We give improved deterministic algorithms solving sparse instances of MAX-SAT and MAX- k -CSP. For instances with n variables and cn clauses (constraints), we give algorithms running in time \poly ( n ) 2 n (1 − ) for \begin{itemize} \item = ( 1 c ) and polynomial space solving MAX-SAT and MAX- k -SAT, \item = ( 1 c ) and exponential space solving MAX-SAT and MAX- k -SAT, \item = ( 1 c k 2 ) and polynomial space solving MAX- k -CSP, \item = ( 1 c k 3 ) and exponential space solving MAX- k -CSP. \end{itemize} The previous MAX-SAT algorithms have savings = ( 1 c 2 log 2 c ) for running in polynomial space~\cite{SST14} and = ( 1 c log c ) for exponential space~\cite{DW06}. We also give an algorithm with improved savings for satisfiability of depth-2 threshold circuits with cn wires.