10 research outputs found
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
Connectivity of Boolean Satisfiability
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. For this
implicitly defined graph, we here study the st-connectivity and connectivity
problems.
Building on the work of Gopalan et al. ("The Connectivity of Boolean
Satisfiability: Computational and Structural Dichotomies", 2006/2009), we first
investigate satisfiability problems given by CSPs, more exactly CNF(S)-formulas
with constants (as considered in Schaefer's famous 1978 dichotomy theorem); we
prove a computational dichotomy for the st-connectivity problem, asserting that
it is either solvable in polynomial time or PSPACE-complete, and an aligned
structural dichotomy, asserting that the maximal diameter of connected
components is either linear in the number of variables, or can be exponential;
further, we show a trichotomy for the connectivity problem, asserting that it
is either in P, coNP-complete, or PSPACE-complete.
Next we investigate two important variants: CNF(S)-formulas without
constants, and partially quantified formulas; in both cases, we prove analogous
dichotomies for st-connectivity and the diameter; for for the connectivity
problem, we show a trichotomy in the case of quantified formulas, while in the
case of formulas without constants, we identify fragments of a possible
trichotomy.
Finally, we consider the connectivity issues for B-formulas, which are
arbitrarily nested formulas built from some fixed set B of connectives, and for
B-circuits, which are Boolean circuits where the gates are from some finite set
B; we prove a common dichotomy for both connectivity problems and the diameter;
for partially quantified B-formulas, we show an analogous dichotomy.Comment: PhD thesis, 82 pages, contains all results from the previous papers
arXiv:1312.4524, arXiv:1312.6679, and arXiv:1403.6165, plus additional
findings. arXiv admin note: text overlap with arXiv:cs/0609072 by other
author
Reconfiguring Triangulations
The results in this thesis lie at the confluence of triangulations and reconfiguration. We make the observation that certain solved and unsolved problems about triangulations can be cast as reconfiguration problems. We then solve some reconfiguration problems that provide us new insights about triangulations. Following are the main contributions of this thesis:
1. We show that computing the flip distance between two triangulations of a point set is NP-complete. A flip is an operation that changes one triangulation into another by replacing one diagonal of a convex quadrilateral by the other diagonal. The flip distance, then, is the smallest number of flips needed to transform one triangulation into another. For the special case when the points are in convex position, the problem of computing the flip distance is a long-standing open problem.
2. Inspired by the problem of computing the flip distance, we start an investigation into computing shortest reconfiguration paths in reconfiguration graphs. We consider the reconfiguration graph of satisfying assignments of Boolean formulas where there is a node for each satisfying assignment of a formula and an edge whenever one assignment can be changed to another by changing the value of exactly one variable from 0 to 1 or from 1 to 0. We show that computing the shortest path between two satisfying assignments in the reconfiguration graph is either in P, NP-complete, or PSPACE-complete depending on the class the Boolean formula lies in.
3. We initiate the study of labelled reconfiguration. For the case of triangulations, we assign a unique label to each edge of the triangulation and a flip of an edge from e to e' assigns the same label to e' as e. We show that adding labels may make the reconfiguration graph disconnected. We also show that the worst-case reconfiguration distance changes when we assign labels. We show tight bounds on the worst case reconfiguration distance for edge-labelled triangulations of a convex polygon and of a spiral polygon, and edge-labelled spanning trees of a graph. We generalize the result on spanning trees to labelled bases of a matroid and show non-trivial upper bounds on the reconfiguration distance
Coloring Reconfiguration Problems and Their Generalizations
Tohoku University周暁課