Revision #2 Authors: Patrick Scharpfenecker

Accepted on: 16th November 2015 10:24

Downloads: 363

Keywords:

In this work we extend the study of solution graphs and prove that for boolean formulas in a class called CPSS, all connected components are partial cubes of small dimension, a statement which was proved only for some cases in [Schwerdtfeger 2013]. In contrast, we show that general Schaefer formulas are powerful enough to encode graphs of exponential isometric dimension and graphs which are not even partial cubes.

Our techniques shed light on the detailed structure of $st$-connectivity for Schaefer and connectivity for CPSS formulas, problems which were already known to be solvable in polynomial time. We refine this classification and show that the problems in these cases are equivalent to the satisfiability problem of related formulas by giving mutual reductions between ($st$-)connectivity and satisfiability. An immediate consequence is that $st$-connectivity in (undirected) solution graphs of Horn-formulas is P-complete while for $2SAT$ formulas $st$-connectivity is NL-complete.

Revision #1 Authors: Patrick Scharpfenecker

Accepted on: 10th August 2015 13:27

Downloads: 447

Keywords:

In this work we extend the study of solution graphs and prove that for boolean formulas in a class called CPSS, all connected components are partial cubes of small dimension, a statement which was proved only for some cases in [Schwerdtfeger 2013]. In contrast, we show that general Schaefer formulas are powerful enough to encode graphs of exponential isometric dimension and graphs which are not even partial cubes.

Our techniques shed light on the detailed structure of $st$-connectivity for Schaefer and connectivity for CPSS formulas, problems which were already known to be solvable in polynomial time. We refine this classification and show that the problems in these cases are equivalent to the satisfiability problem of related formulas by giving mutual reductions between ($st$-)connectivity and satisfiability. An immediate consequence is that $st$-connectivity in (undirected) solution graphs of Horn-formulas is P-complete while for $2SAT$ formulas $st$-connectivity is NL-complete.

Added reference to FCT 2015 publication.

Clarified Corollary 5.

Corrected a typo in Corollary 6.

In this work we extend the study of solution graphs and prove that for boolean formulas in a class called CPSS, all connected components are partial cubes of small dimension, a statement which was proved only for some cases in [Schwerdtfeger 2013]. In contrast, we show that general Schaefer formulas are powerful enough to encode graphs of exponential isometric dimension and graphs which are not even partial cubes.

Our techniques shed light on the detailed structure of $st$-connectivity for Schaefer and connectivity for CPSS formulas, problems which were already known to be solvable in polynomial time. We refine this classification and show that the problems in these cases are equivalent to the satisfiability problem of related formulas by giving mutual reductions between ($st$-)connectivity and satisfiability. An immediate consequence is that $st$-connectivity in (undirected) solution graphs of Horn-formulas is P-complete while for $2SAT$ formulas $st$-connectivity is NL-complete.