Approximability of GROUND STATE Problem for Tridimensional Ising Spin Glasses Alberto Bertoni1 1

Paola Campadelli1

Roberto Posenato2

Massimo Santini1

Dipartimento di Scienze dell’Informazione, Universit`a degli Studi di Milano 2 Facolt` a di Scienze MM FF NN, Universit`a degli Studi di Verona

7th April 1998 RI-DSI 217-98

1 Introduction Spin glasses represent one of the most challenging problems for solid state and statistical physics. The prototype of a spin glass is a dilute magnetic alloy, such as 1% of Mn or Fe embedded in Cu or Au. Many models have been proposed to describe the behaviour of these systems. In this paper we refer to the Edwards-Anderson model where elements are placed on the vertices of a regular lattice, the magnetic interactions hold only for nearest neighbours [EA75] and every element has only two states (Ising spin glasses [Bar82]). One of the most interesting problems about this model is the determination of the minimal-energy states (GROUND STATE problem). Bieche et al. [BMRU80] solved in polynomial time the GROUND STATE problem for an Ising spin glass on a planar lattice, where the interactions can have only two values. Barahona [Bar82] proved that GROUND √ STATE is NP-hard even for the simple tridimensional Ising spin glass on a two-level planar grid with O( n) vertical connections, where n is the number of vertices. Barahona’s result makes it necessary to sacrifice optimality and look for approximation algorithms which run in polynomial time. For an Ising spin glass on a two-level grid such that the number of vertical connections is at most n γ (γ < 1), a polynomial-time approximation algorithm for the GROUND STATE problem with absolute error O(n γ ) has been given [BCM94, BCGP97]. Moreover, this algorithm has been proved to be optimal (up to a multiplicative constant) under the conjecture P6=NP as consequence of results on the approximability of NP-hard optimization problems [ALM+ 92] and the reducibility among them [PY91]. In this paper, we consider the case of arbitrary two-level grids. First of all, we discuss the design of “good” polynomial-time approximation algorithms for GROUND STATE. To be more precise, we prove that, for every ε > 0, there is a parallel algorithm on PRAM CRCW with absolute error O(n/(ε lg n)) and computation time O((n ε+1 lg n)/ p + lg n), where p is the number of processors. Then, we estimate lower bounds on the absolute error for polynomial-time approximation algorithms. Under the conjecture P 6= NP, we can only prove bounds of the kind ∞ (n γ ) (γ < 1) [BCGP97], so in order to obtain tighter lower bounds, we assume a weaker conjecture. We observe that, despite the efforts of the last 20 years, the best algorithms for SAT [MS85, Sch93] work in time 2cm (for some c > 0) and that every exact algorithm for SAT, designed using a wide class of techniques, takes 2∞ (m) time [Gal76], where m is the length of the formula. We relate the error of polynomial-time algorithms for GROUND STATE with the computation time required to solve SAT. We can state the lower bound ∞ (n/ lg2 n) on the absolute error for every polynomial-time approximation algorithm. In Section 2 we recall some basic definitions and in Section 3 we present the GROUND STATE problem. A polynomial-time approximation algorithm for GROUND STATE on arbitrary two-level grids is presented in Section 4. Results about lower bounds are given in Section 5. In Section 6 a reduction from SAT to GROUND STATE is shown.

2 Preliminary Definitions and Results In this section some basic definitions and a few recent results about the approximability of “difficult” combinatorial optimization problems are summarized. Definition 2.1 An NP optimization problem (NPO) 5 is defined by the quadruple hI n, Sol, w, T ypei where I n denotes the set of the instances, Sol is a mapping that, given an instance I ∈ I n, provides the set of feasible solutions, w is the objective function that associates a non negative rational number (solution value) with every couple hI, Si where I ∈ I n and S ∈ Sol(I ); finally T ype ∈ {max, min}. It is assumed that there is a “natural” notion of size |I | for every instance I and of size |S| for every feasible solution S ∈ Sol(I ) and it is required that: 1. the size of every feasible solution S ∈ Sol(I ) is polynomially bounded in the size of the instance I , i.e. there is a polynomial p such that: |S| ≤ p(|I |); 2. the predicate S ∈ Sol(I ) can be decided in time polynomial in |I | and |S|; 3. the objective function w can be computed in polynomial time. The optimal solution w∗ (I ) for a given instance I ∈ I n is defined, respectively according to T ype, as = max{w(I, S) : S ∈ Sol(I )} or w∗ (I ) = min{w(I, S) : S ∈ Sol(I )}. For any NPO problem different formulations can be given: a decision version, an evaluation version, and a constructive version [ACP95]. It is well known (see, for instance, [GJ79]) that, if the decision version of a maximization problem is NP-complete, there is no hope of finding “efficient” (i.e. polynomial-time) algorithms to solve both the constructive and the evaluation version exactly, unless P=NP. Thus, one can try and find only “good” approximate solutions in polynomial time; two measures of the quality of an approximate solution are the absolute error and the relative error. w∗ (I )

