Approximability of GROUND STATE Problem on Tridimensional Ising Spin Glasses (extended abstract)
Alberto Bertoni, Paola Campadelli Roberto Posenatoy and Massimo Santini
Dipartimento di Scienze dell'Informazione Universita degli Studi di Milano via Comelico, 39/41 I-20135 Milano (Italy)
1 Introduction and basic de nitions 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-levels planar grid with O(pn) vertical connections, where n is the number of vertices; this result can help in understanding why spin glasses have very long relaxation time. If P 6= NP,
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Barahona's result makes it necessary to sacri ce optimality and look for approximation algorithms which run in polynomial time. In this paper, we consider the GROUND STATE problem for arbitrary twolevels grids; we discuss the design of \good" polynomial-time approximation algorithms and estimate lower bounds of the absolute error made by polynomial-time algorithms. We now start giving some basic de nitions then will introduce the GROUND STATE problem. (For notational simplicity, de nitions are given only for maximization problems.)
De nition 1.1 A maximization problem is de ned by the triple hIn; Sol; wi where In denotes the set of the instances, Sol is a mapping that, given an instance I 2 In, provides the set of feasible solutions (such that the size jS j of every feasible solution S 2 Sol(I) is polynomially bounded respect to jI j and the predicate S 2 Sol(I) can be decided in polynomial time respect to jI j and jS j), w is the objective function (computable in polynomial time) that associates a non negative rational number (solution value) with every couple hI; S i where I 2 In and S 2 Sol(I). (It is assumed that there is a \natural" notion of size jI j for every instance I and of size jS j for every feasible solution S 2 Sol(I).) If a maximization problem is NP-hard, then one can try and nd only \good" approximate solutions in polynomial time, unless P = NP; two measures of the quality of an approximate solution are the absolute error and the relative error.
De nition 1.2 Given a maximization problem = hIn; Sol; wi, let S 2 Sol(I); the absolute error e(I; S) is: e(I; S) = w (I) ? w(I; S) and the relative error is: ) err(I; S) = ew(I;S (I ) where w (I) = maxS 2Sol(I ) fw(I; S)g. An approximation algorithm for a maximization problem = hIn; Sol; wi is an algorithm A that, having as input a problem instance I, outputs a solution A(I) 2 Sol(I). We say that A is an approximation algorithm of level " > 0 (equivalently, an "-approximation algorithm) if, for every instance I, err(I; A(I)) ". Let us now present a particular structure of spin glasses that it's used to de ne the GROUND STATE problem.
De nition 1.3 A n m two-levels grid (n; m 2 N) is a graph hV; Ei with V = f1; : : :; ng f1; : : :; mg f1; 2g and E such that fx; yg 2 E only if the Euclidean distance between the nodes x and y, seen as elements of N3 , is 1. The level l 2 f1; 2g is the set of nodes of the type (x1 ; x2; l); an edge of the type f(x1; x2; 1); (x1; x2; 2)g is called vertical edge.
p Consider an Ising spin glass on a pn n two-levels grid hV; E i with 2n vertices. With each node x 2 V , there is associated a variable x with values in f?1; 1g indicating the spin orientation; with each edge fx; yg 2 E there is
associated a weight Jxy , chosen in the set f?1; 0; 1g, indicating the interactions between nearest-neighbour spins. In this way a weighted grid G = hV; E; J i, where J : E ! f?1; 0; 1g, is obtained. P The energy of a spin con guration is given by the Hamiltonian HG() = ? fx;yg2E Jxy xy and the ground states are those con gurations which minimize HG . Given 0 < 1, let G be the class of the weighted grids G = hV; E; J i just described, such that if 2n is the total number of nodes, the number of vertical edges in E is at most n . Finally, the problem of nding the ground state for weighted grids of the class G can be formally de ned as follows: GROUND STATE( ) Instance: a pn pn two-levels weighted grid G = hV; E; J i 2 G . Question: determine a spin con guration that minimizes the function HG : f?1; 1g2n ! Z de ned as HG() = ?
