We give a general complexity classification scheme for monotone computation, including monotone space-bounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple notion of monotone reducibility and exhibit complete problems. This provides a framework for stating existing results and asking new questions. We show that mNL (monotone nondeterministic log-space) is not closed under complementation, in contrast to Immerman's and Szelepcs 'enyi's nonmonotone result [Imm88, Sze87] that NL = co-NL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for st-connectivity. We also consider mBWBP (monotone bounded width branching programs) and study the question of whether mBWBP is properly contained in mNC 1 , motivated by Barrington's result [Bar89] that BWBP = NC 1 . Although we cannot answer t...

This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should

The central observation of this paper is that if n random edges are added to any n-node connected graph or digraph then the resulting graph has diameter O(log n). We apply this to smoothed analysis of algorithms and property testing.

We study the efficient verification of properties of probabilistic systems. We first take the bounded model checking point of view and show how to efficiently approximate IP rob[/], the probability of certain CTL formulas. We define a fragment of probabilistic CTL for which we guarantee such an approximation. We then consider probabilistic programs that take a finite structure such as a graph as input and look for compression schemas that map a (large) structure M (the state structure) into a smaller one M 0 of a different signature and study how a property defined by a formula / (the property to be verified) can also be defined by another formula ' on the much smaller structure M 0 . In the cases of Graph Accessibility and Perfect Matching we give compression schemas where: M 0 j= ' )M j= IP rob[/] ? b i.e. a deterministic statement on M 0 implies a probabilistic one on a (large) M. We conclude with a comparison with OBDD's and prove an exponential lower bound on...