Definition 2.2 Given an NPO problem 5 = hI n, Sol, w, T ypei and I ∈ I n let S ∈ Sol(I ); the absolute error e(I, S) is: e(I, S) = |w∗ (I ) − w(I, S)|, and the relative error is:

w(I, S) w∗ (I ) . , err(I, S) = max w∗ (I ) w(I, S)

An approximation algorithm for a maximization problem 5 = hI n, Sol, wi is an algorithm A that, having as input a problem instance I , outputs a solution A(I ) ∈ Sol(I ). We say that A is an approximation algorithm of level ε > 0 (equivalently, an ε-approximation algorithm) if, for every instance I , err(I, A(I )) ≤ ε. We observe that in literature, as recalled in the NPO definition, the objective function w is restricted to assume only non negative values; if w assumes also negative values the given definition of relative error becames meaningless since, as can be easily shown, in some case better solutions, in the absolute error sense, give rise to worst relative error rates. Thus, when dealing with optimization problem with arbitraty objective functions, a different notion of relative error should be considered as, for instance err ′ (I, S) =

w(I, S) max S∈Sol( I ){w(I, S)} − min S∈Sol( I ){w(I, S)}

which relates the value of a given solution to the whole range of variation of w.

3 The GROUND STATE Problem A n × m two-level grid is a graph hV, Ei such that: • V = {1, . . . , n} × {1, . . . , m} × {1, 2} is the set of nodes, n, m ∈ N; 2

• every node in V can be seen as element in N3 ; • given nodes x and y, if {x, y} ∈ E then the euclidean distance between x and y is 1. The level l ∈ {1, 2} is the set of nodes of the type (x1 , x2 , l); an edge of the type {(x1, x2 , 1), (x1 , x2 , 2)} is called vertical edge (see Figure 1).

Figure 1 √ √ Consider an Ising spin glass on a n × n two-level grid hV, Ei with 2n vertices. To each node x ∈ V is associated a variable σx with values in {−1, 1} indicating the spin orientation; to each edge {x, y} ∈ E is associated a weight Jx y , chosen in the set {−1, 0, 1}, indicating the interactions between nearest-neighbour spins. In this way a weighted grid G = hV, E, Ji, where J : E → {−1, 0, 1}, is obtained. The energy of a spin configuration σ = [σ1 , . . . , σ2n ] is given by the hamiltonian X Jx,y σx σ y , HG (σ ) = − {x,y}∈E

and the ground states are those configurations which minimize HG .

Given 0 < γ ≤ 1, let Gγ be the class of the weighted grids G = hV, E, Ji just described, such that if 2n is the total number of nodes, the number of vertical edges in E is at most n γ . The problem of finding the ground state for weighted grids of the class Gγ is formally defined as follows: GROUND STATE(γ ) √ √ Instance: a n × n two-level weighted grid G = hV, E, Ji ∈ Gγ . Question: determine a spin configuration that minimizes the function HG : {−1, 1}2n → Z defined as X Jx,y σx σ y . HG (σ ) = − {x,y}∈E

Barahona has proved that the decision version of GROUND STATE(1/2) is NP-complete [Bar82]. This implies that, for γ ≥ 1/2, GROUND STATE(γ ) is NP-hard. Hence there is no polynomial-time exact algorithm for the problem, unless P=NP thus we focus our attention to polynomial-time approximation algorithms. Furthermore, since HG (seen as the objective function of the related NPO problem) can assume both positive and negative values, here we are interested in evaluating absolute error, or some “adapted” notion of relative error as the one introduced in the preceeding section. But as one can easily see max HG (σ ) − min HG (σ ) = 2(n) σ

σ

so we restrict our attention only to the absolute error to which the “adapted” relative error can immediatly be related. Let HG∗ denote the minimum energy value of a spin glass on the weighted grid G ∈ Gγ , i.e. HG∗ = minσ HG (σ ); given a polynomial-time approximation algorithm A for the GROUND STATE(γ ) problem, 3

we denote the spin configuration given by the algorithm A on input G by A(G) and the corresponding energy value by HG (A(G)). For every γ ≤ 1, a polynomial-time approximation algorithm for GROUND STATE(γ ) with absolute error O(n γ ) has been designed [BCM94, BCGP97]. Moreover, if γ < 1 this algorithm has been proved to be optimal (up to a multiplicative constant) under the conjecture P 6= NP, as conseguence of results on the approximability of NP-hard optimization problems [ALM+ 92] and the reducibility among them [PY91]. This result can not be extended to the case γ = 1, as we see in the next section where a polynomial-time approximation algorithm with sublinear absolute error O(n/ lg n) is designed.