X
fx;yg2E
Jx;y xy :
Let HG denote the minimum energy value of a spin glass on the weighted grid G 2 G , i.e. HG = min HG(); given a polynomial-time approximation algorithm A for the GROUND STATE( ) problem, we denote the spin con guration given by the algorithm A on input G by A(G) and the corresponding energy value by HG(A(G)).
2 Results We recall that Barahona has proved the decision version of GROUND STATE( 21 ) is NP-complete [Bar82]. This implies that, for 12 , GROUND STATE( ) is NPhard; hence there is no polynomial-time exact algorithm for the problem, unless P = NP. For these reasons, in the following we are interested in estimating bounds on the absolute error of polynomial-time approximation algorithms for this problem. For every 1, a polynomial-time approximation algorithm for GROUND STATE( ) with absolute error O(n ) has been designed [BCM94], [BCGP95]. Moreover, if < 1 this algorithm has been proved to be optimal (up to a multiplicative constant) under the conjecture P 6= NP, as consequence of results on the approximability of NP-hard optimization problems [ALM+ 92] and the reducibility among them [PY91]. This optimality result can not be extended to the case = 1 for which a polynomial-time approximation algorithm with sub linear absolute error can be designed. In particular, we prove that the problem can be solved at this level of error by an ecient parallel algorithm, implemented on the PRAM model [FW78].
Theorem 2.1 For all " > 0, there exists an approximation parallel algorithm A^ on PRAM for GROUND STATE(1) that, for all the weighted grids G = hV; E; J i of ^ with error the class G1, nds a solution A(G) ^ j < 13 jV j : jHG ? HG(A(G)) " lg jV j
jV j"+1 lg jV j
+ lg jV j where p is the number of processors. p Techniques used to obtain the lower bound results in [Bar82] and in [BCM94] are based on the conjecture P 6= NP. In order to obtain tighter lower bounds for the case = 1, we assume a weaker conjecture. We observe that, despite the e orts of the last 20 years, the best algorithms for SAT [MS85, Sch93] work in time 2cjI j (c > 0) and that every exact algorithm for SAT, designed using a wide class of techniques, takes 2 1 (jI j) time [Gal76], where I is an instance of SAT. We then relate the error of polynomial-time algorithms for GROUND STATE with the computation time required to solve SAT as stated in the next: Theorem 2.2 Let be > 0 such that there is a polynomial-time reduction g from SAT to GROUND STATE with jg(I)j = O(jI j ), where I is an instance of SAT. If there is a polynomial-time approximation algorithm for GROUND STATE(1) with k) n O ( j I j . absolute error O( (lg n)k ) then SAT is solvable in time 2 Putting together the assumed conjecture on SAT and this \time-error" relation we obtain the following: Corollary 2.3 If every exact algorithm for SAT takes 2 1(jIj) time, where I is an instance of SAT, and there is a polynomial-time reduction g from SAT to GROUND STATE with jg(I)j = O(jI j ), then every polynomial-time approximation algorithm for GROUND STATE(1) has an absolute error 1 ( (lgnn) ). Then we design a particular reduction g^ from SAT to GROUND STATE(1) such that jg^(I)j = O(jI j2) (i.e. = 2). From the above results we can therefore state the last: Theorem 2.4 If every exact algorithm for SAT takes 2 1(jIj) , where I is in time O
an instance of SAT, then every polynomial-time approximation algorithm for GROUND STATE(1) has an absolute error 1 ( (lgnn)2 ).
In the reduction g^, we use a technique very close to that presented in [Bar82] for embedding a graph with bounded degree into a two layers grid. Since the dimension of the grid 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 similar approach. It is an open problem to nd a di erent technique to reduce MAX WEIGHTED CUT-3 to GROUND STATE.
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