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\begin{document}

\title{\bf CNF Satisfiability Problem}

\author{Andrew Makhorin {\tt<mao@gnu.org>}}

\date{August 2011}

\maketitle

\section{Introduction}

The {\it Satisfiability Problem (SAT)} is a classic combinatorial
problem. Given a Boolean formula of $n$ variables
$$f(x_1,x_2,\dots,x_n),\eqno(1.1)$$
this problem is to find such values of the variables, on which the
formula takes on the value {\it true}.

The {\it CNF Satisfiability Problem (CNF-SAT)} is a version of the
Satisfiability Problem, where the Boolean formula (1.1) is specified
in the {\it Conjunctive Normal Form (CNF)}, that means that it is a
conjunction of {\it clauses}, where a clause is a disjunction of
{\it literals}, and a literal is a variable or its negation.
For example:
$$(x_1\vee x_2)\;\&\;(\neg x_2\vee x_3\vee\neg x_4)\;\&\;(\neg
x_1\vee x_4).\eqno(1.2)$$
Here $x_1$, $x_2$, $x_3$, $x_4$ are Boolean variables to be assigned,
$\neg$ means
negation (logical {\it not}), $\vee$ means disjunction (logical
{\it or}), and $\&$ means conjunction (logical {\it and}). One may
note that the formula (1.2) is {\it satisfiable}, because on
$x_1$ = {\it true}, $x_2$ = {\it false}, $x_3$ = {\it false}, and
$x_4$ = {\it true} it takes on the value {\it true}. If a formula
is not satisfiable, it is called {\it unsatisfiable}, that means that
it takes on the value {\it false} on any values of its variables.

Any CNF-SAT problem can be easily translated to a 0-1 programming
problem as follows.\linebreak A Boolean variable $x$ can be modeled by
a binary variable in a natural way: $x=1$ means that $x$ takes on the
value {\it true}, and $x=0$ means that $x$ takes on the value
{\it false}. Then, if a literal is a negated variable, i.e. $t=\neg x$,
it can be expressed as $t=1-x$. Since a formula in CNF is a conjunction
of clauses, to provide its satisfiability we should require all its
clauses to  take on the value {\it true}. A particular clause is
a disjunction of literals:
$$t\vee t'\vee t''\dots ,\eqno(1.3)$$
so it takes on the value {\it true} iff at least one of its literals
takes on the value {\it true}, that can be expressed as the following
inequality constraint:
$$t+t'+t''+\dots\geq 1.\eqno(1.4)$$
Note that no objective function is used in this case, because only
a feasible solution needs to be found.

For example, the formula (1.2) can be translated to the following
constraints:
$$\begin{array}{c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c}
x_1&+&x_2&&&&&\geq&1\\
&&(1-x_2)&+&x_3&+&(1-x_4)&\geq&1\\
(1-x_1)&&&&&+&x_4&\geq&1\\
\end{array}$$
$$x_1, x_2, x_3, x_4\in\{0,1\}$$
Carrying out all constant terms to the right-hand side gives
corresponding 0-1 programming problem in the standard format:
$$\begin{array}{r@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }r}
x_1&+&x_2&&&&&\geq&1\\
&-&x_2&+&x_3&-&x_4&\geq&-1\\
-x_1&&&&&+&x_4&\geq&0\\
\end{array}$$
$$x_1, x_2, x_3, x_4\in\{0,1\}$$

In general case translation of a CNF-SAT problem results in the
following 0-1 programming problem:
$$\sum_{j\in J^+_i}x_j-\sum_{j\in J^-_i}x_j\geq 1-|J^-_i|,
\ \ \ i=1,\dots,m\eqno(1.5)$$
$$x_j\in\{0,1\},\ \ \ j=1,\dots,n\eqno(1.6)$$
where $n$ is the number of variables, $m$ is the number of clauses
(inequality constraints),\linebreak $J^+_i\subseteq\{1,\dots,n\}$ is
a subset of variables, whose literals in $i$-th clause do not have
negation, and $J^-_i\subseteq\{1,\dots,n\}$ is a subset of variables,
whose literals in $i$-th clause are negations of that variables. It is
assumed that $J^+_i\cap J^-_i=\varnothing$ for all $i$.

\section{GLPK API Routines}

\subsection{glp\_read\_cnfsat --- read CNF-SAT problem data in DIMACS
format}

\synopsis

\begin{verbatim}
   int glp_read_cnfsat(glp_prob *P, const char *fname);
\end{verbatim}

\description

The routine \verb|glp_read_cnfsat| reads the CNF-SAT problem data from
a text file in DIMACS format and automatically translates the data to
corresponding 0-1 programming problem instance (1.5)--(1.6).