4 An Upper Bound on the Error for GROUND STATE(1) In this section, we show a polynomial-time approximation algorithm for GROUND STATE(1) with a sublinear O(n/ lg n) bound on the error. Moreover, we prove that the problem can be solved at this level of error by an efficient parallel algorithm, implemented on the PRAM model [FW78]. Theorem 4.1 For all ε > 0, there exists an approximation parallel algorithm Aˆ on PRAM for GROUND ˆ with error STATE(1) that, for all the weighted grids G = hV, E, Ji of the class G1 , finds a solution A(G) ˆ |HG∗ − HG ( A(G))|

0. Given an integer h > 0, let fh : G1 → G1 be a function such that G ′ = fh (G) is a grid made of h 2 separated copies of the grid G. Given a configuration C of G ′ , we denote with 5s (C) the corresponding configuration of the s t h copy of G. Let A be the following algorithm for GROUND STATE(1): Algorithm A Input:

A two-level weighted grid G = hV, E, Ji ∈ G1 ;

Step 1:

G ′ := f h (G), where n = |V |/2 and h 2 = 2(2an)

1/k

/n;

Step 2: C := A′ (G ′ ); ˆ configuration 5s (C) such that HG (5s (C)) = min1≤ j ≤h 2 HG (5j (C)); Step 3: C:= ˆ Output: C. Let e(C) = |HG∗ ′ − HG′ (C)| and e(5j (C)) = |HG∗ − HG (5j (C))|, it holds that: 2

e(C) ≥

h X j =1

ˆ e(5j (C)) ≥ h 2 e(C).

Since, by hypothesis, e(C) ≤ anh 2 /(lg nh 2 )k , we conclude that: ˆ ≤ e(C)

an = (lg nh 2 )k

an lg 2(2an)

1/k

k =

1 < 1, 2

ˆ = 0. that is e(C) 1/k 1/k It’s easy to verify that Algorithm A works in time O(2l(2an) ), hence in time 2 O(n ) . Since, by hypothesis, SAT is reducible to GROUND STATE(1) by a function g such that |g(I )| = α/k O(|I |α ), the algorithm A can be used to solve SAT on instances I in 2 O(|I | ) time. Corollary 5.2 If every exact algorithm for SAT takes 2∞ (|I |) time and there is a polynomial-time reduction g from SAT to GROUND STATE with |g(I )| = O(|I |α ), then every polynomial-time approximation algorithm for GROUND STATE(1) has an absolute error ∞ (n/ lgα n). In the next section we show a reduction g from SAT to GROUND STATE(1) such that |g(I )| = O(|I |2) (i.e. α = 2). From the above results and the reduction shown in the next section we can therefore state the following Theorem 5.3 If every exact algorithm for SAT takes 2∞ (|I |), then every polynomial-time approximation algorithm for GROUND STATE(1) has an absolute error ∞ (n/ lg2 n). 6

