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The figure shows how the $ M $ possible clauses are grouped. The dark gray region represents clauses that conflict with $r$ and so cannot be selected. The white regions represent clauses that conflict only with $s$ or $s'$. The selected clauses must include $b$ and $b'$ conflicting only with $s$ and $s'$, respectively. 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For comparison, the gray and dashed curves show the scaling for random selection and, approximately, for unstructured search.}}{14}}\newlabel{fig.optimal}{{3}{14}}\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces \relax \fontsize  {10.95}{13.6}\selectfont  \abovedisplayskip 11\p@ plus3\p@ minus6\p@ \abovedisplayshortskip \z@ plus3\p@ \belowdisplayshortskip 6.5\p@ plus3.5\p@ minus3\p@ \def \leftmargin \leftmargini \parsep 5\p@ plus2.5\p@ minus\p@ \topsep 10\p@ plus4\p@ minus6\p@ \itemsep 5\p@ plus2.5\p@ minus\p@ {\leftmargin \leftmargini \topsep 9\p@ plus3\p@ minus5\p@ \parsep 4.5\p@ plus2\p@ minus\p@ \itemsep \parsep }\belowdisplayskip \abovedisplayskip Optimal choices of $\tau $ (black) and $\rho $ (gray) as a function of $\mu =m/n$. Solid curves are for the random 3-SAT ensemble and the dashed curves are for the ensemble with a prespecified solution.}}{14}}\newlabel{fig.parameters}{{4}{14}}\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Best parameter values and scaling behavior for single-step search of 3-SAT problems for random and prespecified solution ensembles.}}{15}}\newlabel{values}{{1}{15}}\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces \relax \fontsize  {10.95}{13.6}\selectfont  \abovedisplayskip 11\p@ plus3\p@ minus6\p@ \abovedisplayshortskip \z@ plus3\p@ \belowdisplayshortskip 6.5\p@ plus3.5\p@ minus3\p@ \def \leftmargin \leftmargini \parsep 5\p@ plus2.5\p@ minus\p@ \topsep 10\p@ plus4\p@ minus6\p@ \itemsep 5\p@ plus2.5\p@ minus\p@ {\leftmargin \leftmargini \topsep 9\p@ plus3\p@ minus5\p@ \parsep 4.5\p@ plus2\p@ minus\p@ \itemsep \parsep }\belowdisplayskip \abovedisplayskip Smallest exponential decay rate $A$ for $\left \delimiter "426830A  { P_{\rm  soln} } \right \delimiter "526930B $ as a function of $\mu =m/n$ for 3-SAT with prespecified solution. For comparison, the gray and dashed curves show the scaling for random selection and, approximately, for unstructured search.}}{16}}\newlabel{fig.prespecified}{{5}{16}}\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces \relax \fontsize  {10.95}{13.6}\selectfont  \abovedisplayskip 11\p@ plus3\p@ minus6\p@ \abovedisplayshortskip \z@ plus3\p@ \belowdisplayshortskip 6.5\p@ plus3.5\p@ minus3\p@ \def \leftmargin \leftmargini \parsep 5\p@ plus2.5\p@ minus\p@ \topsep 10\p@ plus4\p@ minus6\p@ \itemsep 5\p@ plus2.5\p@ minus\p@ {\leftmargin \leftmargini \topsep 9\p@ plus3\p@ minus5\p@ \parsep 4.5\p@ plus2\p@ minus\p@ \itemsep \parsep }\belowdisplayskip \abovedisplayskip Optimal asymptotic behavior of $\left \delimiter "426830A  { P_{\rm  soln} } \right \delimiter "526930B $ for 3-SAT with $\mu =2$ (gray) and 4 (black) on a log-scale vs.\nobreakspace  {}$n$. 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Behavior for random 3-SAT (black curve, up to $\mu =6$), prespecified solution 3-SAT (gray curve, for $\mu $ between 0.1 and 40) and the upper bound based on probability to find the prespecified solution (black curve, for $\mu \ge 20$). 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