaMOSt RB共M兲 is the triple 1. 2.

⌺ = 兵−1 , 1其 M 苹 ␶, ␰␶ = exp共−␤HM 共␶兲兲,

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113302-4

3.

J. Math. Phys. 46, 113302 共2005兲

Luca De Sanctis

˜qr ¯r = 1 / M兺 j␶共rj 1兲 ¯ ␶共rj l兲. 1 l

We choose ˜ =− H ␶

PK␣N

兺 ␯=1

1 1 1 共1 + ˜J␯1␶ j1兲 ¯ 共1 + ˜J␯K−1␶ jK−1兲 共1 + J␯K␴i␯兲, ␯ ␯ 2 2 2 P共K−1兲␣N

ˆ =− H ␶

兺 ␯=1

1 1 共1 + Jˆ␯1␶ j1兲 ¯ 共1 + Jˆ␯K␶ jK兲, ␯ ␯ 2 2

where the independent random variables j.. are uniformly distributed over 1 , . . . , M and ˜J.. , Jˆ.. are independent copies of J. Theorem 2 (Reversed Bound): − ␤ f 艌 lim lim inf GN共RB共M兲兲. N→⬁ M→⬁

Proof: It is clearly enough to show 1 Z M+N − ␤ f = C lim E ln 艌 lim inf lim inf GN共RB共M兲兲, N ZM M N M where the limit C lim is in the Cesàro sense. We can rewrite GN as

冋

˜

1 兺␶,␴e−␤共HM +H兲 ZN+M 共␣⬘兲 ZN+M 共␣兲 Z M 共␣兲 E ln N ZN+M 共␣⬘兲 ZN+M 共␣兲 Z M 共␣兲 兺 e−␤共HM +Hˆ兲 ␶

共4兲

册

and therefore we have four terms. If we now take

冉

␣⬘ = ␣ 1 + 共K − 1兲

N M

冊

we see that the fourth fraction is the same as ZM 共␣兲 / Z M 共␣⬘兲, and it cancels out with the second in the limit of large M 共just like in Ref. 2兲. We also know that the third fraction tends to −␤ f. Now we keep proceeding like in Ref. 2. In the denominator of the first fraction we can split the mean ␣⬘共N + M兲 into the sum of three means such that HN+M splits into the sum of three Hamiltonians with the first depending only on cavity spins ␶, the second containing exactly one spin from the original system in its interactions, the third has the interactions with at least two spins ␴. Hence the three coefficients of the connectivities, up to corrections vanishing when M is large, will be M K , KM K−1, and a negligible third 共of order M K−2兲, respectively. Incidentally, this means that when the cavity is large the added spins do not interact with one another 共asymptotically兲. The choice of ␣⬘ we made guarantees that numerator and denominator contain two 共up to a negligible third in the denominator兲 identical Hamiltonians with the same connectivities. As a 䊐 consequence, the first fraction in GN vanishes in the limit and the theorem is proven. An immediate consequence of the two bounds is clearly the following Theorem 3 (Extended Variational Principle): − ␤ f = lim inf GN . N→⬁ R

All the RaMOSts yielding the correct value of the free energy per site in the thermodynamic limit are called optimal, and they enjoy the same factorization property that is found for both nondilute spin glasses and the Viana-Bray model of dilute spin glass.2 This statement is made precise by factorizing the cavity part of the trial free energy2

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J. Math. Phys. 46, 113302 共2005兲

RaMOSts for Optimization Problems

˜ 兲, c1 ¯ cN = 兺 exp共− ␤H

共5兲

␴

in the next Theorem 4: In the whole region where the parameters are uniquely defined, the following Cesàro limit is linear in N and ¯␣: C lim E ln ⍀ M M

再兺 ␴

冎

ˆ 共¯␣/N兲兲兴 = N共− ␤ f + ␣A兲 + ¯␣A, c1 ¯ cN exp关− ␤共H

共6兲

where A=

兺 n⬎0

冉

共− 1兲n+1 e−␤ − 1 n 2K

冊

n

具共1 + Qn共q兲兲K典.