6 A reduction from SAT to GROUND STATE(1) In this section we will sketch a polynomial time reduction from SAT to GROUND STATE. By an instance I of SAT we mean a formula in conjunctive normal form (CNF) over some set X of variables. For every x ∈ X, m I (x) P denotes the number of occurrences of x or x in I ; the dimension of the instance I is defined as: |I | = x∈X m I (x). With MAX 2-SAT we intend the related maximization problem in which each clause contains at most 2 literals; with MAX WEIGHTED CUT-3 we intend the maximization version of the MAX CUT problem restricted to weighted graphs of degree at most 3. Definition 6.1 Given an integer h ≥ 1, an amplifier A with h handles is a graph with O(h) nodes, a subset H of whom are told handles, such that: |H | = h, any cut hV1 , V2 i of A contains at least min(|V1 ∩ H |, |V2 ∩ H |) edges and every handle has degree at most 2 while any other node has degree at most 3. In [Pap94] it is proved that for any positive integer h an amplifier graph with h handles can be built in polynomial time. Lemma 6.2 There is a polynomial time reduction g2 from MAX 2-SAT to MAX WEIGHTED CUT-3 such that the graph g2 (I ) has weights in {−1, 1} and O(η|I |) nodes, where η is the maximum multiplicity taken over the variables of I . Proof: (sketch) This reduction goes through two steps. First we reduce MAX 2-SAT to MAX CUT for a graph with parallel edges and not bounded degree. Let G = (V, E) be the multigraph so defined: V contains a pair of vertex vx and vx for each variable x in I and a special node T . For each variable x in I , E contains 2m I (x) parallel edges {vx , vx }; for each clause {a, b} in I , E contains the edges: {va , vb }, {vb , T }, {T , va }; for each clause {a} in I , E contains two parallel edges {T , va }. We observe that the degree of T is O(|I |) and the degree of each other node is O(η) where η = maxx∈X m I (x). It is straightforward to verify [PY91] that there exist a truth assignment that satisfies q clauses of I iff there exist a cut of G with at least 2(|I | + q) edges. Now, by means of amplifier graphs, we make the second step of the reduction which transforms the multigraph G in a graph G ′ = (V ′ , E1′ ∪ E2′ ) of degree at most 3. For each node v of G we build an amplifier Av with as many handles as the degree of v; we then collect all the nodes of these amplifiers in V ′ and all their edges in E1′ . For each edge {u, v} in E, E2′ contains an edge between one handle of Au and one handle of Av so that each handle is present in exactly one edge of E2′ (this is made possible since the number of handles is related to the degree). We then assign weight −1 to all the edges in E1′ and weight 1 to all the edges in E2′ . Clearly G ′ has no parallel edges and every one of its nodes has degree at most 3; the amplifier for T has O(|I |) nodes and all other amplifiers, which are at most |I |, have O(η) nodes each, we can thus conclude that |V ′ | ≤ O(|I |) + |I |O(η) = O(η|I |). Finally the graph G has a cut of at least k edges iff the graph G ′ has a cut of weight at least k, in fact: (⇒) if hV1 , V2 i is a cut of G with at least k edges then the cut hV1′ , V2′ i of G ′ such that all the nodes of the amplifier Av are in Vi′ iff v ∈ Vi (i = 1, 2) clearly contains at last k edges from E2′ and no one form E1′ . (⇐) Given a cut hV1′ , V2′ i of G ′ consider for instance an amplifier with m 1 handles in V1′ and m 2 > m 1 handles in V2′ ; changing the position of the handles from V1′ to V2′ does not decrease the weight of the cut since we lose at most m 1 edges of weight 1 of E2′ but, for the amplifier properties, we also lose at least m 1 edges of weight −1 in E1′ . We can thus assume that in G ′ for each cut of fixed weight there is a corresponding cut of at least equal weight, but with the handles of each amplifier all in the same set. Let hV1′ , V2′ i such a cut with weight at least k (and so containing at least k edges of E2′ ) then the cut hV1 , V2 i of G such that v ∈ Vi iff all the nodes of the amplifier Av are in Vi′ (i = 1, 2) clearly contains at last k edges. In [Pap94] a polynomial-time reduction g1 from SAT to MAX 2-SAT such that g1 (I ) has dimension O(|I |) and all the variables have multiplicity at most 5 is presented; we can therefore conclude:

7

Lemma 6.3 There is a polynomial time reduction g ′ = g2 ◦ g1 from SAT to MAX WEIGHTED CUT-3 such that the graph g ′ (I ) has weights in {−1, 1} and O(|I |) nodes. Using a technique very close to that presented in [Bar82, Sec. 4.2] it is easy to obtain an embedding of the graph g ′ (I ) into a two-level grid with O(|I |2) nodes. This embedding together with the results of lemma 6.3 gives the reduction g. Since the dimension of the grid, where a graph with bounded degree can be embedded by such technique, is related to the square of the bipartition number of the graph, the quadratic factor of the reduction can not be easily lowered by means of a similar approach. It is an open problem to find a different technique to reduce MAX WEIGHTED CUT-3 to GROUND STATE.

References [ACP95]

G. Ausiello, P. Crescenzi, and M. Protasi. Approximate Solution of NP Optimization Problems. Theoretical Computer Science, (150):1–55, 1995.

[ALM+ 92] Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and intractability of approximation problems. In Proceedings of the 33rd Annual IEEE Symposium on the Foundations of Computer Science, pages 14–23. IEEE, 1992. [Bar82]

Francisco Barahona. On the computational complexity of Ising spin glass models. Journal Physic A: Math. Gen., (15):3241–3253, 1982.

[BCGP97] Alberto Bertoni, Paola Campadelli, Cristina Gangai, and Roberto Posenato. Approximability of the Ground State Problem for Certain Ising Spin Glasses. Journal of Complexity (13):326– 339 (1997). [BCM94]

Alberto Bertoni, Paola Campadelli, and Giovanna Molteni. On the approximability of the energy function of Ising spin glasses. Journal of Physic A: Math. Gen., (27):6719–6729, 1994.

[BMRU80] I. Bieche, R. Maynard, R. Rammal, and J.P. Uhry. On the ground states of the frustration model of a spin glass by a matching method of graph theory. Journal of Physic A: Math. Gen., (13):2553–2576, 1980. [EA75]

S. F. Edwards and P. W. Anderson. Theory of Spin Glasses. Journal of Physics F., (5):965–978, 1975.

[FW78]

S. Fortune and J. Wyllie. Parallelism in random access machines. In Proceedings Tenth Annual ACM Symposium on Theory of Computing, pages 114–118, New York, 1978. ACM, ACM.

