Phase Transitions and all that

Since the experimental paper of Cheeseman, Kanefsky and Taylor [1] phase transitions in combinatorial problems held the promise to shed light on the “practical” algorithmic complexity of combinatorial problems. However, the connection conjectured in [1] was easily seen to be inaccurate. A much more realistic possible connection has been highlighted by the results (based on experimental evidence and nonrigorous arguments from Statistical Mechanics) of Monasson et al. [2] (see also [3]). These results supported the conjecture that it is first-order phase transitions that have algorithmic implications for the complexity of restricted classes of algorithms, including the important class of Davis-Putnam-Longman-Loveland (DPLL) algorithms [4]. There exists, indeed, a nonrigorous argument supporting this conjecture: phase transitions amount to nonanalytical behavior of a certain order parameter; the phase transition is first order if the order parameter is actually discontinuous. At least for random k-SAT [5] the order parameter suggested by Statistical Mechanics considerations has a purely combinatorial interpretation: it is the backbone of the formula, the set of literals that assume the same value in all optimal assignments. But intuitively one can relate (see e.g. the presentation of this argument by Achlioptas, Beame and Molloy [6]) the size of the backbone to the complexity of DPLL algorithms, when run on random k-SAT instances slightly above the phase transition: All literals in the backbone require well-defined values in order to satisfy the formula. But a DPLL algorithm has very few ways to know what those “right” values are. If w.h.p. the backbone of formulas above the transition contains a positive fraction of the literals that is bounded away from zero as we approach the transition (which happens in a case of a first-order phase transition) then, intuitively, DPLL will misassign a variable having Ω(n) height in the tree representing the behavior of the algorithm, and will be forced to backtrack on the given variable. The conclusion of this intuitive argument is that a first-order phase transition implies a 2Ω(n) lower bound for the running time of any DPLL algorithm, valid with high probability for random instances located slightly above the transition. While previous rigorous results [7, 8, 9], supported these intuitions, to date, the extent of a connection between first-order phase transitions and algorithmic complexity was unclear. The goals of this paper are