Reachability and LTL model-checking problems for flat counter systems are known to be decidable but whereas the reachability problem can be shown in NP, the best known complexity upper bound for the latter problem is made of a tower of several exponentials. Herein, we show that the problem is only NP-complete even if LTL admits past-time operators and arithmetical constraints on counters. For instance, adding past-time operators to LTL immediately leads to complications; an NP upper bound cannot be deduced by translating formulae into Büchi automata. Actually, the NP upper bound is shown by adequately combining a new stuttering theorem for Past LTL and the property of small integer solutions for quantifier-free Presburger formulae. Other complexity results are proved, for instance for restricted classes of flat counter systems such as path schemas. Our NP upper bound extends known and recent results on model-checking weak Kripke structures with LTL formulae as well as reachability problems for flat counter systems.Keywords: linear-time temporal logic, stuttering, model-checking, counter system, flatness, complexity, system of equations, small solution, Presburger arithmetic.Email addresses: demri@lsv.ens-cachan.fr (Stéphane Demri), dhar@liafa.univ-paris-diderot.fr (Amit Kumar Dhar), sangnier@liafa.univ-paris-diderot.fr (Arnaud Sangnier)Supported by ANR project REACHARD ANR-11-BS02-001. This is the completed version of [8].