[Gal76]

Zvi Galil. On enumeration procedures for theorem proving and for integer programming. In International Conference on Automata, Languages and Programming, pages 355–381. Edinburgh University Press, 1976.

[GJ79]

Micheal R. Garey and David S. Johnson. Computers and Intractability. A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., San Francisco, 1979.

[HPR92]

Y. Han, V. Pan, and J. Reif. Efficient Parallel Algorithms for Computing All Pair Shortest Paths in Directed Graphs. In Symposium on Parallel Algorithms and Architectures ’92, pages 353–362. ACM, ACM, 1992.

[MS85]

B. Monien and E. Speckenmeyer. Solving satisfiability in less than 2n steps. Journal of Discrete Applied Mathematics, (10):287–295, 1985.

[Pap94]

Christos H. Papadimitriou. Computational Complexity. Addison-Wesley Publishing Company, Reading, Massachusetts, 1994. 8

[PY91]

Christos H. Papadimitriou and Mihalis Yannakakis. Optimization, Approximation, and Complexity Classes. Journal of Computer and System Sciences, (43):425–440, 1991.

[Sch93]

Ingo Schiermeyer. Solving 3-satisfiability in less than 1, 579n steps. In Computer Scienze Logic. 6th workshop, pages 379–394. Springer-Verlag, 1993.

9

Paola Campadelli1

Roberto Posenato2

Massimo Santini1

Dipartimento di Scienze dell’Informazione, Universit`a degli Studi di Milano 2 Facolt` a di Scienze MM FF NN, Universit`a degli Studi di Verona

7th April 1998 RI-DSI 217-98

1 Introduction Spin glasses represent one of the most challenging problems for solid state and statistical physics. The prototype of a spin glass is a dilute magnetic alloy, such as 1% of Mn or Fe embedded in Cu or Au. Many models have been proposed to describe the behaviour of these systems. In this paper we refer to the Edwards-Anderson model where elements are placed on the vertices of a regular lattice, the magnetic interactions hold only for nearest neighbours [EA75] and every element has only two states (Ising spin glasses [Bar82]). One of the most interesting problems about this model is the determination of the minimal-energy states (GROUND STATE problem). Bieche et al. [BMRU80] solved in polynomial time the GROUND STATE problem for an Ising spin glass on a planar lattice, where the interactions can have only two values. Barahona [Bar82] proved that GROUND √ STATE is NP-hard even for the simple tridimensional Ising spin glass on a two-level planar grid with O( n) vertical connections, where n is the number of vertices. Barahona’s result makes it necessary to sacrifice optimality and look for approximation algorithms which run in polynomial time. For an Ising spin glass on a two-level grid such that the number of vertical connections is at most n γ (γ < 1), a polynomial-time approximation algorithm for the GROUND STATE problem with absolute error O(n γ ) has been given [BCM94, BCGP97]. Moreover, this algorithm has been proved to be optimal (up to a multiplicative constant) under the conjecture P6=NP as consequence of results on the approximability of NP-hard optimization problems [ALM+ 92] and the reducibility among them [PY91]. In this paper, we consider the case of arbitrary two-level grids. First of all, we discuss the design of “good” polynomial-time approximation algorithms for GROUND STATE. To be more precise, we prove that, for every ε > 0, there is a parallel algorithm on PRAM CRCW with absolute error O(n/(ε lg n)) and computation time O((n ε+1 lg n)/ p + lg n), where p is the number of processors. Then, we estimate lower bounds on the absolute error for polynomial-time approximation algorithms. Under the conjecture P 6= NP, we can only prove bounds of the kind ∞ (n γ ) (γ < 1) [BCGP97], so in order to obtain tighter lower bounds, we assume a weaker conjecture. We observe that, despite the efforts of the last 20 years, the best algorithms for SAT [MS85, Sch93] work in time 2cm (for some c > 0) and that every exact algorithm for SAT, designed using a wide class of techniques, takes 2∞ (m) time [Gal76], where m is the length of the formula. We relate the error of polynomial-time algorithms for GROUND STATE with the computation time required to solve SAT. We can state the lower bound ∞ (n/ lg2 n) on the absolute error for every polynomial-time approximation algorithm. In Section 2 we recall some basic definitions and in Section 3 we present the GROUND STATE problem. A polynomial-time approximation algorithm for GROUND STATE on arbitrary two-level grids is presented in Section 4. Results about lower bounds are given in Section 5. In Section 6 a reduction from SAT to GROUND STATE is shown.

