Complexity of hierarchically and 1-dimensional periodically specified problems I: Hardness results

We study the complexity of various combinatorial problems when instances are speci ed using one of the following succinct speci cations: (1) the 1-dimensional nite periodic narrow speci cations (denoted 1-FPN-speci cations) of Ford et al. and Wanke [FF58, Wa93]; (2) the 1-dimensional nite periodic narrow speci cations with explicit boundary conditions (denoted 1-FPN(BC)-speci cations) of Gale [Ga59]; (3) the 2-way in nite 1-dimensional periodic narrow speci cations (denoted 1-PN-speci cations) of Orlin et al. [Or82a]; and (4) the hierarchical speci cations (denoted L-speci cations) of Lengauer et al. [LW87a]. We present three general types of results. First, we give a polynomial time algorithm that, given a 1-FPNor 1-FPN(BC)-speci cation of a graph (or a CNF formula), constructs a level-restricted L-speci cation of an isomorphic graph (or formula). Second, we prove the PSPACEhardness of several basic CNF satis ability problems when instances are speci ed succinctly using 1-FPN-, 1-FPN(BC)-, 1-PNor L-speci cations. Finally, we use our rst two types of results to prove the PSPACEhardness of a number of basic combinatorial problems when instances are so speci ed. These results along with those in [MH+96] include all the problems shown to be PSPACE-hard for 1-PN-speci cations by Orlin [Or82a] and for L-speci cations by Lengauer and Wagner [LW92]. Our results signi cantly extend the hardness results in [Or82a, Pa94, LW92, Wa93]. 1991 Mathematics Subject Classi cation. 68Q25, 68Q15, 68R10.