ECCC-Report TR15-101https://eccc.weizmann.ac.il/report/2015/101Comments and Revisions published for TR15-101en-usMon, 16 Nov 2015 10:24:01 +0200
Revision 2
| On the structure of Solution-Graphs for Boolean Formulas |
Patrick Scharpfenecker
https://eccc.weizmann.ac.il/report/2015/101#revision2In 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.Mon, 16 Nov 2015 10:24:01 +0200https://eccc.weizmann.ac.il/report/2015/101#revision2
Revision 1
| On the structure of Solution-Graphs for Boolean Formulas |
Patrick Scharpfenecker
https://eccc.weizmann.ac.il/report/2015/101#revision1In 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.Mon, 10 Aug 2015 13:27:31 +0300https://eccc.weizmann.ac.il/report/2015/101#revision1
Paper TR15-101
| On the structure of Solution-Graphs for Boolean Formulas |
Patrick Scharpfenecker
https://eccc.weizmann.ac.il/report/2015/101In 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.Fri, 19 Jun 2015 23:14:21 +0300https://eccc.weizmann.ac.il/report/2015/101