The parameter \verb|P| specifies the problem object, to which the
0-1 programming problem instance should be stored. Note that before
reading data the current content of the problem object is completely
erased with the routine \verb|glp_erase_prob|.

The character string \verb|fname| specifies the name of a text file
to be read in. (If the file name ends with the suffix `\verb|.gz|',
the file is assumed to be compressed, in which case the routine
decompresses it ``on the fly''.)

\newpage

\returns

If the operation was successful, the routine returns zero. Otherwise,
it prints an error message and returns non-zero.

\para{DIMACS CNF-SAT problem format}\footnote{This material is based on
the paper ``Satisfiability Suggested Format'', which is publicly
available at {\tt http://dimacs.rutgers.edu/}.}

The DIMACS input file is a plain ASCII text file. It contains lines of
several types described below. A line is terminated with an end-of-line
character. Fields in each line are separated by at least one blank
space.

\para{Comment lines.} Comment lines give human-readable information
about the file and are ignored by programs. Comment lines can appear
anywhere in the file. Each comment line begins with a lower-case
character \verb|c|.

\begin{verbatim}
   c This is a comment line
\end{verbatim}

\para{Problem line.} There is one problem line per data file. The
problem line must appear before any clause lines. It has the following
format:

\begin{verbatim}
   p cnf VARIABLES CLAUSES
\end{verbatim}

\noindent
The lower-case character \verb|p| signifies that this is a problem
line. The three character problem designator \verb|cnf| identifies the
file as containing specification information for the CNF-SAT problem.
The \verb|VARIABLES| field contains an integer value specifying $n$,
the number of variables in the instance. The \verb|CLAUSES| field
contains an integer value specifying $m$, the number of clauses in the
instance.

\para{Clauses.} The clauses appear immediately after the problem
line. The variables are assumed to be numbered from 1 up to $n$. It is
not necessary that every variable appears in the instance. Each clause
is represented by a sequence of numbers separated by either a space,
tab, or new-line character. The non-negated version of a variable $j$
is represented by $j$; the negated version is represented by $-j$. Each
clause is terminated by the value 0. Unlike many formats that represent
the end of a clause by a new-line character, this format allows clauses
to be on multiple lines.

\para{Example.} Below here is an example of the data file in DIMACS
format corresponding to the CNF-SAT problem (1.2).

\begin{footnotesize}
\begin{verbatim}
c sample.cnf
c
c This is an example of the CNF-SAT problem data
c in DIMACS format.
c
p cnf 4 3
1 2 0
-4 3
-2 0
-1 4 0
c
c eof
\end{verbatim}
\end{footnotesize}

\newpage

\subsection{glp\_check\_cnfsat --- check for CNF-SAT problem instance}

\synopsis

\begin{verbatim}
   int glp_check_cnfsat(glp\_prob *P);
\end{verbatim}

\description

The routine \verb|glp_check_cnfsat| checks if the specified problem
object \verb|P| contains a 0-1 programming problem instance in the
format (1.5)--(1.6) and therefore encodes a CNF-SAT problem instance.

\returns

If the specified problem object has the format (1.5)--(1.6), the
routine returns zero, otherwise non-zero.

\subsection{glp\_write\_cnfsat --- write CNF-SAT problem data in DIMACS
format}

\synopsis

\begin{verbatim}
   int glp_write_cnfsat(glp_prob *P, const char *fname);
\end{verbatim}

\description

The routine \verb|glp_write_cnfsat| automatically translates the
specified 0-1 programming problem instance (1.5)--(1.6) to a CNF-SAT
problem instance and writes the problem data to a text file in DIMACS
format.

The parameter \verb|P| is the problem object, which should specify
a 0-1 programming problem instance in the format (1.5)--(1.6).

The character string \verb|fname| specifies a name of the output text
file to be written. (If the file name ends with suffix `\verb|.gz|',
the file is assumed to be compressed, in which case the routine
performs automatic compression on writing that file.)

\returns

If the operation was successful, the routine returns zero. Otherwise,
it prints an error message and returns non-zero.

\subsection{glp\_minisat1 --- solve CNF-SAT problem instance with
MiniSat solver}

\synopsis

\begin{verbatim}
   int glp_minisat1(glp_prob *P);
\end{verbatim}

\description

The routine \verb|glp_minisat1| is a driver to MiniSat, a CNF-SAT
solver developed by Niklas E\'en and Niklas S\"orensson, Chalmers
University of Technology, Sweden.\footnote{The MiniSat software module
is {\it not} part of GLPK, but is used with GLPK and included in the
distribution.}

\newpage

It is assumed that the specified problem object \verb|P| contains
a 0-1 programming problem instance in the format (1.5)--(1.6) and
therefore encodes a CNF-SAT problem instance.