Proof: As expected, the proof is similar to that of the analogous theorem in Ref. 2 and to that of Theorem 2 above, except here we will choose

␣⬘ = ␣ + 共K − 1兲

1 共␣N + ¯␣兲 M

and use only three fractions, re-writing the left-hand side of 共6兲 as

冋

册

˜ 兲兲 Z 兺␶,␴ exp共− ␤共H M + H 1 N+M 共␣⬘兲 ZN+M 共␣兲 E ln . N ZN+M 共␣⬘兲 ZN+M 共␣兲 Z M 共␣兲 Notice that this choice of ␣⬘ reduces to the previous one of Theorem 2 for ¯␣ = 0, as it should do. 䊐 V. REPLICA SYMMETRY BREAKING AND ULTRAMETRIC RaMOSt

Now we are equipped with the main theorems about the random multi-overlap structures in the case of the K-SAT problem, and we are about to extend the results of Ref. 3 to the K-SAT. We want to construct a trial function depending on ultrametric trial multi-overlaps, that fulfills the generalized bound and obeys the factorization property of the optimal structures. Definition 3 (Ultrametric RaMOSt): The R-level replica symmetry breaking ultrametric RaMOSt RU is the triple 1. 2. 3.

⌺ = NR 苹 ␥ = 共␥1 , . . . , ␥R兲; ␰␥共m1 , . . . , mR兲 from the random probability cascades; ˜qr ¯r = 共q ˜ r共1兲,¯,r − ˜qr共0兲¯r 兲␦␥r1,¯,␥rl + ¯ + 共q ˜ r共R兲,¯,r − ˜qr共R−1兲 兲␦␥r1¯␥rl ¯ ␦␥r1¯␥rl is the ultrametric 1 l 1 1 1 1 R R 1 l 1 l 1 l 1,¯,rl 共a兲 multi-overlap, given partitions qr ,¯,r , 0 艋 a 艋 R of the interval [0,1]. 1

l

Denoting by X the map X:˜␣a → ma satisfying 共7兲, we can consider the trial function G共RU兲 as a function G共X兲 of X. ˜, H ˆ satisfying 共2兲 and 共3兲 with ˜q ultrametric and the ultrametric Theorem 5: There exist H trial function G共X兲 satisfying the bound − ␤ f共␤, ␣兲 艋 inf G共X兲 X

as a special case of the generalized bound. Moreover, the Ultrametric RaMOSt enjoys the same factorization property as in Theorem 4. Proof: Take

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J. Math. Phys. 46, 113302 共2005兲

Luca De Sanctis

˜ = H ␥

PK␣N

1

1

兺 ˜u␯␥ 2 共1 + J␯␴i 兲 − ␤ ln cosh共␤˜u␯␥兲, ␯=1 ␯

P共K−1兲␣N

兺 ␯=1

ˆ = H ␥

uˆ␯␥ −

1 ln cosh共␤uˆ␯␥兲 ␤

with ˜u␥ , uˆ␥ defined by tanh共␤˜u␥兲 = 共e−␤ − 1兲 21 共1 + ˜J1W␥1 兲 ¯ 21 共1 + ˜JK−1W␥K−1兲, tanh共␤uˆ␥兲 = 共e−␤ − 1兲 21 共1 + Jˆ1W␥1 兲 ¯ 21 共1 + JˆKW␥K兲 in which W␥ is the same as in Ref. 3 ¯ + ¯ +␻ ¯ ˜ ˜␣ 共␳k 兲J ˜ ˜␣ 共␳k 兲J , W␥ = ␻ 1 ␯ ␥1 R ␯ ␥1,¯,␥R ˜ ˜␣共␳k 兲 is the infinite volume limit of the Boltzmann-Gibbs average of a random spin from where ␻ ␯ an auxiliary system with a Viana-Bray one-body interaction Hamiltonian at connectivity ˜␣3 and ¯J = ± 1 is symmetric. Notice that ˜ ˜␣n 共␳k 兲 = 具q ˜ n典˜␣ = ˜qn共˜␣兲. E␻ . The indices, the bar, the tilde, and the caret mean independent copies of the corresponding variR of the interval 关0, 1兴, ables. Let us now report a comment from Ref. 2. Given any partition 兵xa其a=0 R there exists a sequence 兵˜␣a其a=0 苸 关0 , ⬁兴 such that ˜q1,¯,n共˜␣a兲 = xa − xa−1. In other words, a sequence R 苸 关0 , ⬁兴 generates for each n 苸 N a partition of 关0, 1兴 considered as the set of trial values 兵˜␣a其a=0 of ˜q1,¯,n, provided the ˜␣a are not too large