2 Preliminary Definitions and Results In this section some basic definitions and a few recent results about the approximability of “difficult” combinatorial optimization problems are summarized. Definition 2.1 An NP optimization problem (NPO) 5 is defined by the quadruple hI n, Sol, w, T ypei where I n denotes the set of the instances, Sol is a mapping that, given an instance I ∈ I n, provides the set of feasible solutions, w is the objective function that associates a non negative rational number (solution value) with every couple hI, Si where I ∈ I n and S ∈ Sol(I ); finally T ype ∈ {max, min}. It is assumed that there is a “natural” notion of size |I | for every instance I and of size |S| for every feasible solution S ∈ Sol(I ) and it is required that: 1. the size of every feasible solution S ∈ Sol(I ) is polynomially bounded in the size of the instance I , i.e. there is a polynomial p such that: |S| ≤ p(|I |); 2. the predicate S ∈ Sol(I ) can be decided in time polynomial in |I | and |S|; 3. the objective function w can be computed in polynomial time. The optimal solution w∗ (I ) for a given instance I ∈ I n is defined, respectively according to T ype, as = max{w(I, S) : S ∈ Sol(I )} or w∗ (I ) = min{w(I, S) : S ∈ Sol(I )}. For any NPO problem different formulations can be given: a decision version, an evaluation version, and a constructive version [ACP95]. It is well known (see, for instance, [GJ79]) that, if the decision version of a maximization problem is NP-complete, there is no hope of finding “efficient” (i.e. polynomial-time) algorithms to solve both the constructive and the evaluation version exactly, unless P=NP. Thus, one can try and find only “good” approximate solutions in polynomial time; two measures of the quality of an approximate solution are the absolute error and the relative error. w∗ (I )

Definition 2.2 Given an NPO problem 5 = hI n, Sol, w, T ypei and I ∈ I n let S ∈ Sol(I ); the absolute error e(I, S) is: e(I, S) = |w∗ (I ) − w(I, S)|, and the relative error is:

w(I, S) w∗ (I ) . , err(I, S) = max w∗ (I ) w(I, S)

An approximation algorithm for a maximization problem 5 = hI n, Sol, wi is an algorithm A that, having as input a problem instance I , outputs a solution A(I ) ∈ Sol(I ). We say that A is an approximation algorithm of level ε > 0 (equivalently, an ε-approximation algorithm) if, for every instance I , err(I, A(I )) ≤ ε. We observe that in literature, as recalled in the NPO definition, the objective function w is restricted to assume only non negative values; if w assumes also negative values the given definition of relative error becames meaningless since, as can be easily shown, in some case better solutions, in the absolute error sense, give rise to worst relative error rates. Thus, when dealing with optimization problem with arbitraty objective functions, a different notion of relative error should be considered as, for instance err ′ (I, S) =

w(I, S) max S∈Sol( I ){w(I, S)} − min S∈Sol( I ){w(I, S)}

which relates the value of a given solution to the whole range of variation of w.

3 The GROUND STATE Problem A n × m two-level grid is a graph hV, Ei such that: • V = {1, . . . , n} × {1, . . . , m} × {1, 2} is the set of nodes, n, m ∈ N; 2

• every node in V can be seen as element in N3 ; • given nodes x and y, if {x, y} ∈ E then the euclidean distance between x and y is 1. The level l ∈ {1, 2} is the set of nodes of the type (x1 , x2 , l); an edge of the type {(x1, x2 , 1), (x1 , x2 , 2)} is called vertical edge (see Figure 1).

Figure 1 √ √ Consider an Ising spin glass on a n × n two-level grid hV, Ei with 2n vertices. To each node x ∈ V is associated a variable σx with values in {−1, 1} indicating the spin orientation; to each edge {x, y} ∈ E is associated a weight Jx y , chosen in the set {−1, 0, 1}, indicating the interactions between nearest-neighbour spins. In this way a weighted grid G = hV, E, Ji, where J : E → {−1, 0, 1}, is obtained. The energy of a spin configuration σ = [σ1 , . . . , σ2n ] is given by the hamiltonian X Jx,y σx σ y , HG (σ ) = − {x,y}∈E

and the ground states are those configurations which minimize HG .

Given 0 < γ ≤ 1, let Gγ be the class of the weighted grids G = hV, E, Ji just described, such that if 2n is the total number of nodes, the number of vertical edges in E is at most n γ . The problem of finding the ground state for weighted grids of the class Gγ is formally defined as follows: GROUND STATE(γ ) √ √ Instance: a n × n two-level weighted grid G = hV, E, Ji ∈ Gγ . Question: determine a spin configuration that minimizes the function HG : {−1, 1}2n → Z defined as X Jx,y σx σ y . HG (σ ) = − {x,y}∈E

