Title:

Instance compression of parametric problems and related hierarchies

We define instance compressibility ([13, 17]) for parametric problems in the classes PH and PSPACE.We observe that the problem ƩiCIRCUITSAT of deciding satisfiability of a quantified Boolean circuit with i1 alternations of quantifiers starting with an existential quantifier is complete for parametric problems in the class Ʃp/i with respect to wreductions, and that analogously the problem QBCSAT (Quantified Boolean Circuit Satisfiability) is complete for parametric problems in PSPACE with respect to wreductions. We show the following results about these problems: 1. If CIRCUITSAT is nonuniformly compressible within NP, then ƩiCIRCUITSAT is nonuniformly compressible within NP, for any i≥1. 2. If QBCSAT is nonuniformly compressible (or even if satisfiability of quantified Boolean CNF formulae is nonuniformly compressible), then PSPACE ⊆ NP/poly and PH collapses to the third level. Next, we define Succinct Interactive Proof (Succinct IP) and by adapting the proof of IP = PSPACE ([11, 6]) , we show that QBCNFSAT (Quantified Boolean Formula (in CNF) Satisfiability) is in Succinct IP. On the contrary if QBCNFSAT has Succinct PCPs ([32]) , Polynomial Hierarchy (PH) collapses. After extending the notion of instance compression to higher classes, we study the hierarchical structure of the parametric problems with respect to compressibility. For that purpose, we extend the existing definition of VChierarchy ([13]) to parametric problems. After that, we have considered a long list of natural NP problems and tried to classify them into some level of VChierarchy. We have shown some of the new wreductions in this context and pointed out a few interesting results including the ones as follows. 1. CLIQUE is VC1complete (using the results in [14]). 2. SET SPLITTING and NAESAT are VC2complete. We have also introduced a new complexity class VCE in this context and showed some hardness and completeness results for this class. We have done a comparison of VChierarchy with other related hierarchies in parameterized complexity domain as well. Next, we define the compression of counting problems and the analogous classification of them with respect to the notion of instance compression. We define #VChierarchy for this purpose and similarly classify a large number of natural counting problems with respect to this hierarchy, by showing some interesting hardness and completeness results. We have considered some of the interesting practical problems as well other than popular NP problems (e.g., #MULTICOLOURED CLIQUE, #SELECTED DOMINATING SET etc.) and studied their complexity for both decision and counting version. We have also considered a large variety of circuit satisfiability problems (e.g., #MONOTONE WEIGHTEDCNFSAT, #EXACT DNFSAT etc.) and proved some interesting results about them with respect to the theory of instance compressibility.