兺 ˜q1,¯,n共˜␣a兲 艋 1.

共7兲

a艋R

We limit our trial multi-overlaps to belong to partitions generated in this way. This implies that the points of the generated partitions tend to get closer to zero as n increases. This is good, ˜ n典 decreases as n increases and therefore the probability integral since in any probability space 具q distribution functions tend to grow faster near zero. Now put inductively ˜ ˜ 共␶共r1兲 ¯ ␶共rl兲兲 = ˜q ␣a兲 = ˜qr共a兲,¯,r − ˜qr共a−1兲 E⍀ ␣a k r1,¯,rl共˜ ,¯,r , k ␯

␯

1

l

1

˜qr共0兲,¯,r = 0,

l

1

l

then an elementary calculation shows that E tanhn共␤˜u␥兲 =

冉

E tanhn共␤uˆ␥兲 =

e −␤ − 1 2K−1

冉

冊

e −␤ − 1 2K

n

˜ 兲兲K−1 , 共1 + Qn共q

冊

n

˜ 兲兲K 共1 + Qn共q

with ˜q ultrametric,3 i.e., r r r r ˜ r共R兲,¯,r − ˜qr共R−1兲 ˜qr ,¯,r = 共q ˜ r共1兲,¯,r − ˜qr共0兲,¯,r 兲␦␥r1¯␥rl + ¯ + 共q ,¯,r 兲␦␥ 1¯␥ l ¯ ␦␥ 1¯␥ l . 1 l 1

l

1

l

1

1

1

l

1

l

1

1

R

R

Hence we reproduced the setting of the generalized bound in the particular case of ultrametric multi-overlap, and therefore we just proved that

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RaMOSts for Optimization Problems

− ␤ f 艋 inf G共X兲. X

At this point we only need to make sure that the ultrametric RaMOSt passes the invariance test prescribed by Theorem 4. Notice that G共X兲 does not depend on N 共see Lemma 1 in Appendix ˜ and W ˆ are chosen to be independent, therefore the factorization property A of Ref. 3兲. Moreover W of the optimal RaMOSts holds: ˆ 共¯␣/N兲兲兴 = NB + ¯␣A E ln ⍀␰关c1 ¯ cN exp共− ␤H for some B, and we are again using the definition 共5兲.

䊐

VI. CONCLUSIONS

The RaMOSt is the minimal generalization of the ROSt, and what we showed here and in Ref. 2 is that the minimal generalization is enough to formulate the variational principle and also exhibit a concrete RaMOSt analogous to the Parisi one for SK. As a consequence, it is enough to restrict the space of trial functions to those expressible in terms of fixed multi-overlaps 共i.e., a set of numbers, not random variables to be averaged兲. Unfortunately the case of odd K still escapes our approach.