If the problem instance has been successfully solved to the end, the
routine \verb|glp_minisat1| returns 0. In this case the routine
\verb|glp_mip_status| can be used to determine the solution status:

\begin{itemize}
\item {\tt GLP\_OPT} means that the solver found an integer feasible
solution and therefore the corresponding CNF-SAT instance is
satisfiable;

\item {\tt GLP\_NOFEAS} means that no integer feasible solution exists
and therefore the corresponding CNF-SAT instance is unsatisfiable.
\end{itemize}

If an integer feasible solution was found, corresponding values of
binary variables can be retrieved with the routine
\verb|glp_mip_col_val|.

\returns

\begin{retlist}
0 & The MIP problem instance has been successfully solved. (This code
does {\it not} necessarily mean that the solver has found feasible
solution. It only means that the solution process was successful.)\\

{\tt GLP\_EDATA} & The specified problem object contains a MIP
instance which does {\it not} have the format (1.5)--(1.6).\\

{\tt GLP\_EFAIL} & The solution process was unsuccessful because of
the solver failure.\\
\end{retlist}

\subsection{glp\_intfeas1 --- solve integer feasibility problem}

\synopsis

\begin{verbatim}
   int glp_intfeas1(glp_prob *P, int use_bound, int obj_bound);
\end{verbatim}

\description

The routine \verb|glp_intfeas1| is a tentative implementation of
an integer feasibility solver based on a CNF-SAT solver (currently
it is MiniSat; see Subsection 2.4).

If the parameter \verb|use_bound| is zero, the routine searches for
{\it any} integer feasibile solution to the specified integer
programming problem. Note that in this case the objective function is
ignored.

If the parameter \verb|use_bound| is non-zero, the routine searches for
an integer feasible solution, which provides a value of the objective
function not worse than \verb|obj_bound|. In other word, the parameter
\verb|obj_bound| specifies an upper (in case of minimization) or lower
(in case of maximization) bound to the objective function.

If the specified problem has been successfully solved to the end, the
routine \verb|glp_intfeas1| returns 0. In this case the routine
\verb|glp_mip_status| can be used to determine the solution status:

\begin{itemize}
\item {\tt GLP\_FEAS} means that the solver found an integer feasible
solution;

\item {\tt GLP\_NOFEAS} means that the problem has no integer feasible
solution (if {\tt use\_bound} is zero) or it has no integer feasible
solution, which is not worse than {\tt obj\_bound} (if {\tt use\_bound}
is non-zero).
\end{itemize}

\newpage

If an integer feasible solution was found, corresponding values of
variables (columns) can be retrieved with the routine
\verb|glp_mip_col_val|.

\para{Usage Notes}

The integer programming problem specified by the parameter \verb|P|
should satisfy to the following requirements:

\begin{enumerate}
\item All variables (columns) should be either binary ({\tt GLP\_BV})
or fixed at integer values ({\tt GLP\_FX}).

\item All constraint and objective coefficients should be integer
numbers in the range\linebreak $[-2^{31},\ +2^{31}-1]$.
\end{enumerate}

Though there are no special requirements to the constraints,
currently the routine \verb|glp_intfeas1| is efficient mainly for
problems, where most constraints (rows) fall into the following three
classes:

\begin{enumerate}
\item Covering inequalities
$$\sum_{j}t_j\geq 1,$$
where $t_j=x_j$ or $t_j=1-x_j$, $x_j$ is a binary variable.

\item Packing inequalities
$$\sum_{j}t_j\leq 1.$$

\item Partitioning equalities (SOS1 constraints)
$$\sum_{j}t_j=1.$$
\end{enumerate}

\returns

\begin{retlist}
0 & The problem has been successfully solved. (This code does
{\it not} necessarily mean that the solver has found an integer
feasible solution. It only means that the solution process was
successful.) \\

{\tt GLP\_EDATA} & The specified problem object does not satisfy
to the requirements listed in Paragraph `Usage Notes'. \\

{\tt GLP\_ERANGE} & An integer overflow occured on translating the
specified problem to a CNF-SAT problem. \\

{\tt GLP\_EFAIL} & The solution process was unsuccessful because of
the solver failure. \\
\end{retlist}

\end{document}