Barahona has proved that the decision version of GROUND STATE(1/2) is NP-complete [Bar82]. This implies that, for γ ≥ 1/2, GROUND STATE(γ ) is NP-hard. Hence there is no polynomial-time exact algorithm for the problem, unless P=NP thus we focus our attention to polynomial-time approximation algorithms. Furthermore, since HG (seen as the objective function of the related NPO problem) can assume both positive and negative values, here we are interested in evaluating absolute error, or some “adapted” notion of relative error as the one introduced in the preceeding section. But as one can easily see max HG (σ ) − min HG (σ ) = 2(n) σ

σ

so we restrict our attention only to the absolute error to which the “adapted” relative error can immediatly be related. Let HG∗ denote the minimum energy value of a spin glass on the weighted grid G ∈ Gγ , i.e. HG∗ = minσ HG (σ ); given a polynomial-time approximation algorithm A for the GROUND STATE(γ ) problem, 3

we denote the spin configuration given by the algorithm A on input G by A(G) and the corresponding energy value by HG (A(G)). For every γ ≤ 1, a polynomial-time approximation algorithm for GROUND STATE(γ ) with absolute error O(n γ ) has been designed [BCM94, BCGP97]. Moreover, if γ < 1 this algorithm has been proved to be optimal (up to a multiplicative constant) under the conjecture P 6= NP, as conseguence of results on the approximability of NP-hard optimization problems [ALM+ 92] and the reducibility among them [PY91]. This result can not be extended to the case γ = 1, as we see in the next section where a polynomial-time approximation algorithm with sublinear absolute error O(n/ lg n) is designed.

4 An Upper Bound on the Error for GROUND STATE(1) In this section, we show a polynomial-time approximation algorithm for GROUND STATE(1) with a sublinear O(n/ lg n) bound on the error. Moreover, we prove that the problem can be solved at this level of error by an efficient parallel algorithm, implemented on the PRAM model [FW78]. Theorem 4.1 For all ε > 0, there exists an approximation parallel algorithm Aˆ on PRAM for GROUND ˆ with error STATE(1) that, for all the weighted grids G = hV, E, Ji of the class G1 , finds a solution A(G) ˆ |HG∗ − HG ( A(G))|

0. Given an integer h > 0, let fh : G1 → G1 be a function such that G ′ = fh (G) is a grid made of h 2 separated copies of the grid G. Given a configuration C of G ′ , we denote with 5s (C) the corresponding configuration of the s t h copy of G. Let A be the following algorithm for GROUND STATE(1): Algorithm A Input:

A two-level weighted grid G = hV, E, Ji ∈ G1 ;

Step 1:

G ′ := f h (G), where n = |V |/2 and h 2 = 2(2an)

1/k

/n;

Step 2: C := A′ (G ′ ); ˆ configuration 5s (C) such that HG (5s (C)) = min1≤ j ≤h 2 HG (5j (C)); Step 3: C:= ˆ Output: C. Let e(C) = |HG∗ ′ − HG′ (C)| and e(5j (C)) = |HG∗ − HG (5j (C))|, it holds that: 2

e(C) ≥

h X j =1

ˆ e(5j (C)) ≥ h 2 e(C).

Since, by hypothesis, e(C) ≤ anh 2 /(lg nh 2 )k , we conclude that: ˆ ≤ e(C)

an = (lg nh 2 )k

an lg 2(2an)

1/k

k =

1 < 1, 2

ˆ = 0. that is e(C) 1/k 1/k It’s easy to verify that Algorithm A works in time O(2l(2an) ), hence in time 2 O(n ) . Since, by hypothesis, SAT is reducible to GROUND STATE(1) by a function g such that |g(I )| = α/k O(|I |α ), the algorithm A can be used to solve SAT on instances I in 2 O(|I | ) time. Corollary 5.2 If every exact algorithm for SAT takes 2∞ (|I |) time and there is a polynomial-time reduction g from SAT to GROUND STATE with |g(I )| = O(|I |α ), then every polynomial-time approximation algorithm for GROUND STATE(1) has an absolute error ∞ (n/ lgα n). In the next section we show a reduction g from SAT to GROUND STATE(1) such that |g(I )| = O(|I |2) (i.e. α = 2). From the above results and the reduction shown in the next section we can therefore state the following Theorem 5.3 If every exact algorithm for SAT takes 2∞ (|I |), then every polynomial-time approximation algorithm for GROUND STATE(1) has an absolute error ∞ (n/ lg2 n). 6