ACKNOWLEDGMENTS

The author gratefully acknowledges the hospitality of the Department of Physics at University of Rome “La Sapienza” 共and in particular Giovanni Jona-Lasinio兲. APPENDIX: THE p-XOR-SAT

The Hamiltonian of the random p-XOR-SAT coincides with the one of the diluted p-spin glass P␣N

H=−

兺 J ␯␴ i

␯=1

1 ␯

¯ ␴i p . ␯

It is therefore elementary, assuming p even, to extend all the results of Refs. 2 and 3 to this case, also when in the presence of an external field. Since it is easy to show2 d 1 1 p E ln 兺 exp共− ␤H兲 = 兺 典兲, E tanh2n共␤J兲共1 − 具q2n d␣ N ␴ n⬎0 2n the structure of the model is the same RaMOSt valid for the case of the Viana-Bray model, but the above-mentioned equality suggests to try and get the non-negative convex function x p − pxy p−1 ˜ and + 共p − 1兲y p whenever we got the square x2 − 2xy + y 2 in the Viana-Bray case. That is why here H ˆ are chosen such that H d p−1 ˜ 兲 = p 兺 1 E tanh2n共␤J兲共1 − 具q ˜ 2n E ln 兺 ␰␥ exp共− ␤H 典兲, . d␣ n⬎0 2n ␥ d 1 p ˆ 兲 = 共p − 1兲 兺 1 E tanh2n共␤J兲共1 − 具q ˜ 2n E ln 兺 ␰␥ exp共− ␤H 典兲 d␣ N 2n ␥ n⬎0 and plugged into

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J. Math. Phys. 46, 113302 共2005兲

Luca De Sanctis

N ˜ ˜ ,H ˆ 兲 = 1 E ln 兺␴,␥␰␥ exp共− ␤兺i=1Hi␴i兲 . GN共R,H N ˆ兲 兺 ␰ exp共− ␤H

␥ ␥

The generalized bound clearly holds, the Boltzmann RaMOSt is the one with ␰␶ = exp共−␤H M 兲 and

˜ =− H ␶

P p␣N

兺

␯=1

P共p−1兲␣N

˜J ␶ 1 ¯ ␶ p−1␴ , ␯ j j i␯ ␯

␯

ˆ =− H ␶

兺

␯=1

Jˆ␯␶ j1 ¯ ␶ j p ␯

␯

共same couplings as the original system兲 and it is optimal so that we can also state the extended variational principle. The broken replica symmetry ultrametric RaMOSt 共which includes as a trivial case the replica symmetric one兲 relies on the weights ␰␥ of the random probability cascades as in Ref. 7 and on

˜ =− H ␥

P p␣N

兺 ␯=1

冉

P共p−1兲␣N

ˆ =− H ␥

兺 ␯=1

冊

cosh共␤J兲 ˜ ␥ 1 ln + J ␯ ␴ i␯ , ␤ cosh共␤˜J␥兲 ␯

冉

cosh共␤J兲 ˆ ␥ 1 ln + J␯ ␤ cosh共␤Jˆ␥兲 ␯

冊

with

˜1¯W ˜ p−1 , tanh共␤˜J␥兲 = tanh共␤J兲W ␥ ␥

˜1¯W ˜ p, tanh共␤Jˆ␥兲 = tanh共␤J兲W ␥ ␥ ˜ . are independent copies of where W ␥ ˜ 共J ¯ ¯ + ¯ +␻ ¯ ˜ ˜␣ 共␳k 兲J ˜ ˜␣ 共␳k 兲J W ␥ ,k␯兲 = ␻ 1 ␯ ␥1 R ␯ ␥1,¯,␥R with ¯J. = ± 1 symmetric. M. Aizenman, R. Sims, and S. L. Starr, Phys. Rev. B 68, 214403 共2003兲. L. De Sanctis, J. Stat. Phys. 117, 785 共2004兲. 3 L. De Sanctis, ArXiv: cond-mat/0411726. 4 S. Franz and M. Leone, J. Stat. Phys. 111, 535 共2003兲. 5 F. Guerra, ArXiv: cond-mat/0307673. 6 M. Mezard and G. Parisi, Eur. Phys. J. B 20, 217 共2001兲. 7 D. Panchenko and M. Talagrand, Probab. Theory Relat. Fields 共to be published兲. 1 2

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