We identify a new class of hard 3-SAT instances, namely a random 3-SAT problems having exactly one solution and as few clauses as possible. It is numerically shown that the running time of complete methods as well as of local search algorithms for such problems is larger than for random instances around the phase transition point. We therefore provide instances with an exponential complexity in the so-called easy'' region, below the critical value of m n. This puts a new light on the connection between the phase transition phenomenon and NP-completeness.
We present a detailed experimental investigation of the easy-hard-easy phase transition for randomly generated instances of satisfiability problems. Problems in the hard part of the phase transition have been extensively used for benchmarking satisfiability algorithms. This study demonstrates that problem classes and regions of the phase transition previously thought to be easy can sometimes be orders of magnitude more difficult than the worst problems in problem classes and regions of the phase transition considered hard. These difficult problems are either hard unsatisfiable problems or are satisfiable problems which give a hard unsatisfiable subproblem following a wrong split. Whilst these hard unsatisfiable problems may have short proofs, these appear to be difficult to find, and other proofs are long and hard. WC study the complexity of proving unsatisfiability for random k-CNP formulas with clause density A = m n where 111 is number of clauses and n is the number of variables. We prove the first nontrivial general upper bound, giving algorithmo that, in particular, for k = 3 produce refutations almost certainly in time 20t”iA). This is polynomial when  For every choice of positive integers c and k such that k ≥ 3 and c 2 - k ≥ 0.7, there is a positive number e such that, with probability tending to 1 as n tends to ∞, a randomly chosen family of cn clauses of size k over n variables is unsatisfiable, but every resolution proof of its unsatisfiability must generate at least (1 + e) n clauses. Abstract The distribution of hard graph coloring problems as a function of graph connectivity is shown to have two distinct transition behaviors. The first, previously recognized, is a peak in the median search cost near the connectivity at which half the graphs have solutions. This region contains a high proportion of relatively hard problem instances. However, the hardest instances are in fact concentrated at a second, lower, transition point. Near this point, most problems are quite easy, but there are also a few very hard cases. This region of exceptionally hard problems corresponds to the transition between polynomial and exponential scaling of the average search cost, whose location we also estimate theoretically. These behaviors also appear to arise in other constraint problems. This work also shows the limitations of simple measures of the cost distribution, such as mean or median, for identifying outlying cases. 