The Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits

For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. In 2006, Gopalan et al. studied connectivity properties of the solution graph and related complexity issues for CSPs, motivated mainly by research on satisfiability algorithms and the satisfiability threshold. They proved dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connectivity question. Recently, we were able to establish the trichotomy [arXiv:1312.4524]. Here, we consider connectivity issues of satisfiability problems defined by Boolean circuits and propositional formulas that use gates, resp. connectives, from a fixed set of Boolean functions. We obtain dichotomies for the diameter and the two connectivity problems: on one side, the diameter is linear in the number of variables, and both problems are in P, while on the other side, the diameter can be exponential, and the problems are PSPACE-complete. For partially quantified formulas, we show an analogous dichotomy.Comment: 20 pages, several improvement

The Connectivity of Boolean Satisfiability: No-Constants and Quantified Variants

For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. Motivated by research on heuristics and the satisfiability threshold, Gopalan et al. in 2006 studied connectivity properties of the solution graph and related complexity issues for constraint satisfaction problems in Schaefer's framework. They found dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connectivity question that we recently were able to prove. While Gopalan et al. considered CNF(S)-formulas with constants, we here look at two important variants: CNF(S)-formulas without constants, and partially quantified formulas. For the diameter and the st-connectivity question, we prove dichotomies analogous to those of Gopalan et al. in these settings. While we cannot give a complete classification for the connectivity problem yet, we identify fragments where it is in P, where it is coNP-complete, and where it is PSPACE-complete, in analogy to Gopalan et al.'s trichotomy.Comment: superseded by chapter 3 of arXiv:1510.0670

Enclaves of freedom. On „second circulation” publishing in the PRL

„Second circulation publishing”, the broadly defined publishing and cultural movement, independent of the authorities of the PRL and not subject to state censorship, was initiated in the autumn of 1976 by the community which opposed communist party rule in Poland. The author of the article offers a synthesis defining the notion of „second circulation”, indicating its scope of influence and its reach, discussing its significance at the levels of community, cultural and political life, and its influence on the democratic changes in Poland and other Central and East European states in the late-1980s

Taxation and Market Power

We analyze the incidence and welfare e¤ects of unit sales taxes in experimental monopoly and Bertrand markets. We nd, in line with economic theory, that rms with no market power are able to shift a high share of a tax burden on to consumers, independent of whether buyers are automated or human players. In monopoly markets, a monopolist bears a large share of the burden of a tax increase. With human buyers, however, this share is smaller than with automated buyers as the presence of human buyers constrains the pricing behavior of a monopolist.tax incidence;monopoly;Bertrand competition;experiment