Optimization hardness as transient chaos in an analog approach to constraint satisfaction

Boolean satisfiability [1] (k-SAT) is one of the most studied optimization problems, as an efficient (that is, polynomial-time) solution to k-SAT (for k ≥ 3) implies efficient solutions to a large number of hard optimization problems [2, 3]. Here we propose a mapping of k-SAT into a deterministic continuous-time dynamical system with a unique correspondence between its attractors and the k-SAT solution clusters. We show that beyond a constraint density threshold, the analog trajectories become transiently chaotic [4, 5, 6, 7], and the boundaries between the basins of attraction [8] of the solution clusters become fractal [7, 8, 9], signaling the appearance of optimization hardness [10]. Analytical arguments and simulations indicate that the system always finds solutions for satisfiable formulae even in the frozen regimes of random 3-SAT [11] and of locked occupation problems [12] (considered among the hardest algorithmic benchmarks); a property partly due to the systems hyperbolic [4, 13] character. The system finds solutions in polynomial continuous-time, however, at the expense of exponential fluctuations in its energy function. ∗The article appeared in Nature Physics 7, 966 (2011) †E-mail: [email protected] ‡E-mail: [email protected] 1 ar X iv :1 20 8. 05 26 v1 [ cs .C C ] 2 A ug 2 01 2 M. Ercsey-Ravasz and Z. Toroczkai Optimization hardness as transient chaos ... Boolean satisfiability [1] (k-SAT, k ≥ 3) is the quintessential constraint satisfaction problem, lying at the basis of many decision, scheduling, error-correction and bio-computational applications. k-SAT is in NP, that is its solutions are efficiently (polynomial time) checkable, but no efficient (polynomial time) algorithms are known to compute those solutions [2]. If such algorithms would be found for k-SAT, all NP problems would be efficiently computable, since k-SAT is NP-complete [2, 3]. In k-SAT there are given N Boolean variables {x1, . . . , xN}, xi ∈ {0, 1} and M clauses (constraints), each clause being the disjunction (OR, denoted as ∨) of k variables or their negation (x). One has to find an assignment of the variables such that all clauses (called collectively as a formula) are satisfied (TRUE = 1). When the number of constraints is small, it is easy to find solutions, while for too many constraints it is easy to decide that the formula is unsatisfiable (UNSAT). Deciding satisfiability, in the ’intermediate range’, however, can be very hard: the worst-case complexity of all known algorithms for k-SAT is exponential in N . Inspired by the mechanisms of information processing in biological systems, analog computing received increasing interest from both theoretical [14, 15, 16] and engineering communities [17, 18, 19, 20, 21]. Although the theoretical possibility of efficient computation via chaotic dynamical systems has been shown previously [15], nonlinear dynamical systems theory has not been exploited for NP-complete problems in spite of the fact that, as shown by Gu et al.[19], Nagamatu et al. [20] and Wah et al. [21], k-SAT can be formulated as a continuous global optimization problem[19], and even cast as an analog dynamical system [20, 21]. Here we present a novel continuous-time dynamical system for k-SAT, with a dynamics that is rather different from previous approaches. Let us introduce the continuous variables [19] si ∈ [−1, 1] , such that si = −1 if the i-th variable (xi) is FALSE and si = 1 if it is TRUE. We define cmi = 1 for the direct form (xi), cmi = −1 for the negated form (xi), and cmi = 0 for the absence of the i-th variable from clause m. Defining the constraint function Km(s) ≡ 2−k ∏N i=1(1 − cmisi) corresponding to clause m, we have Km ∈ [0, 1] and Km = 0 if and only if clause m is satisfied. The goal would be to find a solution s∗ with si ∈ {−1, 1} to E(s∗) = 0, where E is the energy function E(s) = ∑M m=1Km(s) 2. If such s∗ exists, it will be a global minimum for E and a solution to the k-SAT problem. However, finding s∗ by a direct minimization of E(s) will typically fail due to non-solution attractors trapping the search dynamics. In order to avoid such traps, here we define a modified energy function V (s, a) = ∑M m=1 amKm(s) 2, using auxiliary variables am ∈ (0,∞) similar to Lagrange multipliers [20, 21]. Let us denote by HN the continuous domain [−1, 1]N . Its boundary is the N -hypercube QN = ∂HN with vertex set VN = {−1, 1}N ⊂ QN . The set of solutions for a given k-SAT formula, called solution space, occupies a subset of VN . Solution clusters are formed by solutions that can be connected via single-variable flips, always staying within satisfying assignments [22]. Clearly, V ≥ 0 in Ω ≡ HN × (0,∞)M , and V (s, a) = 0 within VN if and only if s = s∗ ∈ VN is a k-SAT solution, for any a ∈ (0,∞)M . We now introduce a continuous-time dynamical system on Ω through: dsi dt = (−∇sV (s,a))i = M ∑ m=1 2amcmiKmi(s)Km(s) , i = 1, . . . , N , (1) dam dt = amKm(s) , m = 1, . . . ,M , (2)