6 A reduction from SAT to GROUND STATE(1) In this section we will sketch a polynomial time reduction from SAT to GROUND STATE. By an instance I of SAT we mean a formula in conjunctive normal form (CNF) over some set X of variables. For every x ∈ X, m I (x) P denotes the number of occurrences of x or x in I ; the dimension of the instance I is defined as: |I | = x∈X m I (x). With MAX 2-SAT we intend the related maximization problem in which each clause contains at most 2 literals; with MAX WEIGHTED CUT-3 we intend the maximization version of the MAX CUT problem restricted to weighted graphs of degree at most 3. Definition 6.1 Given an integer h ≥ 1, an amplifier A with h handles is a graph with O(h) nodes, a subset H of whom are told handles, such that: |H | = h, any cut hV1 , V2 i of A contains at least min(|V1 ∩ H |, |V2 ∩ H |) edges and every handle has degree at most 2 while any other node has degree at most 3. In [Pap94] it is proved that for any positive integer h an amplifier graph with h handles can be built in polynomial time. Lemma 6.2 There is a polynomial time reduction g2 from MAX 2-SAT to MAX WEIGHTED CUT-3 such that the graph g2 (I ) has weights in {−1, 1} and O(η|I |) nodes, where η is the maximum multiplicity taken over the variables of I . Proof: (sketch) This reduction goes through two steps. First we reduce MAX 2-SAT to MAX CUT for a graph with parallel edges and not bounded degree. Let G = (V, E) be the multigraph so defined: V contains a pair of vertex vx and vx for each variable x in I and a special node T . For each variable x in I , E contains 2m I (x) parallel edges {vx , vx }; for each clause {a, b} in I , E contains the edges: {va , vb }, {vb , T }, {T , va }; for each clause {a} in I , E contains two parallel edges {T , va }. We observe that the degree of T is O(|I |) and the degree of each other node is O(η) where η = maxx∈X m I (x). It is straightforward to verify [PY91] that there exist a truth assignment that satisfies q clauses of I iff there exist a cut of G with at least 2(|I | + q) edges. Now, by means of amplifier graphs, we make the second step of the reduction which transforms the multigraph G in a graph G ′ = (V ′ , E1′ ∪ E2′ ) of degree at most 3. For each node v of G we build an amplifier Av with as many handles as the degree of v; we then collect all the nodes of these amplifiers in V ′ and all their edges in E1′ . For each edge {u, v} in E, E2′ contains an edge between one handle of Au and one handle of Av so that each handle is present in exactly one edge of E2′ (this is made possible since the number of handles is related to the degree). We then assign weight −1 to all the edges in E1′ and weight 1 to all the edges in E2′ . Clearly G ′ has no parallel edges and every one of its nodes has degree at most 3; the amplifier for T has O(|I |) nodes and all other amplifiers, which are at most |I |, have O(η) nodes each, we can thus conclude that |V ′ | ≤ O(|I |) + |I |O(η) = O(η|I |). Finally the graph G has a cut of at least k edges iff the graph G ′ has a cut of weight at least k, in fact: (⇒) if hV1 , V2 i is a cut of G with at least k edges then the cut hV1′ , V2′ i of G ′ such that all the nodes of the amplifier Av are in Vi′ iff v ∈ Vi (i = 1, 2) clearly contains at last k edges from E2′ and no one form E1′ . (⇐) Given a cut hV1′ , V2′ i of G ′ consider for instance an amplifier with m 1 handles in V1′ and m 2 > m 1 handles in V2′ ; changing the position of the handles from V1′ to V2′ does not decrease the weight of the cut since we lose at most m 1 edges of weight 1 of E2′ but, for the amplifier properties, we also lose at least m 1 edges of weight −1 in E1′ . We can thus assume that in G ′ for each cut of fixed weight there is a corresponding cut of at least equal weight, but with the handles of each amplifier all in the same set. Let hV1′ , V2′ i such a cut with weight at least k (and so containing at least k edges of E2′ ) then the cut hV1 , V2 i of G such that v ∈ Vi iff all the nodes of the amplifier Av are in Vi′ (i = 1, 2) clearly contains at last k edges. In [Pap94] a polynomial-time reduction g1 from SAT to MAX 2-SAT such that g1 (I ) has dimension O(|I |) and all the variables have multiplicity at most 5 is presented; we can therefore conclude:

7

Lemma 6.3 There is a polynomial time reduction g ′ = g2 ◦ g1 from SAT to MAX WEIGHTED CUT-3 such that the graph g ′ (I ) has weights in {−1, 1} and O(|I |) nodes. Using a technique very close to that presented in [Bar82, Sec. 4.2] it is easy to obtain an embedding of the graph g ′ (I ) into a two-level grid with O(|I |2) nodes. This embedding together with the results of lemma 6.3 gives the reduction g. Since the dimension of the grid, where a graph with bounded degree can be embedded by such technique, is related to the square of the bipartition number of the graph, the quadratic factor of the reduction can not be easily lowered by means of a similar approach. It is an open problem to find a different technique to reduce MAX WEIGHTED CUT-3 to GROUND STATE.

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[ALM+ 92] Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and intractability of approximation problems. In Proceedings of the 33rd Annual IEEE Symposium on the Foundations of Computer Science, pages 14–23. IEEE, 1992. [Bar82]

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